Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Regularity of solutions and parameter estimation for SPDE's with space-time white noise
(USC Thesis Other)
Regularity of solutions and parameter estimation for SPDE's with space-time white noise
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
REGULARITY OF SOLUTIONS AND PARAMETER ESTIMATION FOR SPDE’S WITH SPACE-TIME WHITE NOISE by Igor Cialenco A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2007 Copyright 2007 Igor Cialenco Dedication To my wife Angela, and my parents. ii Acknowledgements I would like to acknowledge my academic adviser Prof. Sergey V . Lototsky who introduced me into the Theory of Stochastic Partial Differential Equations, suggested the interesting topics of research and guided me through it. I also wish to thank the members of my committee - Prof. Remigijus Mikulevicius and Prof. Aris Protopapadakis, for their help and support. Last but certainly not least, I want to thank my wife Angela, and my family for their support both during the thesis and before it. iii Table of Contents Dedication ii Acknowledgements iii List of Tables v List of Figures vi Abstract vii Chapter 1: Introduction 1 1.1 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Diffusion processes and absolute continuity of their measures . . . . . 4 1.3 Stochastic partial differential equations and their applications . . . . 7 1.4 Itˆ o’s formula in Hilbert space . . . . . . . . . . . . . . . . . . . . . 14 1.5 Existence and uniqueness of solution . . . . . . . . . . . . . . . . . . 18 Chapter 2: Regularity of solution 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Equations with additive noise . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Existence and uniqueness . . . . . . . . . . . . . . . . . . . 29 2.2.2 Regularity in space . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.3 Regularity in time . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Equations with multiplicative noise . . . . . . . . . . . . . . . . . . . 41 2.3.1 Existence and uniqueness . . . . . . . . . . . . . . . . . . . 41 2.3.2 Regularity in space and time . . . . . . . . . . . . . . . . . . 48 2.4 Covariance functional and relation with some known results . . . . . 57 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 3: Parameter estimation problems for some classes of SPDE’s 65 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Special case: space-time white noise . . . . . . . . . . . . . . . . . . 69 3.3 General case. Approximation of the solution . . . . . . . . . . . . . . 75 3.4 The estimate and its properties . . . . . . . . . . . . . . . . . . . . . 79 3.5 Applications to stochastic parabolic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.6 Examples and some numerical results . . . . . . . . . . . . . . . . . 91 References 100 iv List of Tables 2.1 Existence and Regularity of Solution. Summary . . . . . . . . . . . . 64 3.1 Errors of estimated parameter . . . . . . . . . . . . . . . . . . . . . . 94 v List of Figures 2.1 Realization of approximated solution of stochastic heat equation with additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Realization of approximated solution of stochastic heat equation with multiplicative noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1 Estimated Parameter. Example 3.6.1 . . . . . . . . . . . . . . . . . . 93 3.2 Estimated Parameter. Example 3.6.2. . . . . . . . . . . . . . . . . . . 96 3.3 Errorsj ^ µ¡µ 0 j vs number of Fourier coefficients. Example 3.6.2 . . . 97 3.4 Estimators vs number of Fourier coefficients. Example 3.6.3.a . . . . 98 3.5 Estimators vs number of Fourier coefficients. Example 3.6.3.b . . . . 98 3.6 Estimators vs number of Fourier coefficients. Example 3.6.3.c . . . . 99 vi Abstract In this work we discuss two problems related to stochastic partial differential equa- tions (SPDEs): analytical properties of solutions and parameter estimation for SPDE’s. We address the problem of existence, uniqueness and regularity of solutions of some parabolic SPDE’s driven by space-time white noise, either additive or multi- plicative. The novelty in our study is the special form of the noise term which depends on a real parameter. We establish existence and uniqueness of weak solution in the scale of Sobolev spaces. Regularity properties of the solution are stated in terms of the real parameter involved in the noise term and spectral properties of the elliptic operator which generates the scale of Sobolev spaces. We study the parameter estimation problem for some parabolic SPDEs with multi- plicative stochastic part. Maximum Likelihood Estimators of the parameter, based on finite-dimensional approximation of the solution are found. Consistency, both in time and space variables, and asymptotic normality of these estimators are established. All theoretical results are followed by numerical simulations. vii Chapter 1 Introduction In this section we discuss some general and known results which we will need for further presentation. An overview of Sobolev spaces, Kolmogorov’s criterion, absolute continuity of measures generated by the solutions of diffusion processes, Itˆ o’s formula in Hilbert space, existence and uniqueness of solution of parabolic SPDEs and some applications of SPDEs are presented. All results are presented without proofs, except Itˆ o’s formula in Hilbert space for which we present a simpler proof than that from the know literature. 1.1 Sobolev spaces Let M be a d-dimensional compact, orientable,C 1 manifold with a smooth positive measuredx. Let us consider an elliptic positive definite selfadjoint differential operator L of order 2m onM. Then the operator ¤ = (L) 1 2m is well-definite, elliptic of order 1 and generates the scalefH s (M)g s2R of Sobolev spaces onM (see for instance [1], [76]). In particular, for positive integern the spaceH n (M) consists of all generalized functions on M whose derivative up to and including the nth order belong to L 2 (M) and the norm is defined to be kuk H n (M) = 0 @ X j®j·n Z M jD ® uj 2 dx 1 A 1 2 ; 1 where D ® denotes the partial derivative of order ® = (® 1 ;:::;® d ) (see [1],[9] for more details). We will writefH s g s2R instead offH s (M)g s2R if there is no ambiguity in context. In what follows we will usually use an alternative characterizations of Sobolev spaces. It is well-known (see for instance [76], Theorem 8.4) that the operator L has a complete orthonormal system (CONS) of eigenvectors fh k g k2N in the space L 2 (M;dx). Then for everyf 2L 2 (M;dx) we have the representation f(x)= 1 X k=1 hf;h k i 0 h k (x)= 1 X k=1 f k h k (x); (x2M) (1.1) whereh¢;¢i 0 denotes the inner product inL 2 (M;dx), andf k = R M f(x)h k (x)dx; k2N are Fourier coefficients off with respect tofh k g k2N . If ¹ k > 0 is the eigenvalue of the operator L corresponding to the eigenvec- tor h k and ¸ k := ¹ 1 2m k , then for all s ¸ 0 we have H s = ff 2 L 2 (M;dx) : 1 P k=1 ¸ 2s k jf k j 2 < 1g and for s < 0; H s is the closure of L 2 (M;dx) in the norm jjfjj s = ¡ 1 P k=1 ¸ 2s k jf k j 2 ¢ 1=2 . Hence, every element of the space H s ; s 2 R, can be identified with a sequenceff k g k2N such that 1 P k=1 ¸ 2s k jf k j 2 < 1. Note that the space H s , equipped with the inner product hf;gi s = 1 X k=1 ¸ 2s k f k g k (f;g2H s ) is a Hilbert space. Moreover, fors > 0; H ¡s is the dual space ofH s relative to inner product inL 2 (M;dx). It is equivalent to say thatH s =¤ ¡s (L 2 (M;dx)); s2R. Note that ¤ s h k =¸ s k h k and jjh k jj s =jj¤ s h k jj=¸ s k (k2N; s2R): (1.2) 2 Moreover, the functionsh k;s :=¸ ¡s k h k ; k2N, form a CONS inH s fors2R. Similarly, the Sobolev SpacesH s (R d )(s2R) are defined. Namely, H s (R d )= 8 < : f 2L 2 (R d ) : Z R (1+jyj 2 ) s j ^ f(y)j 2 dy <1 9 = ; with normkfk 2 s = R R (1+jyj 2 ) s j ^ f(y)j 2 dy, where ^ f denotes the Fourier Transform in R d , i.e. ^ f(y) = 1 (2¼) d=2 R R d exp(¡ixy)f(x)dx. Also, we can characterize the spaces H s (R d ) using the Laplace operator ¢(u) = P d k=1 u x k x k . Namely, H s (R d ) = (I¡ ¢) ¡ s 2 L 2 (R d ). For more details on structure and some properties of operator (I ¡ ¢) s ; s2R, see for instance [1], [43]. For p ¸ 1 and ° 2 R, we define the space H ° p (R d ) as collection of all gene- ralized functions f such that (I¡¢) ¡°=2 f 2 L p (R d ), and we putkfk °;p := k(I ¡ ¢) ¡°=2 fk Lp(R d ) . In our notationsH ° =H ° 2 . The triple (V;H;V 0 ) of Hilbert spaces is called normal triple if: (a) V ,! H ,! V 0 and both inclusions are dense and continuous; (b) The space V 0 is dual to V relative to inner product in H; (c) There exists a constant C > 0 such that j(h;v) H j · C¢jjvjj V ¢jjujj V 0 (v 2 V; u 2 H). For example the Sobolev spaces (H l+° (R d );H l (R d );H l¡° (R d )); ° > 0; l2R, form a normal triple. Note thatH ° 1 p ½H ° 2 p andk¢k ° 2 ;p ·k¢k ° 1 ;p if° 2 < ° 1 . Also, let us recall that by Sobolev embedding theorem H ° p (R d )½C °¡d=p (R d ) ifp° >d (see [9], Chapter 5). Also, we recall the interpolation theorem for spacesH s [40], [41]. For everyu2 H °+2 and s 2 [°;° + 2], we havekuk s · Ckuk µ °+2 kuk 1¡µ ° , where µ = s¡° 2 and C depends only onM;°;s. We will refer to inequalityab·" jaj p p + jbj q q" p¡1 , as"-inequality, wherea;b2R; 1=p+ 1=q =1; ">0. It follows directly from H¨ older inequality. 3 Another central result in our investigation is Kolmogorov’s criterion. To some extends, Kolmogorov’s criterion is a version of Sobolev embedding theorem. For sake of completeness of our presentation, we will state this result here too. For the proof, see for instance [79], Corollary 1.2, or [8], Theorem 3.3. 1.1.1 Theorem. [Kolmogorov 0 scriterion] Let(;F;P) be a probability space,G½ R d a bounded domain, andX = X(x); x2 G, a measurable map from £G toR so thatX is continuous inx and!. Assume that there exist positive real numbersK;p andq so thatp>1 and E ¯ ¯ ¯X(x)¡X(y) ¯ ¯ ¯ p ·Kjx¡yj d+q : Then there exist a positive real number A and a random variable Y so that, with probability one, ¯ ¯ ¯X(x)¡X(y) ¯ ¯ ¯·Yjx¡yj q=p µ ln A jx¡yj ¶ 2=p ; (1.3) for allx;y2G; q·p. 1.2 Diffusion processes and absolute continuity of their measures For a given potability space (;F;P), we consider a family of random random processes » t (!) on this space with t 2 [0;T], where t usually is understood as time variable, andT is a fixed parameter, called also terminal time. For a fixed!2 , the time function» t (!); t2[0;T] is called a trajectory or realization corresponding to an elementary event!. The¾-algebraF » t :=f» s : s·tg, being the smallest¾-algebra 4 with respect to which the random variables » s ; s · t are measurable, are naturally associated with random process» t .F » t is called the filtration generated by the random process» t . LetfF t g t2[0;T] be a nondecreasing family of ¾-algebras,F s µ F t µ F; s · t. We say that a random process » t ; t 2 [0;T], is adapted to a family of ¾-algebras F t ; t2 [0;T], if for anyt2 [0;T] the random variables» t areF t -measurable. Also, we will say that» t isF t -adapted or nonanticipative. 1.2.1 Definition. Let (;F;P) be a probability space. The random process W = (W(t)); 0 · t · T , is called a standard Brownian Motion on the probability space (;F;P) if: (a)W(0)=0; (b) for every partition0=t 0 <¢¢¢<t i <t i+1 <:::t n = T , the random variablesW(t i+1 )¡W(t i ) andW(t j+1 )¡W(t j ) are independent for every i 6= j (independent increments); (c) W(t)¡ W(s) has a Gaussian standard normal distribution withE(W(t)¡W(s))=0; Var(W(t)¡W(s))=1; (d) for almost all!2 the functionsW(t)=W(t;!) are continuous on the interval0·t·T . For more details about existence of Brownian motion see for instance [13], [47]. The Brownian motion is also called Winer Process. Let(;F;fF t g t2[0;T] ;P) be a stochastic basis,W(t) be a Wiener Process and»(t) a random process, both adapted to the filtration F t . Denote by P » the measure on C(0;T) corresponding to the process» t , P » (B)=Pf! : »(!)2Bg: In this section we will discuss the problem of the absolute continuity of measures P » andP ´ , where» and´ are some diffusion processes. As usually, by P » P´ we denote the Radon-Nykodim derivative of the measureP » w.r.t. the measureP ´ . 5 In what follows all stochastic integrals are understood in sense of Itˆ o. Let » = (» 1 (t);:::;» n (t)) and´ = (´ 1 (t);:::;´ n (t)); t2 [0;T], be vector-processes, with Itˆ o differentials d» t =A t (»)dt+b t (»)dW t ; d´ t =a t (´)dt+b t (´)dW t ; » 0 =´ 0 ; (1.4) whereW t =(W 1 (t);:::;W k (t)) is ak-dimensional Winer process with respect toF t , A t (x) = (A 1 (t;x);:::;A n (t;x)); a t (x) = (a 1 (t;x);:::;a n (t;x)); b t (x) = [b ij (t;x)] is a matrix of ordern£k, and´ 0 =(´ 1 (0);:::;´ n (0)) is a vector of initial values such thatP ¡ n P j=1 j´ j (0)j <1 ¢ = 1. We suppose that equations (1.4) have a unique strong solution (for more details about conditions on the coefficients see [62], Theorem 5.5, or [47], Theorem 4.6). We denote by A ¤ the transpose matrix of a given matrix A. By b + we denote the pseudoinverse matrix with respect to the matrix b. Recall that the matrix b + is called pseudoinverse matrix with respect to the matrixb, ifb + bb + = b + andbb + b = b. Note that if b is a square invertible matrix, then b + = b ¡1 . For more details about pseudoinverse matrices see for instance [4]. The following result about absolute continuity of measuresP » andP ´ holds true: 1.2.2 Theorem. Assume that the system of algebraic equations b t (x)y t (x)=A t (x)¡a t (x) (1.5) 6 has a solution with respect toy t (x) for everyt2[0;T]; x2R, and T Z 0 ³ A ¤ t (x)(b t (x)b ¤ t (x)) + A t (x)+a ¤ t (x)(b t (x)b ¤ (x)) + a t (x) ´ dt<1 P » ¡a:s: ThenP » »P ´ and dP » P ´ (t;´)=exp n T Z 0 ¡ A s (´)¡a s (´) ¢ ¤ ¡ b t (´)b ¤ (´) ¢ + d´(s) ¡ 1 2 T Z 0 ¡ A s (´)¡a s (´) ¢ ¤ ¢¡ b t (´)b ¤ (´) ¢ + ¡ A s (´)+a s (´) ¢ ds o : (1.6) For the proof see for instance Chapter 7 in [47]. More general results are discussed in [26], [27], [28]. 1.3 Stochastic partial differential equations and their applications The theory of Stochastic Differential Equations (SDEs) is dealing with finite- dimensional noise, namely the perturbation consists of finite number of Wiener Processes. In the case of Stochastic Partial Differential Equation (SPDEs) besides of considering functions of multivariables, usually the perturbation part represents an infinite-dimensional noise, i.e. infinitely many Wiener Processes. A general theory of infinite-dimensional noise is based on the Hilbert space-valued Wiener process (see for instance [8], [70]). For sake of completeness we present here briefly the general idea and construction of infinite-dimensional noise. 7 Let H be a separable Hilbert space, and Q a selfadjoint none-negative operator on H. Suppose that Q is a trace (nuclear) operator. Then there exists a complete orthonormal system (CONS) h k ; k ¸ 1, inH, and a sequence of non-negative real numbersfq k g k¸1 such that Qh k = q k h k ; k ¸ 1 and Tr(Q) = P q k < 1. In other words, Q is a selfadjoint, trace operator onH, with eigenvalues q k and eigenvectors h k ; k¸1. 1.3.1 Definition. AH-valued stochastic process W(t); t ¸ 0 is called a Q-Wiener Process if (i)W(0) = 0; (ii)W has continuous trajectories; (iii)W has independent increments; (iv) W(t)¡W(s) » N(0;(t¡s)Q); 0 · s · t, where N denotes a Gaussian process in Hilbert spaceH (for more details see [8], Chapter 1-4). One can show thatE(W(t))=0; Cov(W(t))=tQ; t¸0, and for arbitraryt the following expansion holds W(t)= 1 X k=1 ¸ k h k W k (t); (1.7) where ¸ k = p q k ; k 2 N, W k (t) are real valued standard Wiener processes, mutu- ally independent on some probability space(;F;P), and the last series converges in L 2 (;F;P). For complete proof see for instance [8], Proposition 4.1. It turns out that the representation (1.7) is very convenient to use in the theory of SPDEs. Moreover, we can assume that Q is a bounded, selfadjoint, non-negative operator on H, with pure discrete spectrum (non necessary trace operator). In this case W(t) will take value in a larger Hilbert space, having the same type of repre- sentation (1.7). If P k¸1 ¸ k = 1, then infinite-dimensional Wiener process W(t) is called cylindrical Wiener process or cylindrical Brownian motion. In special case, if we takeH = L 2 (R d ) and¸ k = 1; k ¸ 1, the formal sumdW(t) = P k¸1 h k (x)dW k (t) 8 is called space-time white noise. In this work, we will mainly focuss on the case ¸ k =k ® ; ®2R. Let(V;H;V 0 ) be a triple of Hilbert spaces and(;F;fF t g;P) a stochastic basis. Suppose thatA(t) : V ! V 0 ; M k (t) : V ! H are linear bounded operators for all k 2N; t 2 [0;T], and for some finite terminal time T . A general Linear Stochastic Partial Differential Equation is written as du(t)=(Au(t)+f(t))dt+(M k u(t)+g k (t))dW k (t); u(0)=u 0 ; (1.8) wheret2 [0;T]; u 0 2 L 2 (;H) andu 0 isF 0 -measurable. We assume thatW k (k 2 N) are independent standard Wiener processes, f;g k (k 2N) areF t -adapted random processes such thatf 2L 2 (£(0;T);V 0 ) andg k 2L 2 (£(0;T);H);k2N. Here and in what follows the summation over repeated indices is assumed. Depending on the noise term the equation (1.8) is classified as follows: - Equations with additive noise, ifM k =0; - Equations with multiplicative noise, ifM k 6=0. Similar to classical deterministic PDEs, the solution of equation (1.8) can be spec- ified in different ways. Since the solution is a stochastic process, besides of PDE part, where we have classical solution, strong/weak generalized solution and mild solution, also we can specify strong and weak probabilistic solution. In this work we consider only solution which are strong in probabilistic sense and weak/strong/mild in PDE sense. 9 We say that F t -adapted function u 2 L 2 (£ (0;T);V) is a weak solution of equation (1.8), if for every'2V and allt2[0;T], the equality (u(t);') H =(u 0 ;') H + t Z 0 [Au(s)+f(s);']ds+ t Z 0 (M k u(s)+g k (s);') H dW k (s) (1.9) holds with probability one. We say that F t -adapted function u 2 L 2 (£ (0;T);V) is a mild solution of equation (1.8) if u(t)=S(t)u 0 + t Z 0 S(t¡s)f(s)ds+ t Z 0 S(t¡s)(M k u(s)+g k (s))dW k (s); t2[0;T]; holds true with probability one, whereS is a strongly continuous semigroup with infin- itesimal generator A (for definition of semigroup and infinitesimal operator see for instance [20]). As we observed, stochastic evolution equations in infinite dimensions are natural generalizations of classical PDEs and Systems of Stochastic Ordinary Differential Equations. The theory related to all these equations has motivations coming from physics, chemistry, biology, medicine, finance etc. Although, the theory of SPDEs is already established and widely developed field in mathematics, and problems arising in this theory represent an interest for mathematics itself, we will mention here several classical problems related to some particular SPDEs which we are going to study latter on in Chapter 2 and 3. 