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Statistical inference for stochastic hyperbolic equations

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Statistical inference for stochastic hyperbolic equations

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Content STATISTICAL INFERENCE FOR STOCHASTIC HYPERBOLIC EQUATIONS by Wei Liu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) December 2010 Copyright 2010 Wei Liu Dedication To my parents. ii Acknowledgements First, I would like to thank my advisor Prof. Sergey Lototsky, who seems to unerring know when and where I need encouragement most. He introduced me into this interesting research filed and taught me how to ask questions and how to solve the problems. Without the techniques or theories that he taught me, without his guidance on research and writing, it is impossible for me to finish this dissertation. His passion for science always encourages me to explore more while his great personality makes it more enjoyable - he is always one of the best mathematicians that I want to work with. Especially, I appreciate the momentum and degrees of freedom that he gave me. I am thankful to my father for the intense inspiration he has always been for me throughout my life. In addition, I would like to thank my mother for her advice, in my life and study. In addition, I want to thank my significant other, Charlie Xia, for being there for me through the inevitable ups and down of a problem. My deepest appreciation to the rest of my thesis committee: Cheng Hsiao, who taught me the knowledge besides mathematics. Jianfeng Zhang, who gave insightful comments and hard questions. Igor Kukavica, for the numerous talks and invaluable advice on plan- ning. Remi Mikulevicius, for his constant guidance and support. iii I am also greatly indebted to many professors in the past: Larry Goldtein for getting me interested in probability and statistics. Chunming Wang, for introducing me into the numerical solutions for partial differential equations. Michael Magill, for his significant assistance in offering valuable advice and getting started on mathematical finance. Last but not the least, let me also say ”thank you” to my friends in the mathematics department at USC for helping me out of trouble and hanging out. iv Table of Contents Dedication ii Acknowledgements iii List of Tables vii List of Figures viii Abstract ix Chapter 1 Introduction 1 1.1 Motivation: The one-dimensional stochastic wave equation. 1 1.2 The Statistical Estimation 7 1.3 Definitions and Notations 14 1.4 Theorems 17 Chapter 2 Second Order in Time Stochastic Ordinary Differential Equations 28 2.1 Introduction 28 2.2 Analysis of Ordinary Differential Equation 30 2.2.1 b=0 30 2.2.2 b 2 ·a 2 37 2.2.3 b 2 >a 2 43 Chapter 3 Stochastic Partial Differential Equations 46 3.1 Introduction 46 3.2 The setting 48 3.3 Diagonalizable Stochastic Hyperbolic Equations 56 v Chapter 4 The Estimation of Stochastic Hyperbolic Equations 63 4.1 Statistical models 63 4.2 Analysis of The Stochastic Wave Equation on The Interval 65 4.2.1 Estimateµ 1 , assuming thatµ 2 is known 76 4.2.2 Estimateµ 2 , assuming thatµ 1 is known 82 4.2.3 Estimateµ 1 andµ 2 simultaneously. 84 4.3 Parameter Estimation in General Stochastic Hyperbolic Equations 88 4.3.1 Analysis of Estimators: Algebraic Case 88 4.3.2 Analysis of Estimators: General Case 112 Chapter 5 Other Statistical Inference for Stochastic Hyperbolic Equations 118 Bibliography 120 vi List of Tables 4.1 Asymptotes for equations (4.3.59), (4.3.60) and (4.3.61). 111 4.2 Asymptotes for equations (4.3.62), (4.3.63) and (4.3.64). 112 vii List of Figures 4.1 Estimates vs true values whenµ 1 =1 andµ 2 =¡1. 87 4.2 Estimates vs true values whenµ 1 =1 andµ 2 =1. 88 4.3 Estimates vs true values whenµ 1 =1 andµ 2 =0. 89 viii Abstract A parameter estimation problem is considered for a stochastic wave equation and a lin- ear stochastic hyperbolic driven by additive space-time Gaussian white noise. The damp- ing/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymp- totic properties of the estimators are studied as the number of the Fourier coefficients increases, while the observation time and noise intensity are fixed. ix Chapter 1 Introduction 1.1 Motivation: The one-dimensional stochastic wave equation. Consider the following stochastic equation u tt =µ 1 u xx +µ 2 u t + _ W(t;x); 0<t·T;x2(0;¼); (1.1.1) with zero initial and boundary conditions, wherejµ 1 j¸ 1 andjµ 2 j· 1 are unknown real numbers and _ W(t;x) is the noise term. With precise definitions to come later, at this point we interpret _ W(t;x) as a formal sum _ W(t;x)= X k¸1 h k (x) _ w k (t); 1 where h k (x) = p 2=¼sinkx, k ¸ 1, and w k (t) are independent standard Brownian motions. We look for the solution of(1:1) as a Fourier series u(t;x)= X k¸