10 Interesting and important examples represent Zakai and Kushner equations (see [8], [70], [80]) which come from diffusion filtering problem. Let X(t) be the unob- servedd-dimensional process, andY(t) be ther-dimensional observed process defined by the following SODEs dX(t)=b(X(t))dt+¾(X(t))dW(t)+½(X(t))dV(t); dY(t)=h(X(t))dt+dV(t); 0<t·T; X(0)=X 0 ; Y(0)=0; (1.10) whereb(x)2R d ; ¾(x)2R d£r ; h(x)2R r , andX 0 has a densityu 0 . Letf = f(x) be a scalar measurable function onR d such that sup 0·t·T Ejf(X(t))j 2 < 1. The filtering problem is to find the best square mean estimate ~ f of f(X(t)); t 2 [0;T], given the observationsY(s); s2[0;t]. One can show that ~ f =E(f(X(t))jF Y t ) and ~ f(t)= R R d f(x)u(t;x)ds R R d u(t;x)dx ; whereu satisfies the following equation du(t;x)=Au(t;x)dt+ r X k=1 B k u(t;x)dY k (t); 0<t·T; x2R d ; u(0;x)=u 0 (x); (1.11) where A k ;B k are some partial differential operators determined by the functions b;¾;½;h. Equation (1.11) is called Zakai filtering equation, which is a particular case of equation (1.8). If we put p(t;x) = u(t;x)= R R d u(t;x)dx then p satisfies Kushner filtering equation which is a non-linear SPDE. Helmholtz equation du(t;x)=(i¢u(t;x)+au(t;x))dt+iu(t;x) 1 X k=1 q k h k (x)dW k (t); 11 wherex2R 2 ; t > 0; a > 0;q k > 0;k¸ 1; h k CONS inL 2 (R 2 ), is used to describe and study random media (see [70]). The equation of the form du(t;x) = (F(t;x;u(t;x))u x (t;x)) x dt + G(t;x;u(t;x))dW(t;x) appears as a limit of a branching diffusion model, and has applications in biology [10], chemistry and theory of super-processes [58], [59], [63]. The simple SPDEv t =v xx ¡v+dW with proper initial values and boundary con- ditions was proposed to model electrical potential at point(x;t) generated by neurons (see for instance [79]). We conclude this section with some application of SPDEs to finance. The great success of stochastic calculus in description of stock markets and valuation of options on stocks strongly influenced the research related to fixed income market, in particular modelling the term structure of interest rates. Before going into details, let us recall some notions from bond market (for more details see for instance [5], [6], [24]). A zero coupon bond with maturity dateT , is a contract which guarantees the holder 1 dollar to be paid on the dateT . We denote by byp(t;T) the price of a bond at time t. We assume thatP(t;T) exists for everyT > t,P(t;T) > 0; P(t;t) = 1, and there exists@P(t;T)=@T . Instantaneous forward rates at time t for all time to maturity x > 0; f(t;x), are defined by f(t;x)=¡ @logP(t;t+x) @x ; (1.12) which is the rate that can be contracted at timet for instantaneous borrowing or lending at timet+x. The spot interest rate at timet is defined as r(t)=f(t;0): (1.13) 12 If we want to make a model for bond market, it suffices to specify the dynamics for one of these variables: Bond Price P(t;T), forward rate f(t;x), spot rate r(t). Knowing the dynamics of one of them, by the Itˆ o’s formula (see for instance [62]) we can get the dynamics of rest of them (see [5], Chapter 15, page 230). There are many models for spot rate, among which we want to mention Vasiˇ cek model dr = (b¡ar)dt+¾dW; a> 0, Cox-Ingersoll-Ross (CIR) modeldr =a(b¡r)dt+¾ p rdW , Ho-Lee modeldr =µ(t)dt+¾dW . A popular model for forward rate is Heath-Jarrow- Morton (HJM) model by which the forward rate follows the dynamics df(t;x)= µ @f(t;x) @x +a(t;x) ¶ dt+¾(t;x)dW(t); (1.14) where a(t;x) = ¾(t;x)( R x 0 ¾(t;y)dy +'(t)), and ' is a known function (actually ' is the market price of risk). We note that in all these models the shock is a one dimen- sional Brownian motion, which means that the same set of shocks affect all forward rates, which constrains the correlations between bond prices. A natural mathematical generalization of the HJM model for forward rates, since we have a function of two variables, is to consider the noise term to be an infinite-dimensional Brownian motion W(t;x). In other words, this means that every instantaneous forward rate is driven by its own noise, and the equation becomes df(t;x)= µ @f(t;x) @x +a(t;x) ¶ dt+¾(t;x)dW(t;x); (1.15) This idea was proposed by Kennedy [34], [35], where the forward rate was modelling as a Gaussian field, and Goldstein [14], where the noise term is a specific space-time white nose. P. Santa-Clara and D. Sornette [71] generalize this approach, presenting various types of noise term, as well as, application to pricing bond derivatives. 13 Motivated by statistical properties of interest rates, R. Cont [7] propose a model in which the forward rate curve is decomposed in factors as follows f(t;x)=f(t;x min )+s(t)(Y(x)+u(t;x)) (1.16) where x min ;x max are shortest and longest maturity available on the market, s(t) = f(t;x min )¡f(t;x max ); Y is a deterministic shape function defining the average profile of the term structure, andu(t;x) an adapted process describing the random deviations of the term structure from its long term average shape. Empirical studies identify the level of interest rates, the steeptness (slope) of term structure and its curvature as three significant parameters in the geometry of the yield (see for instance [49]). Then, under some technical assumptions on the market properties, R. Cont [7] deduces that u satisfies the following parabolic SPDE du(t;x)= µ @u(t;x) @x +b(t;x;u(t;x))+ k 2 @ 2 u(t;x) @x 2 ¶ dt +¾(t;x;u(t;x))dW(t;x); (1.17) whereb and¾ are some functions, andk a real parameter. In [7], author discussed the caseb´0; ¾´const. 1.4 Itˆ o’s formula in Hilbert space In this section we will present a version of the Itˆ o’s formula in Hilbert spaces. Although this result is known and can be found, for example, in [70], Chapter 2, Theorem 4.2, we are going to present here a simpler proof, using the one-dimensional Itˆ o’s formula. 14 LetF = (;F;fF t g t2[0;T] ;P) be a stochastic basis. A random variable ¿ taking values inR + is called a stopping time with respect to the filtrationF t ; t2R + , if the random eventf! : ¿(!) · tg 2 F t , for every t 2R + . The real-valued stochastic process M(t) is called martingale relative to the filtration F t , if it F t -adaptive, P- integrable, andE[M(t)jF s ] = M(s) for every s;t 2 R + ; s · t. The real-valued process M(t) is called local martingale relative to the filtration F t , if there exists a sequence of stoping times f¿ n g n2N ; ¿ n " 1, such that for every n, the process M(t^¿ n ) is a martingale relative toF t . We denote byM c loc (R + ;R) the set of all real- valued continuous local martingales. One can show that for everyM 2 M c loc (R + ;R), there exists a unique (up to a version) continuous increasing stochastic processhMi t such thatjM(t)j 2 ¡hMi t 2 M c loc (R + ;R). The processhMi t is called the quadratic variation process of the local martingale M(t). For a predictable process X(t) one may define the integral t R 0 X(t)dM(t) in sense of Itˆ o. For more on these definitions, properties and proofs see for instance [48], [75]. All these definitions can be extended naturally to the stochastic processes with val- ues in some Hilbert space (for more details see Chapter 2, Section 1.4 in [70]). For sake of completeness we briefly recall some of them. Let V be a separable Hilbert space and we denote by V 0 its conjugate space. An V -process M(t) is called martingale (local martingale) relative to the filtrationF t if (a)EjjM(t)jj V <1, for everyt2R + (EjjM(t^¿ n )jj, for everyt2R + and some sequence of stoping timesf¿ n g n2N such that ¿ n " 1); (b) for every h ¤ 2 V 0 , the process h ¤ M(t) is a martingale (local mar- tingale) relative to the filtrationF t . The set of all continuous local martingales taking values in V will be denoted by M c loc (R + ;V). Similarly to real-valued local martin- gales, given M(t) 2 M c loc (R + ;V), we denote byhMi t the increasing process such 15 thatkM(t)k 2 ¡hMi t 2 M c loc (R + ;V). We say thathMi t is the quadratic variation of theM(t). Let us consider a triplet of Hilbert spaces (V;H;V 0 ) and suppose that M(t) is a continuous local martingale relative to filtration fF t g, which takes values in the Hilbert space V . Since V is a separable Hilbert space, there exists a CONSfh k g k2N in this space. We denote by hM i i t the quadratic variation of the local continuous martingaleM k (t):=(M(t);h k ). To prove the main result of this section we will need the following result. 1.4.1 Lemma. For all!2 1 ½, whereP( 1 )=1, hMi t = dimV X k=1 hM i i t : For the proof see Lemma 8, Section 2.1.8 in [70]. For a predictableV -processf we put t Z 0 (f(s);dM(s)):= lim N!1 N X k=1 t Z 0 (f(s);h k )dM k (s): The last integral is well-defined [70]. Suppose thatx(!;t) andy(!;t) are given functions on£[0;T] and taking values in V and V 0 respectively. We assume that these functions are predictable and satisfy the inequality T Z 0 ¡ kx(!;t)k 2 V +ky(!;t)k 2 V 0 ¢ dt<1 (P¡a:s:): (1.18) Letx(¢;0) be anF 0 -measurable function on, and let¿ be a stopping time. 16 1.4.2 Theorem. If for everyv2V the following equality (x(t);v)=(x(0);v)+ t Z 0 [v;y(s)] V;V 0ds+(M(t);v) (1.19) holdsP£B-a.s. on the setf(!;t):t<¿(!)g, then kx(t)k 2 V =kx(0)k 2 V +2 t Z 0 [x(s);y(s)] V;V 0ds+2 t Z 0 (x(s);dM(s))+hMi t (1.20) Proof. SinceV is a separable Hilbert space, there exists a CONSfh k g k2N in this space. Then, equation (1.19) forv =h k implies x k (t)=x k (0)+ t Z 0 [h k ;y(s)]ds+M k (t); (1.21) wherex k (t)=(x;h k ) V ; M k (t)=(M;h k ) V , andk2N. Note that, since x(!;t) : £[0;T] ! V is an adaptive process, and x k (!;t) = (x(!;t);h k ) V , using the continuity of inner product inV , we find thatx k isF t -adapted for every k 2N. By the same arguments, we conclude that M k (t) = (M(t);h k ) V is a local martingale. Also, since[¢;¢] is a bounded bilinear form onV £V 0 we get that [v;y(t)] V;V 0 isF t -adaptive, and by (1.18) we deduce that P 0 @ t Z 0 ¯ ¯ ¯[v;y(s)] ¯ ¯ ¯ 2 ds<1 forallt>0 1 A =1: 17 Thus, we can apply one dimensional Itˆ o’s formula to the process x k given by (1.21), by which we get x 2 k (t)=x 2 k (0)+2 t Z 0 x k (s)[h k ;y(s)]ds+2 t Z 0 x k (s)dM k (s)+hM k (t)i; (k2N): (1.22) Summing up both parts of last equalities with respect tok =1;:::;N, we find N X k=1 x 2 k (t)= N X k=1 x 2 k (0)+2 t Z 0 N X k=1 x k (s)[h k ;y(s)]ds +2 t Z 0 N X k=1 x k (s)dM k (s)+ N X k=1 hM k (t)i: (1.23) In the last equality for every(!;t)2£[0;T] such thatt·¿(!) we pass to the limit withN !1. By Parseval’s equality 1 P k=1 x 2 k (t) =kx(t)k 2 . Similarly we getkx(0)k 2 . Taking into account the continuity of inner product, lim N!1 t R 0 N P k=1 x k (s)[h k ;y(s)]ds = t R 0 [x(s);y(s)] V;V 0ds. By the definition lim N!1 t R 0 N P k=1 x k (s)dM k (s) = t R 0 (x(s);dM(s)). Finally, by Lemma 1.4.1 the last term from LHS of (1.23) converges tohMi t . Hence we resume (1.19). Theorem is proved. 1.5 Existence and uniqueness of solution For every differential equation, deterministic or stochastic, ordinary or with partial derivatives, the first and fundamental question is existence and uniqueness of solution. A general theory of this problem for SPDEs can be found in [8], [43], [70], [79]. In this section we present some known results about existence and uniqueness of solution of parabolic SPDEs, which will need for future studying. 18 Let (V;H;V 0 ) be a triple of Hilbert spaces, (;F;fF t g;P) a stochastic basis, A(t) : V ! V 0 ; M k (t) : V ! H; k ¸ 1; t 2 [0;T] some linear bounded operators. We consider the following SPDE du(t)=(Au(t)+f(t))dt+(M k u(t)+g k (t))dW k (t); (1.24) where t 2 [0;T]; u(0) = u 0 2 L 2 (;H) and u 0 isF 0 -measurable. We assume that W k (k2N) are independent standard Wiener processes,f;g k (k2N) areF t -adapted random processes such thatf 2L 2 (£(0;T);V 0 ) andg k 2L 2 (£(0;T);H)(k2 N). LetV be a dense subset of the Hilbert spaceV relative to the topology generated by the normk¢k V . The following theorem holds true. For complete proof see for instance [70], Chapter 3. 1.5.1 Theorem. In addition to all assumptions mentioned above, suppose that (i) 1 P k=1 T R 0 Ejjg k (t)jj 2 H dt<1; (ii) For every'2V, the processesA'(t) andM k '(t) areF t -adapted; (iii) 1 P k=1 T R 0 EjjM k '(t)jj 2 H dt<1 for all'2V; (iv) There exist a positive constant ± and a real number C 0 such that, for all t 2 [0;T], all!2 and allv2V, we have 2[A'(t);']+ 1 X k=1 jjM k '(t)jj 2 H ·¡±jj'jj 2 V +C 0 jj'jj 2 H ; (1.25) (v) There exist a positive numberC A so that, for allt2[0;T]; !2; '2V, jjA'jj V 0 ·C A jj'jj V : 19 Then there exists a unique weak solutionu of equation (1.24), the solution belongs to L 2 (;C((0;T);H))\L 2 (£[0;T];V) and E ³ sup 0<t<T jju(t)jj 2 H ´ + ± 2 E ³ T Z 0 jju(t)jj 2 V ´ · (1.26) C(C A ;C 0 ;T)E ³ jju 0 jj 2 H +C f (±) T Z 0 jjf(t)jj 2 V 0dt+C g (±) 1 X k=1 T Z 0 jjg k (t)jj 2 H dt ´ : We want to mention that conditions (i)-(iii), (v) mostly are related to the definition of solution and come naturally from the structure of equation (1.24). The condition (iv) represents the (strong or super-) parabolicity or coercivity assumption and the equa- tion in this case is called parabolic SPDE. This is due to the fact that in the case when operatorA is an elliptic differential operator andM k a differential operator subordi- nated, in some sense, toA, the condition (iv) is fulfilled. For example classical heat equation is the simplest deterministic parabolic equation and a stochastic counterpart can be taken asdu(t;x)=au xx dt+¾u x dW(t), which is a particular case of equation (1.24) withA = a @ 2 @x 2 ; f(t) = g(t) = 0; M = ¾ @ @x . This equation is solvable in the triple(H 1 (R);H 0 (R);H ¡1 (R)) if± =2a¡¾ 2 >0 or in equivalent form 2[A'(t);']+jjM'(t)jj 2 H =a Z R ' xx 'dx+ ¾ 2 2 Z R ' 2 x dx =¡(2a¡¾ 2 ) Z R ' 2 x dx=¡±k'k 2 1 +c 0 k'k 2 0 ; which is exactly condition (iv) from Theorem 1.5.1. Note that ± = 0 corresponds to the degenerate case2a=¾ 2 , and although the corresponding SPDE is a parabolic type equation, special studies have to be done in this case. 20 Similarly to deterministic equations the estimate (1.26) of the solution is called energy estimate which implies uniqueness of the solution. For all equations considered in our research, first we will establish the existence and uniqueness of the solution by applying Theorem 1.5.1. Here we will show how this theorem can be applied to stochastic heat equation. Besides the fact that this equation is one of the simplest SPDE, it will also play the role of benchmark in our investigations. 1.5.2 Example. Stochastic heat equation with additive space-time white noise. Let (;F;fF t g;P) be a stochastic basis and assume that ° < ¡ 1 2 . In the triple of Hilbert spaces(H °+1 (0;¼);H ° (0;¼);H °+1 (0;¼)) we consider the following SPDE 8 > > < > > : du(t;x)=u xx dt+ P k¸1 h k (x)dW k (t); u(0;x)=u(t;0)=u(t;¼)=0; x2[0;¼];t2[0;T]; (1.27) where h k (x) = q 2 ¼ sin(k¼x) and W k are independent standard Brownian motions. Note that h k forms a CONS in L 2 (0;¼) and also are all eigenvalues of the Laplace operator ¢ = @ xx with zero boundary conditions on (0;¼). Using the notations from Theorem 1.5.1 we haveV = H °+1 ; H = H ° ; V 0 = H °+1 ; A = ¢; f = 0; M k = 0; g k = h k . Now let us check the conditions (i)-(v) from Theorem 1.5.1. Since ° < ¡ 1 2 we have P k¸1 kh k k 2 ° = P k¸1 k 2° < 1, and hence (i) is satisfied. Opera- torsA andM k are deterministic and hence (ii) holds true. M k = 0 and (iii) obvi- ously is satisfied. Coercivity condition (iv): 2[A';'] = 2[' xx ;'] = ¡2[' x ;' x ] = ¡2k' x k 2 ° ·¡±k'k °+1 , with± =2, so (iv) is verified. SinceA is a differential opera- tor of second order, from general theory of Sobolev spaces, we have that the operator A : H s ! H s¡2 is a linear bounded operator for every s 2 R. Thus assumption 21 (v) is verified, and we conclude that for every ° · ¡ 1 2 there exists a unique solution u 2 L 2 (;C((0;T);H ° ))\L 2 (£[0;T];H °+1 ) of equation (1.27), and it satisfies the following estimate (energy type estimate) E 0 @ sup t2(0<t) ku(t)k 2 ° + T Z 0 ku(t)k 2 °+1 dt 1 A ·C : We conclude this chapter with a version of Burkholder-Davis-Gundy (BDG) inequality [8]. Let W(t) be a standard Brownian Motion, andF t the filtration gen- erated by this Brownian Motion. 1.5.3 Theorem. Let » be a square integrableF t adapted process. For every p > 0 there exists constantsc p andC p , such that c p E 0 B @ ¯ ¯ ¯ ¯ ¯ ¯ T Z 0 j»(s)j 2 ds ¯ ¯ ¯ ¯ ¯ ¯ p 2 1 C A ·E 0 @ sup 0<t<T ¯ ¯ ¯ ¯ ¯ ¯ t Z 0 »(s)dW(s) ¯ ¯ ¯ ¯ ¯ ¯ p 1 A ·C p E 0 @ T Z 0 j»(s)j 2 ds 1 A p 2 : (1.28) 22 Chapter 2 Regularity of solution 2.1 Introduction For a given evolution equation, besides existence and uniqueness of solution, it is important to know regularity properties of solution. By regularity or smoothness we mean continuity, H¨ older continuity, differentiability, etc. of the solution with respect to both time and space variables. This problem have been widely studied by many authors, and various classes of SPDEs have been considered. Here we recall some results related to parabolic evolution equations. As an application to neurophysiology, J.B.Walsh [79] considered the stochastic heat equation on a finite interval, namely the equation of the form 8 > > > > > > < > > > > > > : @v @t = @ 2 v @x 2 ¡v+f(v;t) _ W t> 0; 0<x<L; @v @x (0;t)= @v @x (L;t)=0 t> 0; v(x;0)=v 0 (x); x2[0;L]; (2.1) wherev 0 isF 0 -measurable,E(v 2 0 (x)) is bounded andf is uniformly Lipschitz contin- uous. In [79] was proven that the mild solution of equation (2.1) (t;x) ! v(t;x) is H¨ older continuous int with exponent 1 4 ¡", and inx with exponent 1 2 ¡", for all">0. A general linear parabolic evolution equation of the form du=(a ij u x i x j +b i u x i +cu+f)dt+(¾ ik u x i +º k u+g k )dW k (t); t>0; (2.2) 23 with proper conditions on the coefficients, boundary values and initial conditions, has been considered in [37]-[45], [50],[51]. In these papers the authors studied the Cauchy problem for the whole space(x2R d ) and half space(x2R d ;x 1 > 0), the Dirichlet problem for a bounded domain, developingL p theory for these equations. ByL p the- ory we mean that suitable Sobolev spaces and sufficient conditions on the coefficients are found such that equation (2.2) has a unique solution in these spaces. From this theory, in particular, one can conclude about regularity properties of the solution. For example, N.V .Krylov [42] studied the Cauchy problem for equation (2.2) is case of whole spaceR d , and proved, as an application of L p theory, that for d = 1, the solu- tion is H¨ older continuous inx of order1=2¡" and int of order1=4¡". The Dirich- let problem on bounded domain for the same equation was discussed by N.V .Krylov, S.Lototsky in [44],[45], where the weighted Sobolev spaces were constructed. In par- ticular the equation (2.1) is considered, up to the boundary conditions (Dirichlet in [45] and Neumann in [79]), and the result ford=1 agrees with that obtained in [79]. In R.Mikulevicius [54] and R.Mikulevicius, H.Pragarauskas [55] the second-order linear parabolic SPDE with deterministic leading coefficients driven by a cylindrical Brownian motion in some Hilbert space were considered. Following the ideas given in B.Rozovskii [69], R.Mikulevicius [54] studied the Cauchy problem on the whole space (x 2 R d ) in the scale of H¨ older spaces, while in [55] the Cauchy-Dirichlet problem on the half-space in suitable weighted H¨ older spaces is investigated. Since solution belongs to H¨ older space the conclusion about the smoothness of the solution follows. Finally we want to mention the results presented in DaPrato [8], a general linear evolution equations is investigated. More precisely, the coefficients of equation are some linear operators which generates a strongly continuous semigroup, and the noise, either additive (Chapter 5) or multiplicative (Chapter 6), is driven by a cylindrical 24 Brownian Motion in some Hilbert space. It is shown that the weak (mild) solution has a continuous path and some particular examples are considered too. In this chapter we will study the regularity properties of equation (1.24) driven by space-time white noise, either additive or multiplicative. Similar to Chapter 1 we consider M being a d-dimensional compact orientableC 1 manifold with a smooth positive measure dx. Let A be a differential, elliptic operator of order 2m on M, formally selfadjoint and semi-lower bounded. Without loss of generality, using the notations from Section 1.1, we consider L = ¡A and apply all results mentioned therein. The operator ¤ = (¡A) 1 2m is well-definite, elliptic of order 1 and generates the scalefH s (M)g s2R of Sobolev spaces. We denote byh k and¸ k ; k2N; the eigen- functions and corresponding eigenvalues of the operator ¤, and by l k the eigenvalues of the operatorA. Recall thatfh k g k2N forms a CONS inL 2 (M;dx) (for more details, see Section 1.1). In the triple(H °+m ;H ° ;H °¡m ), for some°2R, we consider the following equa- tion 8 > > > > > > < > > > > > > : du(t)=Au(t)dt+ 1 P k=1 ® k B k (u)h k (x)dW k (t); u(t;¢)j M =0; 0<t<T; u(0;x)=u 0 (x); x2M; (2.3) where t 2 [0;T]; x 2 M; u 0 2 C 1 (0;¼); M k are operators acting in H ° and ® k = k ® for some real parameter ®. Using results from Chapter 1, we will find sufficient conditions on parameters ® and °, and operatorsA; B k , which guarantee existence and uniqueness of solution in the scale of Sobolev spacesH ° (M). A major novelty is the factor ® k in the noise term; in all previous works ® = 0. The specific form of this factor is motivated by the asymptotic behavior of the eigenvalues of the operatorA. It turns out that the parameter® controls the smoothness 25 of the solution u of equation (2.3). As one may expect, as ® decreases the solution becomes smoother both in time and space. We state the regularity properties (number of derivatives and H¨ older continuity) of the solution in both space and time in terms of the parameter®, theL 1 norm and some regularity properties of the eigenfunctions of the operatorA. In connection with this we make the following reasonable assumptions (A1) kh k k 1 ·cjl k j ½ , for some½2R + and everyk2N; (A2) jh k (x)¡h k (y)j·jl k j ±(¯) jx¡yj ¯ , where0<¯·1; x;y2M; k2N, and± is an increasing function, with positive values. We rely on some results from spectral theory of selfadjoint differential operators, which is a separate topic in mathematics, with many fundamental results. Even the particular question of pointwise bounds of eigenfunctions have been studied widely. There are many papers written on this subject, where various operators are considered, and this question represents a well explored field. We refer the reader to classical work [61], and as well as to some recent papers [15], [19], [17], [18], [29], [30], [31], [32], [72], [73]. Of course the estimates vary from case to case and depend on the operatorA, as well as on the geometry of the domainM and boundary conditions. One general result holds true. 2.1.1 Remark. Without making any geometric assumptions, ifA is the Laplacian on compact manifolds with boundary, (A1) holds true with ½ = d¡1 4 (see the proof in [15]). In particular,kh k k 1 <cjl k j d¡1 4 holds true for a general bounded domain inR d . This result is sharp for the sphereM =S d . 26 We recall that regardless of the domain M, the eigenvalues ¹ k of the operatorA with zero boundary conditions have the following asymptotic l k »k 2m=d ; (2.4) and consequently¸ k »k 1=d (see for instance [76]). We want to mention that generally speaking the conditions (A1) and (A2) are hard to check, and represent difficult mathematical problems, however in many applications the eigenvalues and eigenfunctions are computable explicitly and these conditions are verified by simple arithmetic evaluations. Let us consider the one dimensional Laplace operator with zero boundary conditions. 2.1.2 Example. One dimensional Laplacian. Assume that d = 1;Au = u xx ; m = 1; M = [0;¼] and zero boundary conditions. Thenh k (x)= 2 ¼ sin(kx), and hencekh k k 1 ·const. Thus (A1) is satisfied with½=0. Also note, thatl k =¡k 2 , and thus (2.4) also holds true. Using well-known inequality jsin(x)j·jxj, and a trivial trigonometric identity, we havejh k (x)¡h k (y)j ¯ ·k ¯ jx¡ yj ¯ for all x;y 2 [0;¼] and every ¯ > 0. Obviously we have thatjh k (x)¡h k (y)j· 2; x;y2[0;¼]. From here we conclude jh k (x)¡h k (y)j=jh k (x)¡h k (y)j ¯ jh k (x)¡h k (y)j 1¡¯ (2.5) ·k ¯ jx¡yj ¯ 2 (1¡¯) ·cl ¯ 2 k jx¡yj ¯ ; wherec is a constant independent of¯. Hence (A2) is verified with±(¯)= ¯ 2 . More concrete examples will be considered at the end of this chapter. 27 2.1.3 Remark. Conditions (A2) in fact represents the H¨ older continuity of the eigen- functions h k . Since h k 2 C 1 , the constant ³ from inequality jh k (x)¡ h k (y)j · ³ ¯ jx¡yj ¯ corresponds to the Lipschitz constant of the functionh k , which can be asso- ciated with the ”first derivative”. On the other hand, ”the first derivative” essentially is ¤ which is a differential operator of order one. Since¤(h k )=¸ k h k and¸ k =l 1 2m k , we get ±(¯) = ¯ 2m . These completely heuristic arguments turn out to hold true in many concrete example. However, we do not know if this is true in general. The results obtained here agree with those known and mentioned above. We would like to note that equation (2.3) is a particular case of equations studied in [50], however, it turns out that to check all abstract conditions on coefficients and spaces of solution is not easier than establishing these results directly. Also, we want to note that equation (2.3) does not belong to the class of equations considered in [54] (whole space), [55] (half plane). Not only because we consider the bounded domain, which to some extend is just a technical problem as long as there are theories for the whole space and half plane, but also the noise term is different. A similar approach to control the smoothness of the solution through some para- meters of the noise term are considered in [60] where the nonlinear SPDE du = 1 2 ¢udt+ p udW(t;x) is studied. Mytnik, Perkins and Sturm in a recent paper [60] study regularity properties of the solution of the equation du = 1 2 ¢udt+ p udW(t;x), where x 2 R; t 2 [0;T] and W(t;x) is a space-time white noise onR + £R. The method of duality is used to establish the existence and uniqueness of the solution. The solution itself repre- sents the density function for one-dimensional super-Brownian motion. The regularity properties of the solution are described in terms of asymptotic behavior of the kernel of the covariance functional on the main diagonal y = x. It turns out that there is 28 a connection between our results and results stated in [60]. We will discuss these in more details in Section 2.2. In terms of regularity properties, the results obtained here are the same for both additive and multiplicative noise, however, the spaces in which equation (2.3) has solu- tion differ. Also, the technical evaluations in multiplicative case are more involved. Thus, we consider separately the equations with additive noise, Section 2.2, and equa- tions with multiplicative noise in Section 2.3. The main results of this chapter are contained in Theorem 2.2.5, 2.2.11 and 2.3.8. Every theorem will be followed by a simple example, usually stochastic heat equation. In the last section of this chapter, as applications of general results, we consider various concrete examples. 2.2 Equations with additive noise 2.2.1 Existence and uniqueness In this section we consider equation (2.3) with additive noise, which means that we takeB k u´1. Thus equation (2.3) becomes du(t)=Au(t)dt+ 1 X k=1 ® k h k (x)dW k (t); (2.6) where t 2 [0;T]; x 2 M, the same initial values and boundary conditions as in equation (2.3), and® k =k ® for some real parameter®. Initially we will establish some results about existence and uniqueness of the solu- tion. For this, we make the following assumption: (A3) Assume that®; °2R such that° <¡d(®+ 1 2 ). 29 2.2.1 Theorem. If the assumption (A3) is satisfied, then equation (2.6) has a unique weak solution u2L 2 ((0;T)£;H °+m )\L 2 (;C((0;T);H ° )): Proof. We apply Theorem 1.5.1 with f = 0; M k = 0, and g k = ® k h k ; k 2 N. Conditions (ii), (iii), (v) of this theorem are obviously satisfied. In order to check condition (i) we write 1 X k=1 T Z 0 Ejjg k (t)jj 2 ° dt=c 1 X k=1 ® 2 k kh k k 2 ° =c 1 X k=1 k 2® ¸ 2° k : Since¸ k »k 1 d , from the last equality we deduce 1 X k=1 T Z 0 Ejjg k (t)jj 2 ° dt=c 1 X k=1 k 2® k 2° =d : The last series converges in virtue of Assumption (A3). Finally, we verify condition (iv). It suffices to check it for basis functionsh k , which leads to 2[Ah k ;h k ]=2l k [h k ;h k ]=2l k kh k k 2 ° =2l k ¸ ¡2m k kh k k 2 °+m =¡2kh k k 2 °+m : Thus (iv) from Theorem 1.5.1 is satisfied with± =2>0. Theorem is proved. 2.2.2 Remark. Note that for® =0, which corresponds to standard space-times white noise, the solution exists if ° < ¡d=2. In particular, for d = 1, we have that the solution exists if° <¡1=2 (this covers well know result, see for instance [79]). Also, we want to mention that the equations driven by additive space-time white noise have 30 solution in someH ° regardless of space dimension d. Namely, for every d 2N and every®2R there exists° 2R such thatu2 L 2 ((0;T)£;H ° ). As will see latter on, this is not the case for the equations with multiplicative space-time white noise. 2.2.3 Example. Assume that d = 1; M = [0;¼] and let A be the one dimen- sional Laplace operator with zero boundary conditions. In the triple of Hilbert spaces (H °+1 ;H ° ;H °+1 ) let us consider the following SPDE du(t;x)=u xx (t;x)dt+ X k¸1 k ¡ 2 3 sin(kx)dW k (t); (2.7) where x2 [0;¼]; t2 [0;1] and W k are independent standard Brownian motions. As we already mentioned (see Example 1.5.2), the functionsh k are the eigenfunctions of the operatorA and forms a CONS in L 2 . Note that ® = ¡ 2 3 . By Theorem 2.2.1, for every° < 1 6 , there exists a unique strong solution of equation (2.7) inH °+1 . Looking forward for regularity properties of the solutionu, we present here the plot of one realizations of the field u given by (2.6) for different values of the parameter ®. In the Figure 2.1, Panel (a), the solution corresponds to the equation discussed in Remark 2.2.2, i.e. ® = 0; d = 1. The solution of equation (2.7) from Example 2.2.3 (® =¡ 2 3 ) is shown in Panel (b). We used Crack-Nicolson finite difference scheme (see for instance [38], [57]) to simulate the fieldu. Note that the solutionu looks smoother, Panel (b), for® =¡ 2 3 than that for the parameter® =0, Panel (a). How smother it is, we will answer in the next subsections. 31 0 1 2 3 0 0.5 1 −1 −0.5 0 0.5 1 x Panel (a). α=0 t u(t,x) 0 2 4 0 0.5 1 −0.4 −0.2 0 0.2 0.4 x Panel (b). α= −2/3 t u(t,x) Figure 2.1: Realization of approximated solution of stochastic heat equation with addi- tive noise In what follows we will use a convenient representation for the solutionu of equa- tion (2.6). Denote by G t (x;y) the fundamental solution of the corresponding deter- ministic parabolic equation u t = Au(t) with the same boundary values and initial conditions as (2.3). Then (see for instance [9], Chapter 2,7) G t (x;y)= X k¸1 h k (x)h k (y)e l k t : (2.8) Similarly to deterministic case, it can be shown that the weak solution of equation (2.6) satisfies the equality u(t;x)= ¼ Z 0 G t (x;y)u 0 (y)dy+ X k¸1 t Z 0 Z M ® k G t¡s (x;y)h k (y)dW k (s): (2.9) The proof can be found in [8], Section 5.2, page 121. Note that by (2.9),u actually is a mild solution of equation (2.6), and by specific structure of this equation it turns out that the mild solution coincides with weak solution, existence of which is provided by Theorem 2.2.1. 32 From (2.9) we see that without loss of generality we can suppose thatu 0 = 0 and study the regularity of the second term in (2.9). Thus using (2.8), from (2.9) we get u(t;x)= X k¸1 t Z 0 Z M ® k X j¸1 h j (x)h j (y)e l j (t¡s) h k (y)dydW k (s); and taking into account thatfh k g k¸1 forms an orthonormal basis in L 2 (M), we have the representation u(t;x)= t Z 0 X k¸1 ® k h k (x)e l k (t¡s) dW k (s): (2.10) 2.2.2 Regularity in space To establish the regularity properties of solution u in space variable x, we will prove an auxiliary result. 2.2.4 Lemma. Under assumption (A2), for any¯2(0;1] and®2R such that ®< m(1¡2±(¯)) d ¡ 1 2 ; (2.11) the solutionu of equation (2.6) satisfies the following inequality E ¯ ¯ ¯u(t;x)¡u(t;y) ¯ ¯ ¯ p ·Cjx¡yj ¯p ; (2.12) wherex;y2M; t2[0;T]; p> 2. Proof. From (2.10) one deduces E ¯ ¯ ¯u(t;x)¡u(t;y) ¯ ¯ ¯ p =E ¯ ¯ ¯ ¯ ¯ ¯ t Z 0 X k¸1 ³ h k (x)¡h k (y) ´ ® k e l k (t¡s) dW k (s) ¯ ¯ ¯ ¯ ¯ ¯ p : 33 From here by the BDG inequality (1.28), we continue E ¯ ¯ ¯u(t;x)¡u(t;y) ¯ ¯ ¯ p · ¯ ¯ ¯ ¯ ¯ ¯ X k¸1 t Z 0 ¯ ¯ ¯h k (x)¡h k (y) ¯ ¯ ¯ 2 ® 2 k e 2l k (t¡s) ds ¯ ¯ ¯ ¯ ¯ ¯ p=2 =c p ¯ ¯ ¯ ¯ ¯ X k¸1 ¯ ¯ h k (x)¡h k (y) ¯ ¯ ¯ 2 ® 2 k ¡2l k ³ 1¡e 2l k t ´ ¯ ¯ ¯ ¯ ¯ p=2 : Note thatl k <0; k2N, and thus1¡e 2l k t is uniformly bounded int andk, hence E ¯ ¯ ¯u(t;x)¡u(t;y) ¯ ¯ ¯ p ·c p ¯ ¯ ¯ ¯ ¯ X k¸1 ® 2 k jl k j jh k (x)¡h k (y)j 2 ¯ ¯ ¯ ¯ ¯ p=2 : (2.13) From here, by assumption (A2), we get E ¯ ¯ ¯u(t;x)¡u(t;y) ¯ ¯ ¯ p ·c p jx¡yj ¯p ¯ ¯ ¯ ¯ ¯ X k¸1 k 2® jl k j ¡1 jl k j 2±(¯) ¯ ¯ ¯ ¯ ¯ p=2 : Finally, sincel k »k 2m=d , we find that the last series converges if 2®+ 2m d (2±(¯)¡1)<¡1; which essentially is assumption (2.11) on parameters¯ and®. Lemma is proved. Now we are ready to prove the theorem about regularity of the solution in space variablex. 34 2.2.5 Theorem. Suppose that Assumptions (A2) and (A3) are satisfied, and®<¡ 1 2 + m d . Let u be the solution of equation (2.6) and r the biggest integer such that there exists¯ 0 2(0;1] which satisfies the following inequality 2d®+2r+4m±(¯ 0 )¡2m<¡d: Then¤ r (u) is H¨ older continuous inx of order ³ :=sup n ¯j±(¯)<¡ d 4m ¡ d® 2m + 1 2 ¡ r 2m ; ¯2(0;1] o ¡"; for every">0, as long as³ > 0; Proof. By the initial assumption, the solutionu of the equation (2.6) exists and satisfies Lemma 2.2.4 for some¯ > 0. Letr be the integer as stated in the theorem. Note that ®<¡ 1 2 + m d implies thatr¸0. By (2.10) we have ¤ r (u(t;x))=¤ r h t Z 0 X k¸1 ® k h k (x)e l k (t¡s) dW k (s) i (2.14) = X k¸1 ® k ¤ r ³ h k (x) ´ t Z 0 e l k (t¡s) dW k (s) = X k¸1 ® k ¸ r k h k (x) t Z 0 e l k (t¡s) dW k (s) as long as the last series converges absolutely. Since¸ k »k 1 d , we apply Lemma 2.2.4 to¤ r u with ~ ® =®+ r d and conclude that ¯ ¯ ¯¤ r (u(t;x))¡¤ r (u(t;y)) ¯ ¯ ¯ p ·c p jx¡yj ¯p ; (2.15) 35 for every¯ >0 such that±(¯)< 1 2 ¡ r 2m ¡ d® 2m ¡ d 4m . Taking into account the definition of³ we have that the inequality (2.15) is satisfied for every¯ · ³. By Kolmogorov’s criterion (Theorem 1.1.1) with d := d;p := p;q := ¯p¡d, we get that the process ¤ r u(t;x) satisfies the following inequality ¯ ¯ ¯¤ r u(t;x)¡¤ r u(t;y) ¯ ¯ ¯·Yjx¡yj ¯¡ d p ³ ln A jx¡yj ´ 2=p (x;y2M; p>2) whereY is a positive random variable, andA is a positive real number. For every">0 the functionjxj " ³ ln A jxj ´ 2=p is continuous at x = 0. Hence, the process ¤ r u(t;x) is H¨ older continuous inx of order¯¡ 1 p ¡", for everyp > 2 and" > 0. Consequently, ¤ r u(t;x) is H¨ older continuous inx of order less than¯·³. Theorem is proved. 2.2.6 Remark. As we already mentioned, the operator ¤ r ; r 2 N, plays the role of the differential operator of orderr. Generally speaking, Theorem 2.2.5 tells how many derivatives has the solutionu and what is the order of continuity of ther-th derivative. Moreover, the theorem remains true if we take any other operator ª instead of¤ r , as long as one can show thatªh k =c¸ r k h k (see equality (2.14)). One special case that occurs in many application is ±(¯) = ¯ 2m (for more moti- vations about this see also Remark 2.1.3). For this particular situation Theorem 2.2.5 implies: 2.2.7 Theorem. Suppose that Assumptions (A2) and (A3) are satisfied, and assume that® <¡ 1 2 + m d and±(¯) = ¯ 2m ; ¯ 2 (0;1]. Letu be the solution of equation (2.6), and denote by ´ := m¡d®¡ d 2 and r := b´c (b´c denotes the biggest integer less than or equal to´). (i) if ´ 6= N then ¤ r (u) is H¨ older continuous of order ³ = ´¡r¡", for every ">0, as long as³ >0; 36 (ii) if´2N, then¤ r¡1 (u) is H¨ older continuous of order³, for every³ 2(0;1). 2.2.8 Example. As an application of Theorem 2.2.7 we will consider the following SPDE: du(t;x) = u xx (t;x)dt + P k¸1 k ¡ 2 3 sin(kx)dW k (t) under the same setup as in Example 2.2.3, i.e. the stochastic heat equation with Dirichlet boundary conditions, zero initial values, ® = ¡ 2 3 ; d = 1; x 2 M = [0;¼]; t 2 [0;1]. By Example 2.2.3 we have that the solution u 2 H °+1 for every ° < 1 6 . As we concluded in Example 2.1.2, Assumption (A2) is verified and±(¯)= ¯ 2 . Hence we can apply Theorem 2.2.7. Recall thath k (x) = 2 ¼ sin(kx) and¸ k = k. Since @ r @x r h k = c¸ r k ~ h k , where ~ h k is either sin(kx) orcos(kx), which essentially obey the same regularity properties, by Remark 2.2.6, we can take ¤ = @ @x . Hence, by Theorem 2.2.7 with ´ = 7 6 we conclude: the solution u has one derivative in x, and the first derivative u x is H¨ older continuous of order less than 1 6 . By the same arguments, we have that the solution u(t;x) of equation du(t;x) = u xx (t;x)dt+ P k¸1 sin(kx)dW k (t) (see Example 1.5.2) is H¨ older continuous inx of order less than ´ = m¡d®¡ 1 2 = 1¡0¢1¡ 1 2 = 1 2 . This resume the classical result by Walsh [79]. Compare the graphs of these two stochastic fields shown on Figure 2.1. 2.2.9 Remark. We want to mention that the Theorem 2.2.5 about regularity of the solution can not be obtained by directly the Sobolev embedding theorems. For exam- ple, by Theorem 2.2.7 we have that the solution exists, is unique andu2 H °+m , for every° <¡d®¡ d 2 , i.e. u2H m¡ d 2 ¡d®+" for every">0. By Sobolev embedding the- orem (see for instance see [9], Chapter 5)H s ½C s¡ d 2 , and henceu2C m¡d®¡d¡" . We proved that u 2 C m¡d®¡ d 2 ¡" , which obviously is a stronger result than that obtained by Sobolev embedding theorems. However, if one can develop the L p theory for this equation, namely to show that the solution exists in H °+m p for every p > 2, then by Sobolev embedding theorem u 2 H m¡ d p ¡d®+" ½ C m¡ d p ¡d®+"¡ d p . Taking p large 37 enough we obtain our result, namely u 2 C m¡d®¡ d 2 ¡" for every " > 0, as long as m¡d®¡ d 2 ¡" > 0. This is not of big surprise, since the main tool in our proof has been the Kolmogorov’s criterion 1.1.1, which to some extends is a version of Sobolev embedding theorem. In other words, one may conclude that we actually developed implicitly theL p theory. 2.2.3 Regularity in time Regularity in time is obtained in the same way as regularity in space. Initially we will prove an auxiliary result similar to Lemma 2.2.4. 2.2.10 Lemma. Under Assumption (A1), for every ® 2 R and ¯ 2 (0;1] such that ¯ < 1¡2½¡ d(1+2®) 2m , the solutionu of equation (2.6) satisfies the following inequality E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p ·Cjt 1 ¡t 2 j ¯p 2 ; (2.16) wheret 1 ;t 2 2[0;T]; x2M; p> 2. Proof. Let0<t 1 <t 2 <T . By (2.10) E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p = E ¯ ¯ ¯ ¯ ¯ ¯ t 1 Z 0 X k¸1 h k (x)® k e l k (t 1 ¡s) dW k (s)¡ t 2 Z 0 X k¸1 h k (x)® k e l k (t 2 ¡s) dW k (s) ¯ ¯ ¯ ¯ ¯ ¯ p = E ¯ ¯ ¯ ¯ ¯ ¯ t 2 Z 0 X k¸1 h k (x)® k ³ e l k (t 2 ¡s) ¡e l k (t 1 ¡s) I (s·t 1 ) ´ dW k (s) ¯ ¯ ¯ ¯ ¯ ¯ p : 38 By the BDG inequality, Theorem 1.5.3, we continue E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p · E ¯ ¯ ¯ ¯ ¯ ¯ t 2 Z 0 X k¸1 h 2 k (x)® 2 k ¯ ¯ ¯e l k (t 2 ¡s) ¡e l k (t 1 ¡s) I (s·t 1 ) ¯ ¯ ¯ 2 ds ¯ ¯ ¯ ¯ ¯ ¯ p=2 · E ¯ ¯ ¯ ¯ ¯ ¯ X k¸1 h 2 k (x)® 2 k t 2 Z 0 ¯ ¯ ¯e l k (t 2 ¡s) ¡e l k (t 1 ¡s) I (s·t 1 ) ¯ ¯ ¯ 2 ds ¯ ¯ ¯ ¯ ¯ ¯ p=2 : After direct evaluations one shows t 2 Z 0 ¯ ¯ ¯e l k (t 2 ¡s) ¡e l k (t 1 ¡s) I (s·t 1 ) ¯ ¯ ¯ 2 ds = 1 ¡2l k µ 2 ³ 1¡e l k (t 2 ¡t 1 ) ´ ¡ ³ e l k t 2 ¡e l k t 1 ´ 2 ¶ · 1¡e l k (t 2 ¡t 1 ) ¡l k : Thus, E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p ·c ¯ ¯ ¯ ¯ ¯ X k¸1 h 2 k (x)® 2 k 1¡e l k (t 2 ¡t 1 ) jl k j ¯ ¯ ¯ ¯ ¯ p=2 : Sincej1¡e ¡¹jt 2 ¡t 1 j j·¹ ¯ jt 2 ¡t 1 j ¯ for every0<¯·1 and¹>0, using Assumption (A1) the last inequality can be continued · ¯ ¯ ¯ ¯ ¯ X k¸1 h 2 k (x)® 2 k l ¯ k jt 2 ¡t 1 j ¯ l ¡1 k ¯ ¯ ¯ ¯ ¯ p=2 ·jt 2 ¡t 1 j ¯p 2 ¯ ¯ ¯ ¯ ¯ X k¸1 l 2½ k ® 2 k l ¯¡1 k ¯ ¯ ¯ ¯ ¯ p=2 : 39 By asymptotic property (2.4), and the definition of ® k = k ® , from the last inequality we have the following estimation E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p ·cjt 2 ¡t 1 j ¯p 2 X k¸1 k 2®+ 2m d (¯¡1+2½) : (2.17) The initial assumption¯ <1¡2½¡ d 2m (1+2®) implies that the last series converges, hence the lemma is proved. Now we are ready to prove the Theorem about regularity int of the solutionu(t;x) 2.2.11 Theorem. Suppose that ® < m d ¡ 2m½ d ¡ 1 2 , and assume that the assumptions (A1) and (A3) are satisfied. Then the solutionu of equation (2.6) is H¨ older continuous int of order less than´ :=minf 1 2 ; 1 2 ¡½¡ d(1+2®) 4m g. Proof. Assumptions (A1) and (A3) imply that the equation (2.6) has a solution. Let ¯ = 2¿. By initial assumption® < m d ¡ 2m½ d ¡ 1 2 we have that´ > 0, and hence¯ 2 (0;1]. Consequently, by Lemma 2.2.10 the solution u of equation (2.6) satisfies the inequalityE ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p ·Cjt 1 ¡t 2 j ¯p 2 , wheret 1 ;t 2 2[0;T]; x2M; p> 2. Applying the Kolmogorov’s criterion (Theorem 1.1.1) withd:=d;p :=p;q := ¯p 2 ¡d, we get ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯·Yjt 1 ¡t 2 j ¯ 2 ¡ d p ³ ln A jt 1 ¡t 2 j ´ 2=p : (2.18) By the same arguments as in Theorem 2.2.7 we conclude that the processu is H¨ older continuous int of any order less than ¯ 2 =´. Theorem is proved. Taking A = ¢, and without making any geometric assumption on M, a direct consequence of Theorem 2.2.11 gives: 40 2.2.12 Corollary. Suppose thatA is the Laplacian on a compact manifold with bound- ary, and assume that° <¡d®¡ d 2 and® < 3 2d ¡1. Then the solutionu of equation (2.6) is H¨ older continuous int of order less than´ =minf 1 2 ; 3 4 ¡ d® 2 ¡ d 2 g. Proof. By Remark 2.1.1 ½ · d¡1 4 . We apply the previous theorem, with m = 1 and above mentioned½ and conclude that for every®< m d ¡ 2m½ d ¡ 1 2 = 3 2d ¡1 the solution u is H¨ older continuous in t of any order less then ´ = minf 1 2 ; 1 2 ¡½¡ d(1+2®) 4m g = minf 1 2 ; 3 4 ¡ d 2 ¡ d® 2 g. 2.2.13 Example. Let us consider the same equation as in Example 2.2.8: du(t;x) = u xx (t;x)dt+ P k¸1 k ¡ 2 3 sin(kx)dW k (t), i.e. the stochastic heat equation with Dirichlet boundary conditions, zero initial values,® =¡ 2 3 ; d = 1; x2 M = [0;¼]; t2 [0;1]. Recall that the solutionu2H °+1 for every° · 1 6 . Sinceu xx is the Laplace operator, by Corollary 2.2.12 we have that the solutionu is H¨ older continuous in time variable t, and the order of continuity is´ =minf 1 2 ; 7 12 g¡"= 1 2 ¡" for every"2(0;1=2). 2.2.14 Remark. Theorem 2.2.7 implies that order of continuity of the solutionu can not exceed 1 2 . This property comes natural since the Brownian motion W is a con- tinuous, nowhere differentiable process, H¨ older continuous of order 1 2 ¡" for every 0<"< 1 2 . 2.3 Equations with multiplicative noise 2.3.1 Existence and uniqueness In this section we study the equation (2.3) in the case of multiplicative noise. For simplicity of writing we takeB k u = u, but the results remain valid forB k u = g(u) where g is global Lipschitz continuous. Generally speaking, the methods used here 41 can be applied to general operatorsB k : H °+m ! H ° , however we postpone these discussion for our future research and the results will be published somewhere else. Thus, keeping the same notations and boundary conditions as in the previous sec- tion, we consider the following equation du(t)=Au(t)dt+ 1 X k=1 ® k u(t)h k (x)dW k (t); (2.19) wheret2[0;T]; x2M; a2R; ® k =k ® and®2R ¡ . First, we will establish the existence and uniqueness of the solution. For this, we make the following assumptions (A4) Assume that®; °2R are such that¡ m 2 <° <¡ d 2 ¡d® . We will prove an auxiliary results. 2.3.1 Lemma. For every" > 0 and° 0 ; ° 1 ; ° 2 2R such that° 0 2 [° 2 ¡2;° 2 ];° 1 · ° 2 ¡2, there existsM " 2R such that kvk 2 ° 0 ·"kvk 2 ° 2 +N " kvk 2 ° 1 ; (2.20) for everyv2H °¡2 . Proof. By interpolation theorem (see Section 1.1 for more details) kvk 2 s · C 1 kvk 2µ 1 s 1 +2 kvk 2(1¡µ 1 ) s 1 , where s 2 [s 1 ;s 1 + 2]; µ 1 = s¡s 1 2 and C 1 is a constant which depends only onM;s;s 1 . Consequently, by"-inequality (see Section 1.1) with ":=" 1 ; p:= 1 µ 1 ; q = 1 1¡µ 1 we have kvk 2 s ·C 1 " 1 kvk 2 s 1 +2 +C 1 " ¡ 1¡µ 1 µ 1 1 kvk 2 s 1 : (2.21) 42 Applying inequality (2.21) tokvk 2 s 1 withs 1 2[s 2 ;s 2 +2] we get kvk 2 s ·C 1 " 1 kvk 2 s 1 +2 +C 1 C 2 " ¡ 1¡µ 1 µ 1 1 " 2 kvk 2 s 2 +2 +C 1 C 2 " ¡ 1¡µ 1 µ 1 1 " ¡ 1¡µ 2 µ 2 2 kvk 2 s 2 : Note thats 1 ¸s 2 , hencekvk s 2 +2 ·kvk s 1 +2 , and consequently by the above inequality we deduce kvk 2 s · ³ C 1 " 1 +C 1 C 2 " ¡ 1¡µ 1 µ 1 1 " 2 ´ kvk 2 s 1 +2 +C 1 C 2 " ¡ 1¡µ 1 µ 1 1 " ¡ 1¡µ 2 µ 2 2 kvk 2 s 2 : Similarly, by indication withs 1 >s 2 >:::>s n , one has kvk 2 s · ³ C 1 " 1 +:::+C 1 ¢¢¢C n " ¡ 1¡µ 1 µ 1 1 ¢¢¢" ¡ 1¡µ n¡1 µ n¡1 n¡1 " n ´ kvk 2 s 1 +2 +C 1 ¢¢¢C n " ¡ 1¡µ 1 µ 1 1 ¢¢¢" ¡ 1¡µn µn 2 kvk 2 s n : (2.22) Note that " 1 ;:::;" n are mutually independent, so we can take " s ; s = 1;:::;n such that C 1 " 1 + ::: + C 1 ¢¢¢C n " ¡ 1¡µ 1 µ 1 1 ¢¢¢" ¡ 1¡µ n¡1 µ n¡1 n¡1 " n · ". Also, we put s 1 = ° 2 ¡ 2, and continue with s 2 ;:::;s n until s n · ° 1 . Finally denote by N " := C 1 ¢¢¢C n " ¡ 1¡µ 1 µ 1 1 ¢¢¢" ¡ 1¡µ n µ n 2 . Lemma is proved. 2.3.2 Theorem. If Assumption (A4) is satisfied and u 0 2 L 2 (;H ° ), then equation (2.19) has a unique solutionu2L 2 ((0;T)£;H °+m )\L 2 (;C((0;T);H ° )) . Proof. Similar to Section 2.2, the existence and uniqueness is established by Theorem 1.5.1. See also [79], Theorem 3.2, pag. 313. For the sake of completeness, let us verify conditions (iii) and (iv) of Theorem 1.5.1. In our case, (iii) becomes I 1 = X k¸1 k 2® kh k vk 2 ° <1 forall v2V =H °+m : (2.23) 43 For everyv 1 2H s and everyv 2 2H jsj+" we havekv 1 v 2 k s ·kv 1 k s ¢kv 2 k jsj+" for every">0 (see for instance [78] Corollary 2.8.2). We continue (2.23) I 1 · 1 X k=1 k 2® kh k k 2 ° ¢kvk 2 j°j+" : Sincev2H °+m ; kvk j°j+" is finite if°+m>j°j which holds true by initial assump- tion. Hence, I 1 ·c 1 X k=1 k 2® kh k k 2 ° : Recall thatkh k k ° =¸ ° k , and¸ k »k 1 d (see notes on page 2), and we continue I 1 ·c 1 X k=1 k 2® k 2° d =C : By assumption (A4) this series converges andC <1. Finally let us check the condition (iv). Assume thatv2H °+m . Then, I 2 =2[Av;v]+ X k¸1 ® 2® k jjh k vjj 2 ° =¡2[¤ 2m v;v]+ X k¸1 k 2® kh k vk 2 ° =¡2kvk 2 °+m + X k¸1 k 2® kh k vk 2 ° : Using the estimates forI 1 established above, we continue I 2 ·¡2kvk 2 °+m +Ckvk 2 j°j+" 1 : 44 Sincej°j·°+m, we can take" 1 >0 such that°+m¡2<j°j+"<°+m and apply Lemma 2.3.1 tokvk 2 j°j+" 1 with° 0 =j°j+" 1 ; ° 1 = minf°;° +m¡2g;° 2 =° +m. From here we have I 2 ·¡2kvk 2 °+m +C"kvk 2 j°j+m +N " kvk 2 ° 1 ; and since° 1 ·°, we continue I 2 · ¡ ¡2+C" ¢ kvk 2 j°j+m +N " kvk 2 ° : Put ± = 2¡C", and take " > 0 small enough, such that ± > 0. Thus condition (iv) from Theorem 1.5.1 is fulfilled. Theorem is proved. Note that ° < ¡ d 2 ¡d® is Assumption (A3), which guaranties the existence and uniqueness of the solution for equation with additive noise. It should be mentioned that for d = 1; m = 1, the conditions on ® and ° from Theorem 2.3.2 imply¡ 1 2 < ° < ¡®¡ 1 2 . Consequently, ® < 0. In other words, Theorem 2.3.2 ”almost” cover the results from Walsh [79] where® =0 is considered. However, we will present here the corresponding result for® = 0 andA being the Laplacian. Although the result is known [79], our proof is different and simpler in our opinion. 2.3.3 Theorem. Suppose that¡1·° <¡ d 2 andu 0 2L 2 (;H ° ). Then the equation du(t;x)=¢u(t;x)dt+ P k¸1 h k (x)u(x;t)dW k (t) has a unique solutionu2L 2 ((0;T)£ ;H °+1 )\L 2 (;C((0;T);H ° )). 45 Proof. Similar to previous proof, we will apply Theorem 1.5.1. Condition (iii) gives I 1 = X k¸1 kh k vk 2 ° = X j¸1 ¸ 2° j X k¸1 ¯ ¯ ¯hh k v;h j i 0 ¯ ¯ ¯ 2 = X j¸1 ¸ 2° j jjvh j jj 2 0 · X j¸1 ¸ 2° j jjh j jj 2 1 jjvjj 2 0 : Since° >¡1 andv2H °+1 , we havejjvjj 0 <1. By Remark 2.1.1jjh k jj 1 ·c¸ d¡1 2 k . Hence, we continue I 1 ·cjjvjj 2 0 X j¸1 ¸ 2°+d¡1 j : Since° <¡ d 2 , the last series converges. Condition (iii) from Theorem 1.5.1 is verified. Now let us check coercivity condition (iv) from Theorem 1.5.1. Suppose thatv 2 H °+1 . Then, by the same arguments as above and using that¤= p ¡¢, we get I 2 =2[¢v;v]+ X k¸1 kh k vk 2 ° ·¡2kvk 2 °+1 +ckvk 2 0 X k¸1 ¸ 2°+d¡1 k : The last series converges, since° <¡ d 2 . Hence I 2 ·¡2kvk 2 °+1 +Ckvk 2 0 : Note that°¡1<¡1<° < 0°+1. By Lemma 2.3.1 we have I 2 ·¡2kvk 2 °+1 +C"kvk 2 °+1 +N " kvk 2 °¡1 · ¡ ¡2+C" ¢ kvk 2 °+1 +N " kvk 2 ° : Take " > 0 such that ± := 2¡C" > 0, and conditions (iv) is verified. Theorem is proved. 46 2.3.4 Remark. Recall that for the equations with additive noise, the solution exists regardless of the space dimension and the order of the differential operator A (see Remark 2.2.2). The situation is different for the equations with multiplicative noise. It is known that the equation du = ¢udt+udW(t;x) has solution only for dimension d = 1. This also follows from Theorem 2.3.3. Indeed, the condition¡1 · ° · ¡ d 2 implies d < 2. For initial equation (2.19) the restrictions are given by Assumption (A4), which also imply that the existence of the solution depends on both space dimen- sion and order of the differential operatorA. However, for any m; d 2 N, we can choose ® ¿ ¡1 such that Assumption (A4) is satisfied, and hence the solution of equation (2.19) exists. We conclude this subsection by presenting the multiplicative counterpart of equa- tions from Example 2.2.3. 2.3.5 Example. Assume that d = 1; M = [0;¼] and let A be the one dimen- sional Laplace operator with zero boundary conditions. In the triple of Hilbert spaces (H °+1 ;H ° ;H °+1 ) let us consider the following SPDE du(t;x)=u xx (t;x)dt+ X k¸1 k ¡ 2 3 sin(kx)u(t;x)dW k (t); (2.24) wherex2[0;¼]; t2[0;1] andW k are independent standard Brownian motions. Note that® =¡ 2 3 . By Theorem 2.3.2, for every¡ 1 2 < ° < 1 6 , there exists a unique strong solution of equation (2.7) inH °+1 (0;¼). In Figure 2.2 we show realizations of the solution u of some SPDE’s with multi- plicative noise (compare to Figure 2.1, where additive noise is discussed). Panel (a) corresponds to the equationdu=u xx dt+h k udW k , investigated in Theorem 2.3.3. One realization of the solution of equation (2.24) is shown Panel (b). For both equations we 47 considered zero boundary conditions and the initial valuesu(0;x)=sin(3x)+1; x2 [0;¼]. To simulate the filedu we used Crack-Nicolson finite difference scheme (see for instance [38], [57]). Similar to additive noise (see Figure 2.1 and Example 2.2.3), the solutionu looks smoother for® =¡ 2 3 than the solution of equation with correspond- ing noise parameter® =0. Actually for a fixed parameter®, the solutions of equations with additive and multiplicative noise, if exist, have the same order of continuity. We will show this in the next subsection. 0 1 2 3 0 0.5 1 0 1 2 3 x Panel (a). α=0 t u(t,x) 0 1 2 3 0 0.5 1 0 1 2 3 x Panel (b). α=−2/3 t u(t,x) Figure 2.2: Realization of approximated solution of stochastic heat equation with mul- tiplicative noise 2.3.2 Regularity in space and time Similar to additive case, to establish the regularity properties of the solutionu, we will prove an auxiliary result. In what follows we will assume thatu 0 2L 2 (;H 0 (M)). 2.3.6 Lemma. Suppose that Assumptions (A2) and (A4) are satisfied and assume that ° >¡m¡d®. Then for any¯2(0;1] and®2R such that ®< m(1¡2±(¯)) d ¡ 1 2 ; (2.25) 48 the solutionu of equation (2.6) satisfies the following inequality E ¯ ¯ ¯u(t;x)¡u(t;y) ¯ ¯ ¯ p ·Cjx¡yj ¯p ; (2.26) wherex;y2M; t2[0;T]; p> 2. Proof. Following the same arguments as in the previous section, the solution u of equation (2.19) can be represented as u(t;x)= Z M G t (x;y)u 0 (y)dy+ X k¸1 t Z 0 Z M G t¡s (x;y)h k (y)® k u(s;y)dydW k (s) (2.27) wherex2M; t2[0;T]. Sinceh k 2 C 1 (M) andu 0 2 L 2 (£M) the first term is a smooth function and (2.36) is satisfied. Hence we will study the regularity properties of the second term. Using (2.27), we get Eju(t;x)¡u(t;z)j p ·E ¯ ¯ ¯ X k¸1 t Z 0 Z M ® k e G(t¡s;x;y;z)h k (y)u(s;y)dydW k (s) ¯ ¯ ¯ p ; where e G(t¡s;x;y;z)=G t¡s (x;y)¡G t¡s (z;y). By BDG (1.5.3), we continue Eju(t;x)¡u(t;z)j p ·c p E ¯ ¯ ¯ X k¸1 t Z 0 ¯ ¯ ¯ Z M e G(t¡s;x;y;z) ¢® k h k (y)u(s;y)dy ¯ ¯ ¯ 2 ds ¯ ¯ ¯ p 2 : (2.28) 49 Define I 1 = X k¸1 ¯ ¯ ¯ ¯ ¯ ¯ Z M ³ G t¡s (x;y)¡G t¡s (z;y) ´ ® k h k (y)u(s;y)dy ¯ ¯ ¯ ¯ ¯ ¯ 2 = X k¸1 ¯ ¯ ¯hu e G(t¡s;x;y;z);® k h k i L 2 (M) ¯ ¯ ¯ 2 : Since¸ k »k 1 d ; ¤ s (h k )=¸ s k h k and® k =k ® , we continue I 1 ·c X k¸1 ¯ ¯ ¯hu e G(t¡s;x;y;z);¤ d® (h k )i L 2 (M) ¯ ¯ ¯ 2 : Using the fact that¤ is a selfadjoint operator we get I 1 ·c X k¸1 ¯ ¯ ¯h¤ d® (u e G);h k i 0 ¯ ¯ ¯ 2 : Finally, by Parseval’s equality and property (1.2) of the operator¤ we have I 1 ·c ° ° °¤ d® (u e G) ° ° ° 2 0 =cku e Gk 2 d® ; and thus, from here and (2.28) one deduces Eju(t;x)¡u(t;z)j p ·cE ¯ ¯ ¯ ¯ ¯ ¯ t Z 0 ° ° ° e G(t¡s;x;¢;z)u(s;¢) ° ° ° 2 d® ds ¯ ¯ ¯ ¯ ¯ ¯ p 2 : (2.29) By the same arguments as in Theorem 2.3.2 from (2.28) we have Eju(t;x)¡u(t;z)j p ·cE ¯ ¯ ¯ ¯ ¯ ¯ t Z 0 ° ° ° e G(t¡s;x;¢;z) ° ° ° 2 d® ° ° °u(s;¢) ° ° ° 2 jd®j++" ds ¯ ¯ ¯ ¯ ¯ ¯ p 2 : (2.30) 50 By H¨ older inequality withp 1 := p p¡2 ; q 1 := p 2 , and (2.30) we get Eju(t;x)¡u(t;z)j p (2.31) ·cE 8 > < > : ¯ ¯ ¯ ¯ ¯ ¯ t Z 0 ku(s;¢)k 2q 1 jd®j+" ds ¯ ¯ ¯ ¯ ¯ ¯ p 2 ¢ 1 q 1 9 > = > ; ¢ ¯ ¯ ¯ ¯ ¯ ¯ t Z 0 k e G(t¡s;x;¢;z)k 2p 1 d® ds ¯ ¯ ¯ ¯ ¯ ¯ p 2 ¢ 1 p 1 : By initial assumption ° + m > jd®j which implies that the first factor in the last inequality is finite. Now, we are going to estimate the last term on the right-hand side of (2.31), and show that it is finite too. Using the form (2.8) of the functionG we get I 2 := t Z 0 k e G(t¡s;x;¢;z)k 2p 1 d® ds= t Z 0 à X j¸1 ¯ ¯ ¯h e G(t¡s;x;¢;z);h j i 0 ¯ ¯ ¯ 2 ¸ 2d® j ! p 1 ds = t Z 0 à X j¸1 ¯ ¯ ¯ X k¸1 (h k (x)¡h k (z))e l k (t¡s) hh k ;h j i 0 ¯ ¯ ¯ 2 ¸ 2d® j ! p 1 ds = t Z 0 à X k¸1 ³ h k (x)¡h k (z) ´ 2 e 2l k (t¡s) ¸ 2d® k ! p 1 ds: By H¨ older inequality withp 2 :=p 1 = p p¡2 andq 2 = p 2 we continue I 2 · t Z 0 X k¸1 ³ h k (x)¡h k (z) ´ 2p 2 ¸ 2d®p 2 k e 2l k (t¡s)p 2 à X j¸1 e 2l j (t¡s)q 2 ! p 1 q 2 ds; and consequently I 2 · X k¸1 ³ h k (x)¡h k (z) ´ 2p 2 ¸ 2d®p 2 t Z 0 e l k (t¡s)p 2 ³ X j¸1 e l j (t¡s)q 2 ´ p 1 q 2 ds: (2.32) 51 Let us estimate the inner integral of the right-hand side of (2.32). Again by H¨ older inequality withp 3 ; q 3 such that 1 p 3 + 1 q 3 =1 we get I 3 := t Z 0 e l k (t¡s)p 2 ³ X j¸1 e l j (t¡s)q 2 ´ p 1 q 2 ds (2.33) · ³ t Z 0 e l k (t¡s)p 2 p 3 ds ´ 1 p 3 ³ t Z 0 h X j¸1 e l j (t¡s)q 2 i p 1 q 3 q 2 ds ´ 1 q 3 : Ifq 3 := q 2 p 1 = p¡2 p then the last factor is finite for everyp>2 . Indeed, I 4 = t Z 0 h X j¸1 e l j (t¡s)q 2 i p 1 q 3 q 2 ds= t Z 0 X j¸1 e l j (t¡s)q 2 ds= X j¸1 e l j tq 2 ¡1 l j q 2 : Recall that¡l j »j 2m d and sincel j <0 we conclude that1¡e l j tq 2 <1. Hence, I 4 ·c X j¸1 1 j 2m d ; and sincem> d 2 , we have thatI 4 is finite. Consequently, from (2.33) we get I 3 ·I 4 ³ t Z 0 e l k (t¡s)p 2 p 3 ds ´ 1 p 3 =I 4 h 1¡e l k tp 2 p 3 jl k jp 2 p 3 i 1 p 3 ·c p jl k j ¡ 1 p 3 ; (2.34) wherep 3 = p¡2 p¡4 . 52 From (2.31) using (2.32), (2.33) and (2.34) and Assumption (A2), for ¯ > 0 one deduces E ¯ ¯ ¯u(t;x)¡u(t;z) ¯ ¯ ¯ p (2.35) ·c p ¯ ¯ ¯ ¯ ¯ X k¸1 ¸ 2d®p 2 k jx¡yj 2p 2 ¯ l 2±(¯)p 2 k l ¡ 1 p 3 k ¯ ¯ ¯ ¯ ¯ p 2p 1 ·c p jx¡yj ¯p ¢ ¯ ¯ ¯ X k¸1 k 4m±(¯) d p p¡2 + 2®p p¡2 ¡ 2m d p¡4 p¡2 ¯ ¯ ¯ p¡2 2 ·c p jx¡yj ¯p ¢ ¯ ¯ ¯ ¯ ¯ X k¸1 k p p¡2 ( 4m±(¯) d +2®¡ 2m d p¡4 4 ) ¯ ¯ ¯ ¯ ¯ p¡2 2 ; and since, by initial assumption®<¡ 1 2 + m(1¡2±(¯)) d , we conclude that for sufficiently largep the last series converges. Lemma is proved. Now, we will prove a similar result for the time variablet. 2.3.7 Lemma. Suppose that ° > ¡m¡ d® and let Assumptions (A1) and (A4) be satisfied. Then for every¯2(0;1] such that ¯ <1¡2½¡ d(1+2®) 2m (2.36) the solutionu of the equation (2.19) satisfies the following inequality E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p ·Cjt 1 ¡t 2 j ¯p 2 ; (2.37) wheret 1 ;t 2 2[0;T]; x2M; p> 2. 53 Proof. Suppose that0· t 1 · t 2 · T . The existence of the solutionu is provided by the above assumptions and Theorem 2.3.2, and by representation (2.27) we have J =E ¯ ¯ ¯u(t 1 ;x)¡u(t 2 ;x) ¯ ¯ ¯ p =E ¯ ¯ ¯ t 2 Z 0 X k¸1 Z M ^ G(t 1 ;t 2 ;s;x;y)® k h k (y)u(s;y)dydW k (s) ¯ ¯ ¯ p where b G(t 1 ;t 2 ;s;x;y) := G t 1 ¡s (x;y)I (s·t 1 ) ¡G t 2 ¡s (x;y). By similar evaluations as in Lemma 2.3.6 we deduce J ·C ¯ ¯ ¯ ¯ ¯ ¯ t 2 Z 0 k b G(t 1 ;t 2 ;s;x;¢)k 2p 1 d® ds ¯ ¯ ¯ ¯ ¯ ¯ p 2 ¢ 1 p 1 ; (2.38) where p 1 = p p¡2 and C depends only on p andE(kuk ° ) (compare to (2.31)). Taking into account the definition (2.8) of the functionG and properties of the normk¢k ° , we get J 1 := t 2 Z 0 k b G(t 1 ;t 2 ;s;x;¢)k 2p 1 d® ds= t 2 Z 0 ³ X j¸1 ¯ ¯ ¯h b G(t 1 ;t 2 ;s;x;¢);h j (¢)i 0 ¯ ¯ ¯ 2 ¸ 2d® j ´ p 1 ds = t 2 Z 0 0 @ ¯ ¯ ¯ ¯ ¯ ¯ * X k¸1 ³ e l k (t 1 ¡s) I (s·t 1 ) ¡e l k (t 2 ¡s) ´ h k (x)h k (¢);h j (¢) + 0 ¯ ¯ ¯ ¯ ¯ ¯ 2 ¸ 2d® j 1 A p 1 ds and sinceh j forms a CONS inL 2 (M), we continue J 1 = t 2 Z 0 ¯ ¯ ¯ ¯ ¯ X k¸1 ³ e l j (t 1 ¡s) I (s·t 1 ) ¡e l j (t 2 ¡s) ´ 2 jh j (x)j 2 ¸ 2d® j ¯ ¯ ¯ ¯ ¯ p 1 ds: 54 By Assumption (A1) and H¨ older inequality withp 2 :=p 1 = p p¡2 andq 2 = p 2 we get J 1 ·c t 2 Z 0 X j¸1 jl j j 2½p 1 ¸ 2d®p 1 +" j ³ e l j (t 1 ¡s) I (s·t 1 ) ¡e l j (t 2 ¡s) ´ 2p 1 ³ X k¸1 ¸ ¡"q 2 k ´ 1 q 2 ds: For sufficiently large p and respectively large q 2 the series P k¸1 ¸ ¡"q 2 k converges, and hence J 1 ·c(p;") X j¸1 jl j j 2½p 1 ¸ 2d®p 1 +" j t 2 Z 0 ³ e l j (t 1 ¡s) I (s·t 1 ) ¡e l j (t 2 ¡s) ´ 2p 1 ds (2.39) Direct arithmetic evaluations yield to t 2 Z 0 ³ e l j (t 1 ¡s) I (s·t 1 ) ¡e l j (t 2 ¡s) ´ 2p 1 ds·c p jt 2 ¡t 1 j p 1 ¯ jl k j p 1 (¯¡1) (¯2(0;1)): From here and (2.39) we conclude J 1 ·c p jt 2 ¡t 1 j p 1 ¯ X k¸1 jl j j p 1 (2½+¯¡1) ¸ 2d®p 1 ¡" j : (2.40) Sincejl k j = ¸ 2m k and ¸ k » k 1 d , the last series converges if 2m(2½p 1 +¯¡1) d + 2d®p 1 ¡" d < ¡1. Recall that p 1 = p p¡2 , so lim p!1 p 1 = 1. Hence, the series converges if ¯ < 1¡2½¡ d® m ¡ d 2m , which holds true by our initial assumption. Finally, by (2.38) and (2.40) we conclude J ·c p ¯ ¯ ¯J 1 ¯ ¯ ¯ p 2p 1 ·c p jt 1 ¡t 2 j ¯p 2 : Lemma is proved. Now, we are ready to formulate the main result of this section. 55 2.3.8 Theorem. Assume that Assumption (A4) is satisfied and letu be the solution of the equation (2.19) (i) if Assumption (A2) is fulfilled and r is the biggest integer such that there exists ¯ 0 2(0;1] which satisfies the inequality2d®+2r+4m±(¯ 0 )¡2m<¡d, then ¤ r (u) is H¨ older continuous of order ´ :=sup n ¯j±(¯)<¡ d 4m ¡ d® 2m + 1 2 ¡ r 2m ; ¯2(0;1] o ¡"; for every">0, as long as´ >0; (ii) If Assumption (A1) is fulfilled and ® < m d ¡ 2m½ d ¡ 1 2 , then the solution u is H¨ older continuous int of order less than³ =minf 1 2 ; 1 2 ¡½¡ d(1+2®) 4m g. As long as the solutions of equations (2.2) and (2.19) exist, Lemma 2.3.6 and 2.3.7 coincide with Lemma 2.2.4 and 2.2.10 from additive noise. Under assumption that the solution of the equation 2.19 exists, which is guaranteed by Assumption (A4), the proof of Theorem 2.3.8 is the similar to the proof of Theorem 2.2.5 and 2.2.11. 2.3.9 Example. Let us consider the following SPDE du(t;x)=u xx (t;x)dt+ X k¸1 k ¡ 2 3 sin(kx)u(t;x)dW k (t); where x 2 [0;¼]; t 2 [0;1]; u(t;0) = u(t;¼) = 0 and W k are independent standard Brownian motions. We showed in Example 2.3.5 that for every¡ 1 2 < ° < 1 6 , there exists a unique strong solution u in the triple of Hilbert spaces (H °+1 ;H ° ;H °+1 ). Recall that in this case½=0;® =¡2=3; ±(¯)= ¯ 2 (See Example 2.1.2). By Theorem 2.3.8 we have that the solution u has one derivative in x and the first derivative is H¨ older continuous of order less than´ :=supf¯j±(¯)<¡ d 4m ¡ d® 2m + 1 2 ¡ r 2m ; ¯2 56 (0;1]g=supf¯ j ¯ 2 <¡ 1 4 + 1 3 + 1 2 ¡ 1 2 g= 1 6 . Respectivelyu is H¨ older continuous in time variablet of order less thanminf 1 2 ; 7 12 g= 1 2 . 2.3.10 Remark. In both cases, additive and multiplicative noise, the solutions, if exists, has the same order of regularity. However, the existence and uniqueness of solution, in terms of the scale of Sobolev spaces, is more restrictive in multiplicative case. 2.4 Covariance functional and relation with some known results As we already mentioned the novelty in our approach is the factor® k =k ® in the noise term. In this section we will show how our results are related to some known results, and also we will find the analogous of parameter® in the known literature in terms of the asymptotic of kernel of covariance functional. In recent paper Mytnik, Perkins, Sturm [60] investigated the equation of the form du(t;x)= 1 2 ¢u(t;x)dt+ p u(t;x)dW(t;x); (2.41) where ¢ is Laplacian, x 2R d ; t 2 [0;T] for some finite T . W is space-time white noise onR + £R d , namelyW is defined on a filtrated probability space(;F;fF t g;P) and is a Gaussian martingale measure on R + £R d in sense of Walsh [79] or that introduced in Section 1.3. In [60] study the existence of mild solution of equation (2.41) and it regularity properties. The special feature of this equation is the nonlinear factor p u. The solutionu is the density for one-dimensional super-Brownian motion [63]. Of course this equation is different from those considered in our investigations. 57 Equation (2.41) is nonlinear on the whole space, in our notations M = R d . Our equations are linear SPDE’s, but M is a bounded domain, and we allow operators of arbitrary order (not necessarily Laplacian). Here we will discuss the similarities in terms of noiseW(t;x). Since we are working on different domains, some evaluations will be pure heuristic. Denote byC 1 c (R + £R d ) the space of compactly supported, infinitely differentiable functions onR + £R d , and putW t (') = t R 0 R R d '(s;x)W(dxds). If W(') = W 1 ('), W can be characterize by its covariance functional J ª (';Ã):=E h W(')W(Ã) i = 1 Z 0 Z R d Z R d '(s;x)ª(x;y)Ã(s;y)dxdyds; (2.42) for';Ã2 C 1 c (R + £R d ). The functionª :R 2d !R is called the correlation kernel of W . A general classes of martingale measures, spatially homogeneous noises, can be describe by (2.42) withª(x;y) = e ª(x¡y). We note that the standard space-time white noise P k¸1 h k (x)W k (t) will correspond to the case e ª = ± 0 , where ± is the Dirac function. Suppose that the correlation is bounded by a Riesz kernel jª(x;y)j·c £ jx¡yj ¡# +1 ¤ (x;y2R d ) (2.43) where#>0. In [60] is proved that for every » 2 (0;1¡ # 2 ) the solution of equation (2.41) is uniformly H¨ older continuous on compacts in[0;1)£R d , with H¨ older coefficients » 2 in time and» in space. Now, let us find the correlation functional and kernel that corresponds to the equa- tions in our research. For simplicity of writing we will consider the one-dimensional 58 stochastic heat equationdu(t;x) = u xx (t;x)dt+ P k¸1 k ® u(t;x)sin(kx)dW k (t), where x2 [0;¼]; t2 [0;1]. In this caseW t (') = P k¸1 t R 0 ¼ R 0 '(s;x)h k (x)k ® dxdW k (ts). Con- sequently, J ª (';Ã)=E h X k¸1 1 Z 0 ¼ Z 0 k ® '(s;x)h k (x)dxdW k (s) ¢ X n¸1 1 Z 0 ¼ Z 0 n ® Ã(s;y)h n (y)dydW n (s) i : Taking into account thatW k are independent, we continue J ª 0 (';Ã)=E h X k¸1 k 2® 1 Z 0 h'(s;¢);h k i 0 dW k (s) 1 Z 0 hÃ(s;¢);h k i 0 dW k (s) i Suppose for simplicity that the functions'; à do not depend on time variablet. Then J ª 0 (';Ã)=c X k k 2® h';h k i 0 hÃ;h k i 0 =ch';Ãi ® ; where c is a constant depending on the support of functions ' and Ã. Note that h';Ãi ® =c P k¸1 k 2® ¼ R 0 ¼ R 0 h k (x)h k (y)'(x)Ã(y)dxdy. Hence, ª 0 (x;y)= X k¸1 k 2® h k (x)h k (y)= X k¸1 k 2® sin(kx)sin(ky): If ® < ¡ 1 2 , then the last series converges uniformly, and the covariance kernel is bounded. Hence, in this case there is no connection with the estimates (2.43). Suppose that¡ 1 2 < ® < 0. By Abel’s criterion about convergence of series, one can show that P k¸1 k 2® sin(kx)sin(ky) converges if x 6= y and diverges if x = y. So, the kernel ª seems to have a similar asymptotic behavior to that from (2.43). To find # we note 59 that J ª 0 (';Ã) = h¤ 2® ';Ãi 0 , where ¤ = p ¡¢ (see Chapter 1 for more details). The counter part for the whole line will be the operator ¤ = (I¡¢) 1 2 which can be represented as an integral operator with kernel the Bessel potential. Namely, h¤ ¡2® ';Ãi 0 = Z R Z R µZ 1 0 t ¡®¡ 3 2 e ¡ jx¡yj 2 4t ¡t dt ¶ '(x)Ã(y)dydx: Hence, the correlation kernel e ª 0 (x) = R 1 0 t ¡®¡ 3 2 e ¡ jxj 2 4t ¡t dt. Note that Fourier trans- formF[ e ª 0 ] = (1+jxj 2 ) ® , andF[jxj ¡# ] =jxj ¡1+# (see for instance [77], Chapter 5). From here we conclude that¡2® =1¡#. Hence we can say that the parameter# from (2.43) corresponds to1¡2® where® is parameter involved in our research. Note that ®2(¡ 1 2 ;0) is equivalent to#2(0;1). Finally we will show that regularity properties coincide. By Theorem 2.3.8 the solution of corresponding equation is H¨ older continu- ous in space variablex with the order of continuity less than» = 1 2 ¡® = 1¡ # 2 , and in time variable with order of continuity less than´ = 1 4 ¡ ® 2 = 1 2 ¡ # 4 , which is exactly the statement of Theorem 1.8 from [60]. 2.5 Examples 2.5.1 Example. Stochastic heat equation on[0;¼] with zero boundary conditions. First we consider the classical heat equationdu =u xx dt+f(u)dW(t;x), as a bench- mark for testing our results. We recall that this problem was investigated by Walsh [79]. In this case, the solution exists, is unique and H¨ older continuous in x of order 1 2 ¡", and int of order 1 4 ¡" for every">0 (see Section 2.1 for more details). Let A be the one dimensional Laplace operator on C 1 (0;¼). In our notations d = 1; Au = u xx ; m = 1; M = [0;¼]. It is well-known that the functions 60 h k (x) = 2 ¼ sin(kx); k 2 N, are the eigenfunctions of the operator A, with corre- sponding eigenvaluesl k =¡k 2 . Moreover, Assumptions (A1) and (A2) are satisfied, and½=0; ±(¯)= ¯ 2 (see Section 2.1, Example 2.7, page 27 for the proof). 2.5.1.a. Additive noise Let us consider the heat equation with additive noise term 8 > > > > > > < > > > > > > : du(t;x)=u xx (t;x)dt+ 1 P k=1 k ® h k (x)dW k (t) t2(0;T); 0<x<¼; u(t;0)=u(t;¼)=0 t2(0;T); u(x;0)=0; x2[0;¼]; (2.44) where® is a real parameter. Existence and Uniqueness. By Theorem 2.2.1, if ° < ¡® ¡ 1 2 then the equation (2.44) has a unique solution u 2 L 2 ((0;T) £ ;H °+1 (0;¼)) \ L 2 (;C((0;T);H ° (0;¼))). Note that regardless of® the solution exists in someH ° . Regularity in x. Since ±(¯) = ¯ 2m = ¯ 2 , we will apply Theorem 2.2.7. Suppose that ° < ¡®¡ 1 2 and denote ® < 1 2 . By Remark 2.2.6 we can take the operator ¤ from Theorem 2.2.7 being the partial derivative operator @ @x (see also Example 2.2.8). Let ´ = 1 2 ¡® and denote bybac the integer part of real number a (largest integer smaller than a). If ´ = 2 N, then the solution u has r := b´c derivatives in x and the r-th derivative is H¨ older continuous of order³ := ´¡r¡", for every" > 0 as long as³ > 0. If´2N, thenu hasr :=b´c¡1 derivatives inx and ther-th derivative is H¨ older continuous of order³ for every³ 2(0;1). Regularity in t. Suppose that ® < 1 2 . Then, by Theorem 2.2.11, the solution u of equation (2.44) is H¨ older continuous in time variable t with order of continuity ´ =minf 1 2 ; 1 4 ¡ ® 2 g¡", for every">0 as long as´ >0. 61 If® = 0, we resume the results by Walsh [79]: the solution exists if° <¡ 1 2 , and u(t;x)2C 1 4 ¡"; 1 2 ¡" t;x . If ® decreases, the solution becomes smoother, both in time and space variable. However, the maximum smoothness in time is achieved for ® = ¡ 1 2 . In this case, the solution u has an order of continuity less than 1 2 and remains the same for every ®·¡ 1 2 . For® =¡ 1 2 , the solution is Lipschitz continuous inx. Generally speaking, there arer =b 1 2 ¡®c derivatives inx, andu (r) x 2C 1 2 ¡®¡r . 2.5.1.b. Multiplicative noise. Now, let us consider the equation du(t;x)=u xx (t;x)dt+ 1 X k=1 k ® u(t;x)h k (x)dW k (t) t2(0;T); 0<x<¼ ; with same initial values and boundary conditions as in (2.44). By Theorem 2.3.2 this equation has a solutionu2H °+1 if¡ 1 2 <° <¡ 1 2 ¡®. If the solution exists, then the regularity in time and space are the same as for additive noise, discussed above. 2.5.2 Example. Stochastic heat equation on torusT 2 ¼ . We consider the operatorA being the Laplace operator¢ on two dimensional torusT 2 ¼ . Following our previous notationsd=2; m=1; M =T 2 ¼ . The eigenfunctions of this operator areh j;n (x 1 ;x 2 ) = 1 ¼ e ijx 1 e inx 2 wherex = (x 1 ;x 2 )2 M; j;n2Z; i = p ¡1, with corresponding eigenvaluesl =j 2 +m 2 . Obviously Assumption (A1) is satisfied, jjh k jj 1 · 1 ¼ , so½=0. Let us check Assumption (A2) and find the function±. Denote x=(x 1 ;x 2 ); y =(y 1 ;y 2 ). ThenR :=jh j;k (x)¡h j;k (y)j=je i(jx 1 +nx 2 ) ¡e i(jy 1 +ny 2 ) j= ¯ ¯ ¯2¡2(cos(z 1 )cos(z 2 )+sin(z 1 )sin(z 2 )) ¯ ¯ ¯ 1 2 , wherez 1 = jx 1 +nx 2 ; z 2 = jy 1 +ny 2 . Consequently, R = p 2 ¯ ¯ ¯1¡ cos( z 1 ¡z 2 2 ) ¯ ¯ ¯ 1 2 = 2 ¯ ¯ ¯sin( z 1 ¡z 2 4 ) ¯ ¯ ¯ · cjz 1 ¡z 2 j ¯ , for every 62 ¯ 2 (0;1]. Hence, R · cjj(x 1 ¡y 1 )+n(x 2 ¡y 2 )j ¯ · c(j 2 +n 2 ) ¯ 2 ¡ (x 1 ¡y 1 ) 2 + (x 2 ¡y 2 ) 2 ¢ ¯ 2 =cl ¯ 2 jx¡yj ¯ . Thus,±(¯)= ¯ 2 ; ¯2(0;1]. Let us consider the corresponding equation with additive noise, i.e. du = ¢udt+ P k¸1 h k dW k (t). Note that ±(¯) = 1 2m = ¯ 2 , and hence Remark 2.2.6 holds true. By Theorem 2.2.1 there exists a unique solution u 2 H °+1 if ° < ¡2®¡1. Theorem 2.2.5 implies: if¡2® = 2N, then the solution u has r := b¡2®c derivatives in x and ther-th derivativeD r (u) is H¨ older continuous of order´ =¡2®¡b2®c; if¡2®2N, then there existsD r¡1 (u) with order of continuity¿, for every´2(0;1). Regularity in time is given by Theorem 2.2.11. The solution u(¢;x) is H¨ older continuous of order´ =minf 1 2 ;¡®g, for every®·0. 2.5.3 Example. SPDE of order2m. Let consider the equation of the form du(t;x)= @ 2m u(t;x) @x 2m dt+ X k¸1 k ® h k (x)dW k (t) x2[0;¼]; t2[0;T]; where h k (x) = 2 ¼ sin(kx) and m 2 N. We assume zero boundary conditions, and zero (or any smooth) initial values. In fact this equation is similar to the equation (2.44) from Example 1, butA = ¢ m . Since the Laplace operator ¢ is a selfadjoint operator, we conclude that the functions h k are the eigenfunctions of the operatorA, with corresponding eigenvalues l k = ¡k 2m . Thus, Assumptions (A1) and (A2) are satisfied, and½=0; ±(¯)= ¯ 2m . Existence. Applying Theorem 2.2.1, we have: if° <¡®¡ 1 2 then there exists a unique solutionu2L 2 ((0;T)£;H °+1 (0;¼))\L 2 (;C((0;T);H ° (0;¼))) Regularity in x. By Theorem 2.2.5, if ® < ¡ 1 2 + m 2 and m¡®¡ 1 2 = 2 N, then the solutionu hasr =bm¡®¡ 1 2 c derivatives, and the last derivative is H¨ older continuous 63 of every order less thanm¡®¡ 1 2 ¡r. Similar form¡®¡ 1 2 2N. Regularity int. If®<m¡1, then the solutionu is H¨ older continuous in time variable, with order of continuity less thanminf 1 2 ; 1 2 ¡ 1¡2® 4m g. To show the impact of the parameter®, we summarize these results from the exam- ples considered in this Chapter in the Table 2.1: Parameters Existence Reg. inx Reg. int Walsh [79] d=1;m=1;®=0 - 1 2 1 4 multiplicative Ex. 2.1.2, 2.2.3 d=1;m=1 ° < 1 6 7 6 1 2 2.2.8, additive ®=¡ 2 3 Ex. 2.3.5, 2.3.9 d=1;m=1 ¡ 1 2 <° < 1 6 7 6 1 2 multiplicative ®=¡ 2 3 Ex. 2.5.1.a d=1; m=1 ° <¡®¡ 1 2 1 2 ¡® minf 1 2 ; 1 4 ¡ ® 2 g additive Ex. 2.5.1.b d=1; m=1 ¡ 1 2 <° <¡®¡ 1 2 1 2 ¡® minf 1 2 ; 1 4 ¡ ® 2 g multiplicative Ex. 2.5.2 d=2; m=1 ° <¡2®¡1 ¡2® minf 1 2 ;¡ ® 2 g additive Ex. 2.5.3 d=1; m>1 ° <¡2®¡1 m¡®¡ 1 2 minf 1 2 ; 1 2 ¡ 1¡2® 4m g additive Table 2.1: Existence and Regularity of Solution. Summary 64 Chapter 3 Parameter estimation problems for some classes of SPDE’s 3.1 Introduction Stochastic Partial Differential Equations (SPDE’s) are of big interest in last decades, primarily as a modelling of various phenomena from fluid mechanics [74], oceanogra- phy [64], temperature anomalies [11], [65], finance [7], [12], [14], and other domains. Every model is describe by a class of SPDE’s, with some specific properties, which are related to the corresponding phenomena, and usually this is a broad class of equa- tions with some unspecified/unknown coefficients also called parameters. In practice we observe the process, which we believe is described by some equation, and we want to find all unknown parameters of this model by using the past data such that the equa- tion fits and predicts, in some sense, as much as possible the real data. In mathematical terms, we want to solve a parameter estimation problem. The parameter estimation problem, generally speaking, is a particular case of inverse problem when the solution is known (observed) and an inference about the coefficients of the equation is made. Because of general setup of this problem, there are numerous methods in modern literature dedicated to this topic, covering different classes of equations. We want to emphasize that the solution of a Stochastic PDE is a random variable, and the parameter estimation problem needs to be solved by 65 some statistical methods. In contrast, the similar problem for the models described by a deterministic PDE is usually reduced to some inverse spectral problems (see for instance [68] and the references therein). In this chapter we will investigate a parameter estimation problem for evolution equations of the following form 8 > > < > > : du(t)=(A 0 +µA 1 )u(t)dt+(Mu(t)+g(t))dW(t); u(0)=u 0 ; (3.1) where t 2 [0;T]; A 0 ;A 1 ; M are linear operators, g a real-valued function, W is a cylindrical Brownian motion, and µ is the unknown parameter belonging to an open subset of the real line. Parameter estimation for the ordinary stochastic differential equations (SDE’s), finite dimensional counterpart of equation (3.1), has been studied by many authors under general assumptions on coefficients, with very subtle results (see for instance [25], [36], [46], [47] and references therein). Some infinite dimensional models with long time and small noise asymptotics, using system theory, have been studied by Aihara [2], Bagchi and Borkar [3], Ibragimov and Khasminskii [26]-[28] etc. Using projection-based methods, the parameter problem for SPDE (3.1) driven by additive noise was studied for the first time by Khasminskii, Rozovskii and Huebner [22] [23]. The key point is that in reality, one may not assume to observe the whole fieldu(t;x), so a finite dimensional approximation of the solution is used to obtain consistent esti- mates of unknown parameter µ. The idea is to choose a basis in the suitable Hilbert space, such that the finite dimensional projection of the solutionu, i.e. Fourier coeffi- cients w.r.t. this basis, will lead to a system of independent processes. Then, using first 66 N Fourier coefficients, an explicit and simple form of the Maximum Likelihood Esti- mator (MLE) ^ µ N forµ is obtained. As one may expect, ^ µ N converges (in some sense) to the true parameter asN increases. The equation (3.1) driven by additive noise, and withA 0 ;A 1 being elliptic differential selfadjoint operators on a bounded domain in R d , with a common system of eigenfunctions, was studied in [21], [22], [23], [66]. It was shown that if ord(A 1 ) ¸ (ord(A 0 )¡d)=2, then MLE ^ µ N is strongly consistent asN !1. We note here, that operatorsA 0 ;A 1 commute, since they have the same system of eigenfunctions, and hence the random field is diagonalizable. Consequently, direct application of Galerkin type of approximations will lead to a system of ODE’s mentioned above. The equation with none-commuting operators was considered by Lototsky and Ro- zovskii [52],[53]. Even though, applying the same technics, formally computed esti- mator is not an MLE, it was shown that under some technical assumptions the estimator is consistent and asymptotically normal. We would like to mention that for ordinary SDE consistent estimates are obtained if either observation timeT goes to infinity or the noise amplitude tends to zero (for some examples see [36]). Although, for SPDE’s within the above framework consistency is archived on any time interval[0;T] as the dimension of the projection increases. This is due to the fact that generally speaking the probability measures generated by the solution u µ of an ordinary SDE are absolute continues for different µ, while for an SPDE with additive space-time white noise, under some conditions on the order of differential operators, the measures are singular (see for instance [56], [39]). However, the measures generated by the projection are absolute continuous for different values of the parameter, which implies the existence of MLE computed from likelihood ratio (Radon-Nikodym derivative). 67 While the SPDE’s driven by additive noise have been studied by many authors, the corresponding equations with multiplicative noise have not been covered at all (as far as we know), in particular the equation with multiplicative space-time white noise. In this work, we will address this question for several classes of equations, including space-time white noise. Despite similarity of these equations, it turns out that the mea- sures generated by the projection of the solution with multiplicative noise, generally speaking, are not anymore absolute continuous and the likelihood ratio does not exist. For example, this is the case when W is space-time white noise, and we will discuss this in Section 3.2. Regarding this, we consider equations with some noises similar to cylindrical Brownian motion, but for which we are able to find consistent estimates of the unknown parameter, based on finite projection of the solution on some basis. For simplicity, we suppose that operatorsA 0 ;A 1 ;M commute, however the results can be extended to non-commutative ones. In Section 3.3 we set up the problem, and study existence, uniqueness and finite dimensional approximation of the solution. The esti- mate and its properties are discussed in Section 3.4. We find sufficient conditions on operatorsA 0 ;A 1 ;M that implies consistency, asymptotic efficiency, and normality of estimates. We conclude this chapter with some application of abstract results to gen- eral stochastic parabolic equations, presented in Section 3.5. Some numerical results are presented at the end of Section 3.5, where we show some practical applications of theoretical results, and provide some counter-examples related to sufficiency of the conditions imposed in on the original equation. 68 3.2 Special case: space-time white noise In this section we will analyze the simplest SPDE driven by space-time white noise, and give some insides about the problems arising in parameter estimation for equations with multiplicative noise. We will follow the notations and definitions from Chapter 1. Let us consider the stochastic one dimensional heat eqation du(t;x)=µu xx (t;x)dt+u(t;x)dW(t;x); (3.2) wheret2[0;T]; x2[0;1]; u(0)=u 0 2L 2 (0;1) and periodic boundary conditions. The elliptic operatorAu = ¡u xx , with eigenfunctions h k (x) = e ik¼x (k 2Z), generates the scale of Sobolev spacesH ° =H ° (0;1)(°2R). The systemh k ; k2Z; forms a CONS in L 2 (0;1) andAh k = ¡(¸ k ) 2 h k , with ¸ k = k¼. By Theorem 1.5.1, if ° < ¡1=2 then equation (3.2) has a unique solution u 2 L 2 ((0;T)£;H °+1 )\ L 2 (;C((0;T);H ° )) (see also [79]). It is assumed that the observed field u satisfies (3.2) for some unknown but fixed value µ 0 of parameter µ. We suppose that the first N Fourier coefficients of u, rel- ative to fh k g k2Z , are known, and we would like to find a consistent estimate of µ 0 as N ! 1. Denote by fu k g k2Z the Fourier coefficients of the solution u w.r.t. h k ;k 2 Z, and by ¦ N the projection of H s ; s 2 R; on Spanfh k g N k=¡N . Hence, u N (t;x) := ¦ N u = N P k=¡N h k (x)u k (t). We consider the following finite-dimensional approximations of (3.2) dv N =µ(v xx ) N dt+¦ N (v N dW N ); (3.3) which to some extends corresponds to Galerkin-type approximation. 69 It should be mentioned that since operator ¦ N and multiplication operator v N ! v N dW N do not commute,v N from (3.3) does not coincide with projectionu N of orig- inal solutionu. In other words, generally speaking¦ N v N dW N 6=u N dW N . However, we will show that approximation sequencev N = N P k=¡N v N k h k converges to the solution u, asN !1, which means that from practical point of view, we can substitute in our estimates the values ofv n by Fourier coefficientsu n ; n=¡N;:::;N. Taking into account that operators ¦ N and ¢v = v xx have a common system of eigenvalues, hence commute, we rewrite (3.3) as follows N X k=¡N h k (x)dv k (t)=¡µ N X k=¡N ¸ 2 k h k (x)v k (t)dt +¦ N ³ N X k=¡N h k (x)v k N X j=¡N h j (x)dW j (t) ´ : (3.4) Observe that h k h m = h k+m ; k;m 2 Z, which implies that (3.4) is equivalent to a system of2N +1 stochastic ODEs, dv ¡N =¡µ¸ 2 ¡N v ¡N dt+ ³ v ¡N dW 0 +v ¡N+1 dW ¡1 +¢¢¢+v 0 dW ¡N ´ dv ¡N+1 =¡µ¸ 2 ¡N+1 v ¡N+1 dt+ ³ v ¡N dW 1 +v ¡N+1 dW 0 +:::v 1 dW ¡N ´ :::::: dv 0 =¡µ¸ 2 0 v 0 dt+ ³ v ¡N dW N +v ¡N+1 dW N¡1 +¢¢¢+v N¡1 dW ¡N ´ :::::: dv N =¡µ¸ 2 N v N dt+ ³ v 0 dW N +v 1 dW N¡1 +¢¢¢+v N dW 0 ´ (3.5) or, in matrix form dv µ;N =µA(v µ;N )dt+b(v µ;N )dW N t ; (3.6) 70 whereA(v N )=¡[¸ 2 ¡N v ¡N ;:::;¸ 2 N v N ] 0 ; and b(v N )= 0 B B B B B B B B B B B B B B @ v 0 v ¡1 ::: v ¡N+1 v ¡N 0 ::: 0 v 1 v 0 ::: ::: v ¡N+1 v ¡N ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: v N v N¡1 ::: v 1 v 0 v ¡1 ::: v ¡N ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: 0 v N v N¡1 ::: v 0 1 C C C C C C C C C C C C C C A ; withdim(b)=(2N+1)£(2N+1) and the elementb 00 is boldfaced. For conveniens of writing, we index the coordinates inR 2N+1 from¡N toN. As we mentioned above, it is natural to have some convergence of the approxima- tion v N to the real solution u as dimension of approximation increases. To show this we prove an auxiliary result. 3.2.1 Lemma. IfM N =(m kj ) N k;j=¡N is a(2N +1)£(2N +1) real matrix such that m jj = ¡jjj 2 ; j = ¡N;:::;N; m kj = 1 ifjk¡jj · N, and ¹ N is the maximum eigenvalues of the matrixM N , then there exist a constantC such that¹ N < C for all N 2N. Proof. It suffices to show thathM N x;xi· C for some real constantC, and for every x2R 2N+1 withkxk=1, whereh¢;¢i andk¢k are usual (Euclidian) inner product and norm inR 2N+1 . Let us denote® n =m nn ;n=0;1;:::;N andf(x)=hM N x;xi. We rewritef(x) as follows: 71 f(x)=® N x 2 ¡N +2x ¡N (x ¡N+1 +¢¢¢+x 0 ) +® N¡1 x 2 ¡N+1 +2x ¡N+1 (x ¡N+2 +¢¢¢+x 0 +x 1 ) :::::::::::: +® 1 x 2 ¡1 +2x ¡1 (x 0 +¢¢¢+x N¡1 ) +® 0 x 2 0 +2x 0 (x 1 +¢¢¢+x N ) +® 1 x 2 1 +2x 1 (x 2 +¢¢¢+x N ) :::::::::::: +® N¡1 x 2 N¡1 +2x N¡1 x N +® N x 2 N : (3.7) So, we want to maximizef under constrainkxk = 1. Using the method of Lagrange multipliers we find thatx should be a non-trivial solution of the system of linear equa- tionsL ¸ x=0, where L ¸ = 0 B B B B B B B B B B B B B B @ ® N +¸ 1 ::: 1 0 0 ::: 0 0 ® N¡1 +¸ ::: 1 1 0 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: ® 0 +¸ 1 1 ::: 1 ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: 0 0 0 ::: ® N +¸ 1 C C C C C C C C C C C C C C A ; and ¸ is the multiplier parameter. Obviously, L ¸ x = 0 has a non-zero solution, iff ® j +¸=0 for somej. Suppose® N +¸=0, since otherwise, we get the same system, but of smaller dimension. Let a = maxfjx ¡N j;jx N jg. Note that sincekxk = 1 we 72 have a· 1. By direct arithmetic evaluations, one can deduce thatjx §j j· a n+j ; j = ¡N +1;:::;N +1. Hence, by (3.7), we getf(x)·C. Now we can prove the following result 3.2.2 Theorem. For every ° < ¡1=2, the sequence fv N g n¸1 converges weakly in H °+1 to the weak solution of the equation (3.9). Proof. Similar to deterministic case, we will use weak sequential compactness of Hilbert spaces (see for instance [9]). The main part of the proof is to establish that the sequencekv µ;N k 2 L 2 (£[0;T];H ° ) is uniformly bounded. Since° <¡ 1 2 , it suffices to show thatE T R 0 kv µ;N k 2 0 dt is uniformly bounded in N. By Itˆ o’s formula forf(x) = N P k=¡N x 2 k we deduce dY =AYdt; whereY = h E(v 2 ¡N );:::;E(v 2 N ) i 0 and A= 0 B B B B B B B B B B @ 1¡2µ¸ 2 ¡N ::: 1 ::: 0 ::: ::: ::: ::: ::: 1 ::: 1¡2µ¸ 2 0 ::: 1 ::: ::: ::: ::: ::: 0 ::: 1 ::: 1¡2µ¸ 2 N 1 C C C C C C C C C C A : By Lemma 3.2.1 the maximal eigenvalue of the matrixA is uniformly (inN) bounded from above. Consequently, by Theorem 4.2 from [16] the normkv µ;N k 0 is also uni- formly bounded inN. Hence, there exists a subsequence ofv µ;N that converges weakly inL 2 (£[0;T];H °+1 ). Let us denote this limit by~ v. By (3.3), for every'2H °+1 we have(v µ;N ;') = (u N 0 ;')+µ t R 0 [v µ;N xx ;']+ t R 0 (¦ N (v µ;N dW N (s);')). TakingN !1 73 we deduce (~ v;') = (u 0 ;')+µ t R 0 [~ v xx ;']+ t R 0 (~ vdW(s);')). Since the last equality holds for every'2H °+1 , we get ~ v =u. Theorem is proved. Let us denote byP µ;N the measure inX = C([0;T];¦ N (H 0 )), generated by the solution v µ;N (s); 0 · s · T of equation (3.6), i.e. P µ;N (a) = P(v µ;N 2 a) for all a2B T , whereB T is the Borel¾-algebra onX . By Theorem 1.2.2 (see also Theorem 7.6.4 in [47]) the measuresP µ;N andP µ 0 ;N are mutually absolutely continuous if the algebraic system of equations b(x)y =(µ¡µ 0 )A(x) (3.8) has a bounded solution (in y 2 R 2N+1 ) for each t 2 [0;T];x 2 R 2N+1 . It turns out that this algebraic equation with apparently nice structure does not have an obvious solution. Direct evaluations show that forN = 1;2;3, the system (3.8) does not have solution for some values of x, which means that we can not apply the general results about absolute continuity of measures. On the other hand, we can not conclude that these measures are singular for different parametersµ. Some examples and generaliza- tion of Theorem 7.6.4 from [47] are discussed in [36]. In particular, it is shown that the corresponding measures can be singular as well as absolute continuous, if the system (3.8) does not have a finite solution. Unfortunately, these results can not be applied to our system of SDE’s, nor the proofs can be adapted (by best of our knowledge). We want to mention, that simple structure of matrixb is due to the unique property of eigenfunctions,h k h m =h k+m . In general,b will be a full matrix, without any specific properties, which makes impossible to find a solution of equation (3.8). We estimated naively the MLE ^ µ N ofµ, by maximizing the likelihood ratio dP µ;N dP µ 0 ;N (v N;µ 0 ), and used it for some numerical simulations. The results suggest us that 74 the measures look like to be singular, which consequently implies that some different approach needs to be considered for the equations with multiplicative space-time white noise. 3.3 General case. Approximation of the solution In this section we will set up the parameter estimation problem, discuss the existence and uniqueness of the solution of the corresponding equations, and consider the finite dimensional approximation of the solution. Let H be a separable Hilbert space with the inner product (¢;¢) 0 , and the corre- sponding normk¢k 0 . Suppose¤ is a linear operator onH such thatk¤uk 0 ¸ ckuk 0 for every u from the domain of ¤. Then the operators ¤ ° ; ° 2R; are well defined, and generate the spacesH ° as follows: the domain of¤ ° coincides withH ° for every ° ¸ 0; H ° is the completion of H with respect to the norm k¢k ° := k¤ ° ¢k 0 , for all ° < 0 (see Chapter 1 here, or [41]). The set fH ° g °2R is called a Hilbert scale. The spaces(H °+m ;H ° ;H °m ) form a normal triple with canonical bilinear form hu 1 ;u 2 i = (¤ °¡m u 1 ;¤ °+m u 2 ) 0 , whereu 1 2H °¡m ;u 2 2H °+m ;¤ ° (H r ) =H r¡° for every°;r2R. Let (;F;fF t g;P) be a stochastic basis. In the triple (H °+m ;H ° ;H °¡m ), for some°2R; m2N, we consider the following SPDE 8 > > < > > : du(t)= ¡ (A 0 +µA 1 )u(t)+f(t) ¢ dt+ n P j=1 (M j u(t))dW j (t) u(0)=u 0 ; (3.9) 75 whereA 0 ;A 1 ;M s are linear operators, f is a given vector-valued function, W j are independent standard Brownian motions,n2N orn=1,t2[0;T], andµ is a scalar parameter subject to estimation belonging to an open set£½R. The first question we will address is the existence and uniqueness of the solution of equation (3.9). In connection with this, throughout in what follows, we will assume that: (H1) there exists m > 0 such that the linear operatorA µ := A 0 +µA 1 , is bounded fromH °+m toH °¡m for every°2R; (H2) M j (j =1;:::;n) are linear bounded operators acting fromH °+m intoH ° ; (H3) u 0 2H ° ; (H4) f 2L 2 (£(0;T);H °¡m ); (H5) The operators¤;A 1 ;A 0 ;M j have the same system of eigenfunctionsfh k g 1 k=1 , the systemfh k g 1 k=1 is a complete system inH, andh k 2\ °¸0 H ° . Condition (H5) is equivalent to say that operators A 1 ;A 0 ;M commute, which consequently implies that eigenfunctions and eigenvalues of these operators does not depend on the parameter µ. We want to note that this is the case in many applica- tions. Usually the operator¡A µ is a differential, selfadjoint, positive defined operator of order 2m defined on the scale of Sobolev spacesfH ° (M)g °2R , where M is a d- dimensional compact orientableC 1 manifold. In this case ¤ = (¡A µ ) 1 2m and with M j = ¤ ° j the assumption (H5) holds true. We will discuss this example in details later on (see Section 3.5). Most of the results remain true in more general situations, in particular when oper- atorsA 1 ;A 0 ;M do not commute, and will be published elsewhere. 76 Denote by l 1 k ;l 0 k ;¸ k and ¹ j k the eigenvalues of the operators A 1 ;A 0 ;¤ and M j respectively, which correspond to the eigenfunctionh k for everyk2N. Assume that: (H6) l r k <0; ¹ j k 6=0 forr =0;1; j =1;:::;n; k2N, and there exist">0; M 2R such that¸ ¡2m k n P s=1 (¹ s k ) 2 ·"¡2¸ ¡2m k (µl 1 k +l 0 k )·M for everyµ2£; k2N. Supposel r k , are enumerated in order of magnitude l r 1 ¸l r 2 ¢¢¢¸l r k ¸::: ; wherer =0;1. 3.3.1 Theorem. Under Assumptions (H1)-(H6), there exists a unique weak solutionu of equation (3.9), the solution belongs toL 2 (;C((0;T);H ° ))\L 2 (£[0;T];H °+m ) and E µ sup 0<t<T ku(t)k 2 ° ¶ +E 0 @ T Z 0 ku(t)k 2 °+m dt 1 A ·CE 0 @ ku 0 k 2 ° + T Z 0 kf(t)k 2 °+m dt 1 A ; (3.10) for some constantC. Proof. This theorem follows from Theorem 1.5.1, a general result from the theory of SPDEs (see also [70], Theorem 3.1.4). Conditions (i)-(iii) and (v) of Theorem 1.5.1 come naturally from the structure of equation (3.9) and follow immediately from Assumptions (H1)-(H4). 77 Condition (iv) represents strong parabolicity or coercivity assumption. It suffices to check this condition for the vectors h k . Put l k (µ) = l 0 k + µl 1 k , then using above assumptions we have 2[A µ h k ;h k ]+ n X j=1 kM j h k k 2 ° =2l k (µ)[h k ;h k ]+ n X j=1 (¹ j k ) 2 kh k k 2 ° ·2l k (µ)kh k k 2 ° + n X j=1 (¹ j k ) 2 kh k k 2 ° = ³ 2l k (µ)+ n X j=1 (¹ j k ) 2 ´ ¸ ¡2m k kh k k 2 °+m : (3.11) Denote by± =¡max k f¸ ¡2m k (2l k (µ)+ n P j=1 (¹ j k ) 2 )g. By Assumption (H6) we have that ± > 0, thus the condition (v) is verified, and the Theorem is proved. To make an inference about parameter µ we will project the solution u on some finite-dimensional subspace ofH °+m , and essentially reduce the SPDE to a system of SODE’s. As we mentioned, a natural approach is to project on the subspace generated by the eigenfunctionsh k , and use the Fourier coefficients of the solutionu with respect tofh k g 1 k=1 . Suppose that the observed fieldu satisfies (3.9) for some unknown but fixed value µ 0 of the parameter µ. Denote byfu k g k2N the Fourier coefficients of the solution u w.r.t. fh k g k2N , and by ¦ N the projection of H ° ;° 2 R, on Spanfh k g N k=1 . Hence, u N (t;x):=¦ N u= N P k=1 u k (t)h k (x). We consider the following finite-dimensional approximation of (3.9) du N (t;x)=¦ N ((A 0 +µA 1 )u(t;x)+f(t))dt + n X j=1 ¦ N (M j u(t;x))dW j (t): (3.12) 78 3.3.2 Remark. By Theorem 3.3.1,u2L 2 (£[0;T];H °+m ), which implies that lim N!1 E T Z 0 jju N ¡ujj 2 °+m dt=0: Denote by f k the Fourier coefficients of the function f. From (3.12) and by assumption (H6), we deduce that Fourier coefficientu k satisfies the following equation du k (t)= ³ (l 0 k +µl 1 k )u k (t)+f k (t) ´ dt+ n X j=1 ¹ j k u k (t)dW j (t); (3.13) withu k (0)=(u 0 ;h k ) ° , andk2N. 3.4 The estimate and its properties In this section we will find a Maximum Likelihood Estimate (MLE) of the parameter µ by using the approximation (3.12) of the solution to the original equation. Let us fix a numberµ 0 2£, that represents the true value of the parameterµ subject to estimation. In what follows we will suppose that the firstk Fourier coefficients of the fieldu are observed, and we will estimateµ 0 from them. For simplicity of writing we will drop the indexµ 0 where there is no room for ambiguity. Namely, we will writeu k instead ofu µ 0 ;k . LetB T be the Borel¾-algebra on the spaceC([0;T];H °+m ). Denote byP µ k the measure onB T generated by the solutionu µ;k of equation (3.13). Note that the equation (1.5) from Theorem 1.2.2 in this case becomes n P j=1 ¹ j k y = (µ¡ µ 0 )l 1 k , 79 which has a finite solution for every k 2N, and hence the measuresP µ k andP µ 0 k are mutually absolute continuous, with Radon-Nikodym derivative dP µ k dP µ 0 k (u k )=exp n T Z 0 l 1 k (µ¡µ 0 ) n P j=1 (¹ j k ) 2 du k u k ¡ l 0 k l 1 k (µ¡µ 0 )T n P j=1 (¹ j k ) 2 ¡ l 1 k (µ¡µ 0 ) n P j=1 (¹ j k ) 2 T Z 0 f k (s)ds u k (s) ¡ (l 1 k ) 2 (µ 2 ¡µ 2 0 )T 2 n P j=1 (¹ j k ) 2 o : (3.14) The maximum likelihood estimate ^ µ N ofµ 0 is obtained by maximizing the Radon- Nykodim derivative (likelihood ratio) (3.14) with respect to the parameter of interest µ2£. Direct computations yield ^ µ k = 1 l 1 k T T Z 0 du k u k ¡ l 0 k l 1 k ¡ 1 l 1 k T T Z 0 f k (s) u k (s) ds: (3.15) By Itˆ o’s Lemma, we find thatdln(u k )= du k u k ¡ 1 2 n P j=1 (¹ j k ) 2 dt, and hence from (3.15) we get ^ µ k = 1 l 1 k T ln u k (T) u k (0) + 1 2l 1 k n X j=1 (¹ j k ) 2 ¡ l 0 k l 1 k ¡ 1 l 1 k T T Z 0 f k (s) u k (s) ds: (3.16) Also, from (3.15) and taking into account (3.13) we have the following represen- tation for the estimate ^ µ k =µ 0 + 1 l 1 k T n X j=1 ¹ j k W j (T); (3.17) which will play the key role in our future investigations. Note that by (3.15) ^ µ k is an unbiased estimate ofµ 0 . 80 In addition to Assumptions (H1)-(H6), we will assume the following: (H7) There exists the limit lim k!1 n P j=1 (¹ j k ) 2 (l 1 k ) 2 =¹<1. We want to mention that assumption (H7) does not follow from (H6) neither converse is true. For example, take l 0 k = ¡¸ 2m k ; l 1 k = ¡¸ m 2 k ; ¹ k = ¸ m k ; n = 1. Then (H6) is satisfied, while (H7) does not hold. If we putl 0 k = 0; l 1 k =¡¸ 2m k ; ¹ k = ¸ 2m¡1 k ; n = 1;m > 2, then (H7) holds, and ¹ = 0, but (H6) is not satisfied. Both assumptions are related to the order of operatorsA 0 ;A 1 ;M j , when all of them are some pseudo- differential operators. The relationship between orders of operators will be discussed in details in the next section. Denote byL the set of all real-valued, nonnegative functionsw defined onR, which are symmetric, w(0) = 0 and monotone for x > 0. These functions are called loss functions. 3.4.1 Theorem. Under Assumptions (H1)-(H6) the following holds true: (i) lim T!1 ^ µ k =µ 0 ; P-a.s., for everyk2N (consistency in time); Suppose in addition that (H7) is satisfied. Then (ii) if¹ = 0, then lim k!1 ^ µ k =µ 0 ; P-a.s., for everyT > 0 (consistency ink). More- over, in this case, MLE ^ µ k is asymptotic efficient, that is for any loss function w2L lim k!1 Ew 0 B B B B @ jl 1 k j p T s n P j=1 (¹ j k ) 2 ( ^ µ k ¡µ 0 ) 1 C C C C A =Ew(»); were» a Gaussian random variable with zero mean and unit variance; 81 (iii) if ¹ 6= 0, then lim k!1 ^ µ k = µ 0 +»(¹); P-a.s., for every T > 0, where »(¹) » N(0; ¹ T ). Proof. Property (i) follows from (3.17) and the fact that lim t!1 W(t) t =0. If¹ = 0 then lim k!1 ¹ j k l 1 k = 0 for everyj = 1;:::;n. Thus, from equality (3.17) we conclude that lim k!1 ^ µ =µ 0 . Also, by (3.17) we note that » k := jl 1 k j p T s n P j=1 (¹ j k ) 2 ( ^ µ k ¡µ 0 )»»; (3.18) where » » N(0;1). Hence E(w(» k )) = E(w(»)), that consequently implies lim k!1 E(w(» k ))=E(w(»)), and (ii) is proven. Property (iii) follows from (3.18) and Assumption (H7). 3.4.2 Remark. By (3.18) one gets at once that ^ µ k is also asymptotic efficient in T . We live this property apart since the main goal is to get a consistent estimate on every small time interval[0;T]. By the above theorem, we can use every individual ^ µ k as an estimate for µ 0 , and by increasing T to get a good estimate of the true parameter, regardless of Assump- tion (H7). However in practice we would like to findµ 0 with any precision in a small interval of time. As we can see from (ii), Theorem 3.4.1, this is possible by increas- ing the number of Fourier coefficients. This is due to the fact that the measuresP µ u , that correspond to the solution of the original equation (3.9), are singular for differ- ent values of the parameter µ. Comparing (ii) and (iii), we also note that for ¹ = 0 we get consistent estimates, while for ¹ 6= 0, the limit estimate is biased. The speed of convergence is of order s n P j=1 (¹ j k ) 2 jl 1 k j p T . Although for ¹ = 0; ^ µ k is a consistent esti- mate of the true values as k ! 1, we want to mention that in practice the estimates 82 will be evaluated by formula (3.16), that involves the values of u k and f k . Note that u k and f k being Fourier coefficients of some functions inH ° will have at least poly- nomial decay to zero. This implies that for large k we will not be able to observe and compute u k ; f k reliably. However, in many applications f ´ 0, and in this case to apply (3.16) we need to know only the values of log u k (T) u k (0) . In this situation, equation (3.13) becomes the well-known geometrical Brownian motion, with solution u k (t)=u k (0)expf(µ 0 l k (µ)¡ n P j=1 (¹ j k ) 2 =2)t+ n P j=1 ¹ j k W j (t)g. Obviously, log u k (T) u k (0) = ³ µ 0 l k (µ)¡ n X j=1 (¹ j k ) 2 =2 ´ T + n X j=1 ¹ j k W j (T) (3.19) which implies thatlog u k (T) u k (0) has the same magnitude asl k for a reasonable time interval [0;T] , and thus the estimator given by (3.16) is computable for sufficiently large k. As will see later on, it suffices to takek being around ten, to get an approximation of order10 ¡4 . On the other hand, a reasonable and natural approach is to consider a weighted average of the estimates ^ µ k . For example, if n = 1, and the sign of ¹ k varies with k, using (3.17) we observe that the estimates ^ µ k will approach the true value by oscillating around it (depending on the sign ofW and¹ k ). Hence, some proper chosen weighted average of µ k will converge faster to the true value. For the rest of this section we will discuss how to choose the weights such that to have at least the same rate of convergence of new estimates comparative to ^ µ k . Supposef¯ k g 1 1 ½R + , and put ^ µ (N) = N P k=1 ¯ k ^ µ k N P k=1 ¯ k : (3.20) 83 SinceE( ^ µ k ) = µ 0 , by (3.20) obviously we have thatE( ^ µ (N) ) = µ 0 , i.e. ^ µ (N) is an unbiased estimator ofµ 0 . From (3.16), we get ^ µ (N) = N P k=1 ¯ k l 1 k ³ ln u k (T) u k (0) ¡ T R 0 f k (s) u k (s) ds ´ T N P k=1 ¯ k + N P k=1 ¯ k l 1 k n P j=1 (¹ j k ) 2 2 N P k=1 ¯ k ¡ N P k=1 ¯ k l 0 k l 1 k N P k=1 ¯ k ; (3.21) which will be the main formula used in computational tasks. Similarly, by (3.17) we deduce ^ µ (N) =µ 0 + N P k=1 ¯ k n P j=1 ´ j k W j (T) T N P k=1 ¯ k ; (3.22) where´ j k =¹ j k =l 1 k forj =1;:::;n andk =1;:::;N. It turns out, to preserve consistency and normality, it suffices to make the following assumption: (H8) ¯ k ¸0; k2N and lim N!1 N P k=1 ¯ k =1. 3.4.3 Theorem. If Assumptions (H1)-(H8) are fulfilled then: (i) lim T!1 ^ µ (N) = µ 0 ; P-a.e., for every N 2 N (consistency in time) and MLE is asymptotic efficient asT !1; (ii) if ¹ = 0, then lim N!1 ^ µ (N) = µ 0 ; P-a.e., for every T > 0 (consistency in N), and MLE ^ µ (N) is asymptotic efficient; (iii) if¹6= 0, then lim N!1 ^ µ (N) = µ 0 +»(¹); P-a.s., for everyT > 0, where»(¹)» N(0; ¹ T ). 84 Proof. Similar to Theorem 3.3.1, (i) follows from (3.22) and the fact that lim t!1 W(t) t = 0. By Stolz-Cesaro Theorem (discrete version of L’Hospital’s rule, see for instance [33], p.35 or [67], p.17), under Assumption (H8) we have lim N!1 N P k=1 ¯ k ´ j k N P k=1 ¯ k = lim k!1 ´ j k ; (3.23) for everyj =1;:::;n. If ¹ = 0, then lim k!1 ´ j k = 0 for every j = 1;:::;n, hence by (3.23) and (3.24) it follows that lim N!1 ^ µ (N) =µ 0 . Using (3.22) one can show that N P k=1 ¯ k p T N P k=1 ¯ k s n P j=1 (´ j k ) 2 ( ^ µ (N) ¡µ 0 )»N(0;1) (3.24) and similar to Theorem 3.3.1 the rest of the proof follows. In practice, ideally we want to choose the weights¯ k such that the rate of conver- gence of ^ µ (N) ! µ 0 to be at least the same (or faster) than the rate of convergence of ^ µ k ! µ 0 and also, as we mentioned before, ¯ k should offset somehow the fast decay of the Fourier coefficientsu k . It should be mentioned that Assumption (H8) does not guarantee the same rate of convergence. For example, if ´ k = k ¡2 and ¯ k = k, then lim k!1 N P k=1 ¯ k ´ k ´ N N P k=1 ¯ k =1. However, if¯ k =k ± with± >1, or¯ k =exp(k " ) with">0, then the rate of convergence is preserved. These heuristic discourses lead to the conclusion that ¯ k should grow fast enough. On the other hand, we have to keep in mind that 85 ¯ k finally should be computable on PC. Thus, a reasonable choose for ¯ k would be ¯ k = exp(k ± ) with± =2f0:5;1;2;3g and depends on how many Fourier coefficients we are going to use in our computations. Finally, we will prove the following technical lemma about rate of convergence. 3.4.4 Lemma. Supposefa n g 1 n=1 is an increasing sequence of positive numbers such that lim n!1 a n =1; ¯ ¯ ¯ ¢ (2) (a n ) ¢ (1) (an) ¯ ¯ ¯<M, where¢ (k) is thek-th finite difference. Then, N P n=1 a ¡1 n exp(a n ) N P n=1 exp(a n ) »a ¡1 N : Proof. We will apply several times the Stolz-Cesaro Theorem to the sequence a N N P n=1 a ¡1 n b n = N P n=1 b n , where b n := exp(a n ), and show that this sequence has a finite limit. A= lim N!1 a N N P n=1 a ¡1 n b n N P n=1 b n = lim N!1 a N N P n=1 b n a n ¡a N¡1 N¡1 P n=1 b n a n b N =1+ lim N!1 (a N ¡a N¡1 ) N¡1 P n=1 a ¡1 n b n b N =1+ lim N!1 (a N ¡a N¡1 ) N¡1 P n=1 b n a n ¡(a N¡1 ¡a N¡2 ) N¡2 P n=1 b n a n b N ¡b N¡1 =1+ lim N!1 (a N ¡a N¡1 )b N¡1 a N¡1 (b N ¡b N¡1 ) + lim N!1 (a N ¡2a N¡1 +a N¡2 ) N¡2 P n=1 b n an b N ¡b N¡1 : (3.25) 86 We claim that both limits in the last expression are zero. Recall that b n = exp(a n ). Thus, A 1 := lim N!1 a N ¡a N¡1 a N¡1 ¢ b N¡1 b N ¡b N¡1 = lim N!1 a N ¡a N¡1 a N¡1 ¢ 1 exp(a N ¡a N¡1 )¡1 ; and since the sequence a n is increasing, by Taylor series expansion for exponential functions, from the last equality we get A 1 · lim N!1 1 a N¡1 =0; henceA 1 =0. Similarly, A 2 = lim N!1 (a N ¡2a N¡1 +a N¡2 ) N¡2 P n=1 b n an b N ¡b N¡1 = lim N!1 ¢ (2) (a N ) N¡2 P n=1 b n an ¢ (1) (a N )exp(a N ) : By the initial assumption, ¯ ¯ ¯ ¢ (2) (a N ) ¢ (1) (a N ) ¯ ¯ ¯<M, that implies A 2 ·M lim N!1 N¡2 P n=1 b n a n exp(a N ) ; and again by the Stolz-Cesaro Theorem, we continue A 2 ·M lim N!1 exp(a N¡2 ) a N¡2 (exp(a N )¡exp(a N¡1 ) ·M lim N!1 1 a N¡2 =0: Finally by (3.25) we haveA=1, and lemma is proved. 87 Similarly to Lemma 3.4.4, one can prove that N P n=1 a ¡1 n bn N P n=1 bn » a ¡1 N , ifa ¡1 n has polyno- mial growth, andb n grows faster thana n . 3.4.5 Remark. Observe that ^ µ k involves only thek-th Fourier coefficient, while evalu- ating ^ µ (N) we use allu k withk =1;:::;N. One should expect that using ^ µ (N) we will get more precise estimates since more information is used. However, we get actually the same precision in our estimates by applying either simple estimates ^ µ k or weighted average estimates ^ µ (N) , since the rates of convergence are the same. Generally speak- ing, the filtrationF u k t (see Section 1.2) generated by the solutionu k is smaller then the filtrationF t generated by the Brownian MotionsW j (t), and could be different for dif- ferentk’s. In our case, eachu k is a geometrical Brownian motion driven by the same processes W j (t) (j = 1;:::;n), hence are given by the same function with different parameters but with same noise. This impplies that the filtrationsF u k t are the same for all k 2 N. In other words, each Fourier coefficient u k contains the same amount of information aboutµ 0 , that explains why the precisions of ^ µ k and ^ µ (N) are the same. 3.5 Applications to stochastic parabolic differential equations In this section, we will present some applications of Theorems 3.4.1 and 3.4.3 to equa- tion (3.9) withA µ ;M j being some pseudo-differential operators. We will begin with an abstract case, and continue with some particular examples, including some numeri- cal results for the initial parameter estimation problem. If otherwise it is not mentioned all notations from Section 3.3 will be preserved. 88 Let M be a d-dimensional compact, orientable, C 1 manifold with a smooth positive measure dx, fH s (M)g s2R be the scale of Sobolev spaces on M (see for instance[1], [76]), and let(;F;fF t g;P) be a stochastic basis. Suppose that A µ = A 0 + µA 1 , is a differential, elliptic operator of order 2m, formally self-adjoint, and the operator A µ is lower semi-bounded for every µ 2 £ (uniformly inµ). The latter means that there exists a positive number" > 0 such that ¡A µ >"I, for everyµ2£, whereI is the identity operator inH °+m . It is well-known that the operator A µ can be extended to a closed, self-adjoint operator onL 2 (M), the spectrum of this operator is discrete, consisting of eigenvalues of finite multiplicity (for more details see the discussion in Chapter 1 and references mentioned therein). Denote by fh k;µ g the eigenfunctions of A µ , and suppose that fh k;µ g 1 k=1 forms an orthonormal system in L 2 (M). By Hilbert-Schmidt theory, this system is complete in L 2 (M), and h k;µ 2 H ° (M)\C 1 (M) for every ° 2R. Gen- erally speaking, the eigenfunctions h k;µ depend on µ, however, for sake of simplicity we will rule this out, and write h k . From general theory of elliptic operators, under above assumptions, the operatorsA µ : H °+m ! H °¡m are linear bounded operators for every°2R. Thus Assumption (H1) from Section 3.3 is satisfied. We set¤ µ :=("I¡A µ ) 1 2m . For everyµ2£, the operator¤ µ generates the Hilbert scalefH s µ (M)g s2R . The spacesfH s µ (M)g s2R are equivalent for all µ 2 £ (see for instance [76]), namely H s 1 µ (M) = H s 2 µ (M) as sets of functions, and topologies are equivalent. Thus without loss of generality, we will consider the operator¤ = ¤ µ for some fixed value of the parameter µ. Now we are under the setup of general results from section 3.3. Assume thatM j = b j ¤ p j , where p j · m; b j 2 R and j = 1;:::;n. Then the operators M j are pseudo-differential operators of order less than m, and hence are 89 linear bounded operators acting fromH °+m inH ° for every°2R. Thus Assumption (H2) is satisfied. We assume that initial condition u 0 satisfies (H3) and free term f satisfies (H4). Finally, we suppose that the operatorsA 0 andA 1 have the same system of eigenfunc- tionsfh k g k2N , so (H5) is satisfied too. Let m i := ord(A i ); i = 0;1, where by ord(A) we denote the order of differen- tial operator A. From the above assumptions obviously 2m = maxfm 0 ;m 1 g, and ord(M j )=p j ·m. From spectral theory of elliptic operators it follows that the asymptotics of the eigenvaluesl k;i ; ¹ j k and¸ k are given by l k;i »k m i d ; ¹ j k »k p j d ; ¸ k »k 1 d ; (3.26) wherei = 0;1 andj = 1;:::;n. From here we get that lim k!1 l 1 k µ+l 0 k ¸ 2m is finite for every µ 2 £. Obviously, ¹ j k = b j ¸ p j k 6= 0 for every j = 1:::;n; k 2N. Hence, to fulfill condition (H6) it is sufficient to assume the following: (H6.1) There exists " > 0 such that n P j=1 b 2 j ¸ 2(p j ¡m) k · "¡ 2(l 1 k ¿+l 0 k ) ¸ 2m k , where k 2N and ¿ =sup(£). In the nutshell, Assumption (H6) requires max j ford(M j )g·2maxford(A 1 );ord(A 0 )g; plus the coercivity condition on the coefficients. Also, (H6.1) can be replaced with the following, more restrictive but easier to check, condition (H6.2) n P j=1 b 2 j <¡ 2(l 1 k ¿+l 0 k ) ¸ 2m k , where¿ =sup(£); k2N. 90 Under above assumptions, the equation (3.9) has a unique solution in the space L 2 (;C((0;T);H ° ))\L 2 (£[0;T];H °+m ), and estimate (3.10) holds true. Finally, we will discuss Assumption (H7). For the differential operators considered in this section, this condition becomes (H7.1) p:=maxfp 1 ;:::;p n g·m 1 . The equivalence of (H7) and (H7.1) follows directly from (3.26). Since all assumptions (H1)-(H7) are satisfied, we conclude 3.5.1 Proposition. Under above assumptions on differential operators A µ ;M j , the estimates ^ µ k are unbiased and consistent in both time and dimension of the projection (number of Fourier coefficients), namely Theorem 3.4.1 holds true. If in addition we assume (H8), then Theorem 3.4.3 about weighted average follows. Moreover, if p < m 1 , then¹ = 0, and hence the MLE’s from Theorems 3.4.1 and 3.4.3 are asymptotic efficient. 3.6 Examples and some numerical results In this section we will proceed to some particular examples, each followed by numer- ical results. 3.6.1 Example. Stochastic heat equation. Let us consider the following SPDE 8 > > > > > < > > > > > : du(t;x)=µ¢u(t;x)dt+u(t;x)dW(t); t2(0;T]; x2(0;1); u(0;x)=u 0 (x) x2(0;1); u(t;0)=u(t;1)=0 t2[0;T]; (3.27) 91 where ¢u = u xx ; W is a standard Brownian motion, and u 0 2 L 2 (0;T). For sim- plicity of writing we consider T = 1, but all results remain true for an arbitrary time T . Using previous notations we put¤:= p ¡¢ acting inL 2 (0;1) with zero boundary conditions. It is well-known thath k (x)= p 2sin(k¼x); k2N, are the eigenfunctions of the operator ¤, with corresponding eigenvalues ¸ k = ¼k, the operator ¤ satis- fies all conditions mentioned in Section 3.3 and generates the scale of Sobolev spaces fH s (0;1)g s2R (compare to Example 2.2.3 - 2.3.5). The operators corresponding to the original evolution equation (3.9) are as follows: A 1 =¢ andl 1 k =¡(k¼) 2 ; k2N; A 0 =0; f ´0; M=I; n=1 and¹ k =1; k2N. Equation (3.26) makes sense only forµ > 0. Indeed, Assumption (H6.1) leads to the following inequality 1 · "+2µ(k¼) 2 for every k 2N, which obviously is satisfied only for µ > 0. Thus, the existence and uniqueness of the solution is guaranteed in H s (0;1); s2R ifµ >0. The MLE’s (see (3.16) and (3.21)) in this case have the following form ^ µ k =¡ 1 (k¼) 2 µ log u k (1) u k (0) + 1 2 ¶ ; ^ µ (N) =¡ N P k=1 ¯ k (k¼) 2 ³ log u k (1) u k (0) + 1 2 ´ N P k=1 ¯ k ; (3.28) whereu k are the Fourier coefficients of the observed solution of equation (3.26) w.r.t. CONSfh k g k2N , and¯ k ’s satisfy (H8). 92 Since p = 0 < m 1 = 2, we have that Assumptions (H7.1) is satisfied, and hence Theorems 3.4.1 and 3.4.3 hold true. Moreover, ¹ = 0, which consequently implies that estimates (3.28) are consistent, and asymptotically efficient and unbiased. We simulated the Fourier coefficientsu k , using Matlab7, taking the true parameter µ 0 = 1. Applying formulas (3.28), we evaluated the estimates ^ µ k (simple estimates), ^ µ (N) with¯ k = k (weighted polynomial estimates), and¯ k = exp(k) (weighted expo- nential estimates). In Figure 3.1 we present the graph for all three estimators and the true value of the parameter, as functions of the number of Fourier coefficients. 0 5 10 15 20 25 30 1 1.002 1.004 1.006 1.008 1.01 Fourier Coefficient θ estimated θ 0 (true value) θ k (simple estimates) θ (N) (weighted polynomial est), β k =k θ (N) (weighted exponential est), β k =exp(k) Figure 3.1: Estimated Parameter. Example 3.6.1 93 We note that ^ µ k and ^ µ (N) with exponential weights converge to the true parameter, basically with the same speed of convergence. However, as mentioned before, the polynomial weighted estimate converge slower. In the Table 3.1 we present the corresponding errors, i.e. jµ 0 ¡ ^ µj, and as we can see, fork =10 we are within four digits precision, and taking into account the first 20 Fourier coefficients we get an error of magnitude less then 10 ¡5 . We see that indeed kn Error Simple Polynomial Exponential 1 0.00961 0.0096 0.0096 5 0.00038 0.0008 0.0006 10 0.00009 0.0002 0.0001 15 4:3 ¡5 0.0001 4:7£10 ¡5 20 2:4£10 ¡5 6:7£10 ¡5 2:5£10 ¡5 30 1:6£10 ¡5 3:0£10 ¡5 1:1£10 ¡5 Table 3.1: Errors of estimated parameter the rate of convergence is equal tok ¡2 . The numerical results are similar if the noise is driven by several independent Brownian motions, i.e. forn>1. 3.6.2 Example. Now we will consider an example where the unknown parameter is not a factor of the leading differential operator. In our notations this meansord(A 0 )> ord(A 1 ). Suppose thatu satisfies the following SPDE du=(u xx ¡µu)dt+ n X j=1 1 j 2 (1¡¢) q j udW j (t); x2[0;1]; t2[0;1]; (3.29) with periodic boundary conditions, u(t;0) = u(t;1), with initial condition u(0;x) = u 0 (x) 2 L 2 (0;1) and q j < 0. Denote by ¢ the Laplace operator @ @x 2 on L 2 ([0;1]) with periodic boundary conditions. Let us consider ¤ = p 1¡¢, then ¤ generates 94 the Hilbert scalefH s g s2R . The orthonormal system of eigenfunctions of this operator is given by: h 0 (x) = 1, h 2k (x) = p 2cos(2¼kx), h 2k¡1 = sin(2¼kx) for k 2 N. Obviously, the operators A 1 = ¡I; A 0 = ¢; M j = 1=j 2 (1¡ ¢) q j and ¤ have a common system of eigenfunctions. Also, note that the corresponding eigenvalues satisfy the relations l 1 k = ¡1; l 0 2k = l 0 2k¡1 = ¡(2k¼) 2 ; ¸ k = p 1¡(l 0 k ) 2 and ¹ j k = ¸ 2q j k =j 2 . We see that conditions (H1)-(H5) and (H7.1) are satisfied, and now, the objective is to show that (H6.2) holds. Withb j =1=j 2 , (H6.2) becomes n X j=1 1 j 2 <¡2 ¡¿ +l 0 k 1¡l 0 k : Without sake of completeness we take ¿ = 1, and since 1 P j=1 1 j 2 = ¼ 2 6 < 2, we con- clude that (H6.2) is verified. Hence, by Proposition 3.5.1 (see also Theorems 3.4.1 and representation (3.16)) there exists a unique solution of equation (3.29), and the estimators ^ µ k =log u k (0) u k (1) ¡ 1 2 n X j=1 (1¡(2k¼) 2 ) q j j 2 +(2k¼) 2 (k2N) are consistent and asymptotic efficient. Similarly, one may apply the Theorem 3.4.3 and get the results about the weighted average MLE’s. Now we will present some numerical results for this example. We take q j = ¡1 andn=100. Figure 3.2 exhibits the estimated parameters evaluated by three different methods, using the first 30 Fourier coefficients. The solid horizontal line corresponds to the true value of the parameterµ 0 = 1. As we mentioned before, Simple Estimates ^ µ k and Weighted Exponential Estimates ^ µ (N) fork;N ¸10 are almost the same, while 95 polynomial weighted average converge slower to the true value. The errorj ^ µ¡µ 0 j; k = 25;:::;45 are presented in Figure 3.3. 0 5 10 15 20 25 30 1 1.0002 1.0004 1.0006 1.0008 1.001 Fourier Coefficient θ estimated θ 0 (true value) θ k (simple estimates) θ (N) (weighted polynomial est), β k =k θ (N) (weighted exponential est), β k =exp(k) Figure 3.2: Estimated Parameter. Example 3.6.2. 3.6.3 Example. We will conclude this section with some hypothetical examples (by indicating only the eigenvalues but not the operators themselves), that show different rates of converges of the estimates, and also, some counter-examples related to condi- tions (H6) and (H7). In what follows we assume (H1)-(H5) and (H8) hold true. 3.6.3.a. LetA 0 = 0;A 1 = ¢ with zero boundary conditions. Similar to Example 1, l 1 k = ¡(k¼) 2 . Suppose thatM is the operator with eigenvalues ¹ k = (¡1) k p k. It is clear that (H6) and (H7) are satisfied and ¹ = 0. Thus, we can apply Theorems 3.4.1 and 3.4.3. The results are presented in Figure 3.4. We observe that in this case, the weighted exponential estimates perform better than simple or polynomial averaged ones. This is due to the fact that the sign of¹ k alternates. 96 30 35 40 45 50 10 −9 10 −8 10 −7 10 −6 Fourier Coefficient Error (log scale) Simple Estimates Weighted Polynomial Est Weighted Exponential Est Figure 3.3: Errorsj ^ µ¡µ 0 j vs number of Fourier coefficients. Example 3.6.2 3.6.3.b. Letl 0 k =¡(k¼) 2 ; l 1 k = 1; ¹ k = k¼, then Assumption (H6) is satisfied. Since lim k!1 ³ ¹ k l 1 k ´ 2 = lim k!1 (k¼) 2 1 = 1, we have that (H7) is violated. This means that there exists a unique solution, but we can not apply the above results about MLE’s. The numerical results, formally applied to this problem are shown on Figure 3.5. As we can see, all estimators diverge. 3.6.3.c. Now we take l 0 k = 1; l 1 k = ¡(k¼) 4 ; ¹ k = (k¼) 3:5 , then one may check that Assumption (H7) is satisfied, while (H6) does not hold. Note that (H6) is related only to existence and uniqueness and not to the convergence of the estimators. It is known that coercivity condition (H6) is a sufficient and in some sense a necessary condition for the existence and uniqueness of the solution for parabolic SPDE. It turns out that numerical results also show that if only (H6) is violated, the estimators do not converge to the true values of the parameterµ 0 =0. The simulation are presented in Figure 3.6. We see that the error is more than .1 for k=100. 97 5 10 15 20 25 0.985 0.99 0.995 1 1.005 1.01 1.015 Fourier Coefficient θ estimated θ 0 (true value) θ k (simple estimates) θ (N) (weighted polynomial est), β k =k θ (N) (weighted exponential est), β k =exp(k) Figure 3.4: Estimators vs number of Fourier coefficients. Example 3.6.3.a 5 10 15 20 25 0 5 10 15 20 25 30 35 40 45 50 Fourier Coefficient θ estimated θ 0 (true value) θ k (simple estimates) θ (N) (weighted polynomial est), β k =k θ (N) (weighted exponential est), β k =exp(k) Figure 3.5: Estimators vs number of Fourier coefficients. Example 3.6.3.b 98 0 10 20 30 40 50 60 70 80 0.6 0.7 0.8 0.9 1 1.1 Fourier Coefficient θ estimated θ 0 (true value) θ k (simple estimates) θ (N) (weighted polynomial est), β k =k θ (N) (weighted exponential est), β k =exp(k) Figure 3.6: Estimators vs number of Fourier coefficients. Example 3.6.3.c 99 References [1] R. A. Adams, Sobolev spaces, Academic Press, New York-London, 1975, Pure and Applied Mathematics, V ol. 65. [2] S. I. Aihara, Regularized maximum likelihood estimate for an infinite- dimensional parameter in stochastic parabolic systems, SIAM J. Control Optim. 30 (1992), no. 4, 745–764. [3] A. Bagchi and V . Borkar, Parameter identification in infinite-dimensional linear systems, Stochastics 12 (1984), no. 3-4, 201–213. [4] A. Ben-Israel and T. N. E. Greville, Generalized inverses, second ed., CMS Books in Mathematics/Ouvrages de Math´ ematiques de la SMC, 15, Springer- Verlag, New York, 2003, Theory and applications. [5] T. Bj¨ ork, Arbitrage theory in continous time, Oxford University Press, 1998. [6] D. Brigo and F. Mercurio, Interest rate models—theory and practice, Springer Finance, Springer-Verlag, Berlin, 2001. [7] R. Cont, Modeling term structure dynamics: an infinite dimensional approach, Int. J. Theor. Appl. Finance 8 (2005), no. 3, 357–380. [8] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclo- pedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. [9] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. [10] W. H. Fleming, Distributed parameter stochastic systems in population biol- ogy, Control theory, numerical methods and computer systems modelling (Inter- nat. Sympos., IRIA LABORIA, Rocquencourt, 1974), Springer, Berlin, 1975, pp. 179–191. Lecture Notes in Econom. and Math. Systems, V ol. 107. [11] C. Frankignoul, Sst anomalies, planetary waves and rc in the middle rectitudes, Reviews of Geophysics 30 (2000), no. 7, 1776–1789. [12] J. G´ all, G. Pap, and M. C. A. van Zuijlen, Forward interest rate curves in discrete time settings driven by random fields, Comput. Math. Appl. 51 (2006), no. 3-4, 387–396. 100 [13] ˘ I. ¯ I. G¯ ıhman and A. V . Skorohod, The theory of stochastic processes. I, Funda- mental Principles of Mathematical Sciences, vol. 210, Springer-Verlag, Berlin, 1980. [14] R. S. Goldstein, The term structure of interest rates as random field, Review of Financial Studies (2000), no. 13, 365–384. [15] D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1283– 1299. [16] P. Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964. [17] A. Hassell and T. Tao, Upper and lower bounds for normal derivatives of Dirich- let eigenfunctions, Math. Res. Lett. 9 (2002), no. 2-3, 289–305. [18] , L 2 bounds for normal derivatives of Dirichlet eigenfunctions, Interna- tional Conference on Harmonic Analysis and Related Topics (Sydney, 2002), Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 41, Austral. Nat. Univ., Can- berra, 2003, pp. 57–67. [19] A. Hassell and A. Vasy, The resolvent for Laplace-type operators on asymptoti- cally conic spaces, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, 1299–1346. [20] E. Hille and R. S. Phillips, Functional analysis and semi-groups, American Math- ematical Society, Providence, R. I., 1974. [21] M. Huebner, S. Lototsky, and B. L. Rozovskii, Asymptotic properties of an approximate maximum likelihood estimator for stochastic PDEs, Statistics and control of stochastic processes (Moscow, 1995/1996), World Sci. Publishing, 1997, pp. 139–155. [22] M. Huebner, B. Rozovskii, and R. Khasminskii, Two examples of parameter estimation, in Stochastic Processes, ed. Cambanis, Chos, Karandikar, Berlin, Springer, 1992. [23] M. Huebner and B. L. Rozovski˘ ı, On asymptotic properties of maximum likeli- hood estimators for parabolic stochastic PDE’s, Probab. Theory Related Fields 103 (1995), no. 2, 143–163. [24] J. C. Hull, Options, futures, and other derivatives, 5th ed., Prentice Hall, 2003. [25] I. A. Ibragimov and R. Z. Khas 0 minski˘ ı, Statistical estimation, Applications of Mathematics, vol. 16, Springer-Verlag, New York, 1981. 101 [26] , Problems of estimating the coefficients of stochastic partial differential equations. I, Teor. Veroyatnost. i Primenen. 43 (1998), no. 3, 417–438. [27] , Problems of estimating the coefficients of stochastic partial differential equations. II, Teor. Veroyatnost. i Primenen. 44 (1999), no. 3, 526–554. [28] , Problems of estimating the coefficients of stochastic partial differential equations. III, Teor. Veroyatnost. i Primenen. 45 (2000), no. 2, 209–235. [29] V . Ivrii, Eigenvalue asymptotics for Neumann Laplacian in domains with ultra- thin cusps, Seminaire: ´ Equations aux D´ eriv´ ees Partielles, 1998–1999, S´ emin. ´ Equ. D´ eriv. Partielles, ´ Ecole Polytech., Palaiseau, 1999, pp. Exp. No. X, 8. [30] , Sharp spectral asymptotics for operators with irregular coefficients. II. Domains with boundaries and degenerations, Comm. Partial Differential Equa- tions 28 (2003), no. 1-2, 103–128. [31] V . Ivrii and M. Bronstein, Sharp spectral asymptotics for operators with irreg- ular coefficients. I. Pushing the limits, Comm. Partial Differential Equations 28 (2003), no. 1-2, 83–102. [32] H. Iwaniec and P. Sarnak, L 1 norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), no. 2, 301–320. [33] W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis. I, Student Mathematical Library, vol. 4, American Mathematical Society, Providence, RI, 2000, Real numbers, sequences and series, Translated and revised from the 1996 Polish original by the authors. [34] D. P. Kennedy, The term structure of interest rates as a gaussian random field, Mathematical Finance (1994), no. 4, 247–258. [35] , Characterizing Gaussian models of the term structure of interest rates, Mathematical Finance (1997), no. 7, 107–118. [36] R. Khasminskii, N. Krylov, and N. Moshchuk, On the estimation of parameters for linear stochastic differential equations, Probab. Theory Related Fields 113 (1999), no. 3, 443–472. [37] K.-H. Kim and N. V . Krylov, On the Sobolev space theory of parabolic and ellip- tic equations in C 1 domains, SIAM J. Math. Anal. 36 (2004), no. 2, 618–642 (electronic). MR MR2111792 (2005k:35168) [38] P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differential equations, Texts in Applied Mathematics, vol. 44, Springer- Verlag, New York, 2003. 102 [39] S. M. Kozlov, Equivalence of measures in Itˆ o’s linear partial differential equa- tions, Vestnik Moskov. Univ. Ser. I Mat. Meh. (1977), no. 4, 47–52. [40] M. A. Krasnosel 0 ski˘ ı, P. P. Zabre˘ ıko, E. I. Pustyl 0 nik, and P. E. Sobolevski˘ ı, Inte- gral operators in spaces of summable functions, Noordhoff International Pub- lishing, Leiden, 1976. [41] S. G. Kre˘ ın, Yu. ¯ I. Petun¯ ın, and E. M. Sem¨ enov, Interpolation of linear opera- tors, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. [42] N. V . Krylov, On L p -theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal. 27 (1996), no. 2, 313–340. MR MR1377477 (97b:60107) [43] , An analytic approach to SPDEs, Stochastic partial differential equa- tions: six perspectives, Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Prov- idence, RI, 1999, pp. 185–242. [44] N. V . Krylov and S. V . Lototsky, A Sobolev space theory of SPDEs with con- stant coefficients in a half space, SIAM J. Math. Anal. 31 (1999), no. 1, 19–33 (electronic). [45] , A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal. 30 (1999), no. 2, 298–325 (electronic). [46] Yu. A. Kutoyants, Parameter estimation for stochastic processes, Research and Exposition in Mathematics, vol. 6, Heldermann Verlag, Berlin, 1984. [47] R. S. Liptser and A. N. Shiryayev, Statistics of random processes I. General theory, 2nd ed., Springer-Verlag, New York, 2000. [48] R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, 1989. [49] R. Litterman and J. Scheinkman, Common factors affecting bond returns, Journal of Fixed Income (1991), no. 1, 49–53. [50] S. V . Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains, Stochastics Stochastics Rep. 68 (1999), no. 1-2, 145–175. [51] , Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods Appl. Anal. 7 (2000), no. 1, 195–204. [52] S. V . Lototsky and B. L. Rosovskii, Spectral asymptotics of some functionals arising in statistical inference for SPDEs, Stochastic Process. Appl. 79 (1999), no. 1, 69–94. 103 [53] S. V . Lototsky and B. L. Rozovskii, Parameter estimation for stochastic evolu- tion equations with non-commuting operators, in Skorohod’s Ideas in Probability Theory, V .Korolyuk, N.Portenko and H.Syta (editors), Institute of Mathematics of National Academy of Sciences of Ukraine, Kiev, Ukraine, 2000, pp. 271–280. [54] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in H¨ older classes, Ann. Probab. 28 (2000), no. 1, 74–103. [55] R. Mikulevicius and H. Pragarauskas, On Cauchy-Dirichlet problem in half- space for parabolic SPDEs in weighted H¨ older spaces, Stochastic Process. Appl. 106 (2003), no. 2, 185–222. [56] R. Mikulevicius and B. L. Rozovskii, Uniqueness and absolute continuity of weak solutions for parabolic SPDEs, Acta Appl. Math. 35 (1994), no. 1-2, 179– 192. [57] G. N. Milstein, Numerical integration of stochastic differential equations, Math- ematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dor- drecht, 1995. [58] C. Mueller, The heat equation with L´ evy noise, Stochastic Process. Appl. 74 (1998), no. 1, 67–82. [59] C. Mueller, L. Mytnik, and A. Stan, The heat equation with time-independent multiplicative stable L´ evy noise, Stochastic Process. Appl. 116 (2006), no. 1, 70–100. [60] E. Mytnik, L. Perkins and A. Sturm, On pathwise uniqueness for stochastic heat equations with non-lipschitz coefficients, Preprint - arXiv:math.PR/0507545 v1 (2005). [61] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162. [62] B. Øksendal, Stochastic differential equations: An introduction with applica- tions, sixth ed., Universitext, Springer-Verlag, Berlin, 2003. [63] E. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions, Lectures on probability theory and statistics (Saint-Flour, 1999), Lecture Notes in Math., vol. 1781, Springer, Berlin, 2002, pp. 125–324. [64] L. I. Piterbarg, The top Lyapunov exponent for a stochastic flow modeling the upper ocean turbulence, SIAM J. Appl. Math. 62 (2001/02), no. 3, 777–800 (electronic). 104 [65] , Relative dispersion in 2D stochastic flows, J. Turbul. 6 (2005), Paper 4, 19 pp. (electronic). [66] L. I. Piterbarg and B. L. Rozovskii, On asymptotic problems of parameter esti- mation in stochastic PDE’s: discrete time sampling, Math. Methods Statist. 6 (1997), no. 2, 200–223. [67] G. P´ olya and G. Szeg˝ o, Problems and theorems in analysis. I, Classics in Math- ematics, Springer-Verlag, Berlin, 1998, Series, integral calculus, theory of func- tions, Translated from the German by Dorothee Aeppli, Reprint of the 1978 Eng- lish translation. [68] A. G. Ramm, Inverse problems, Mathematical and analytical techniques with applications to engineering, Springer, 2005. [69] B. L. Rozovskii, Stochastic partial differential equations, Mat. Sb. (N.S.) 96(138) (1975), 314–341, 344. [70] , Stochastic evolution systems, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer Academic Publishers Group, Dordrecht, 1990, Linear theory and applications to nonlinear filtering. [71] P. Santa-Clara and D. Sornette, The dynamics of the forward interest rate curve with stocahstic string shocks, The review of financial studies 14 (Spring 2001), no. 1, 149–185. [72] P. Sarnak, Spectra and eigenfunctions of Laplacians, Partial differential equations and their applications (Toronto, ON, 1995), CRM Proc. Lecture Notes, vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 261–276. [73] A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), no. 3, 669–682. [74] S. E. Serrano and T. E. Unny, Random evolution equations in hydrology, Appl. Math. Comput. 38 (1990), no. 3, 201–226. [75] A. N. Shiryaev, Probability, second ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. [76] M. A. Shubin, Pseudodifferential operators and spectral theory, second ed., Springer-Verlag, Berlin, 2001. [77] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. 105 [78] H. Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkh¨ auser Verlag, Basel, 1983. [79] J. B. Walsh, An introduction to stochastic partial differential equations, ´ Ecole d’´ et´ e de probabilit´ es de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. [80] M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeit- stheorie und Verw. Gebiete 11 (1969), 230–243. 106
Abstract (if available)
Abstract
In this work we discuss two problems related to stochastic partial differential equations (SPDEs): analytical properties of solutions and parameter estimation for SPDE's.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Asymptotic problems in stochastic partial differential equations: a Wiener chaos approach
PDF
Statistical inference for second order ordinary differential equation driven by additive Gaussian white noise
PDF
Statistical inference of stochastic differential equations driven by Gaussian noise
PDF
Statistical inference for stochastic hyperbolic equations
PDF
On the non-degenerate parabolic Kolmogorov integro-differential equation and its applications
PDF
Second order in time stochastic evolution equations and Wiener chaos approach
PDF
On spectral approximations of stochastic partial differential equations driven by Poisson noise
PDF
Time-homogeneous parabolic Anderson model
PDF
Parameter estimation in second-order stochastic differential equations
PDF
Tamed and truncated numerical methods for stochastic differential equations
PDF
A fully discrete approach for estimating local volatility in a generalized Black-Scholes setting
PDF
Numerical weak approximation of stochastic differential equations driven by Levy processes
PDF
On stochastic integro-differential equations
PDF
Optimal and exact control of evolution equations
PDF
Optimizing statistical decisions by adding noise
PDF
Linear filtering and estimation in conditionally Gaussian multi-channel models
PDF
Covariance modeling and estimation for distributed parameter systems and their approximations
PDF
Equilibrium model of limit order book and optimal execution problem
PDF
Certain regularity problems in fluid dynamics
PDF
Large deviations rates in a Gaussian setting and related topics
Asset Metadata
Creator
Cialenco, Igor
(author)
Core Title
Regularity of solutions and parameter estimation for SPDE's with space-time white noise
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Applied Mathematics
Defense Date
12/15/2006
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Holder continuity of solutions of SPDE,MLE of the drift term,model of forward rates,OAI-PMH Harvest,regularity of solutions,statistical inference for SPDE,stocahstic PDE
Language
English
Advisor
Lototsky, Sergey V. (
committee chair
), Mikulevicius, Remigijus (
committee member
), Protopapadakis, Aris (
committee member
)
Creator Email
cialenco@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m312
Unique identifier
UC1122431
Identifier
etd-Cialenco-20070307 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-319649 (legacy record id),usctheses-m312 (legacy record id)
Legacy Identifier
etd-Cialenco-20070307.pdf
Dmrecord
319649
Document Type
Dissertation
Rights
Cialenco, Igor
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
Holder continuity of solutions of SPDE
MLE of the drift term
model of forward rates
regularity of solutions
statistical inference for SPDE
stocahstic PDE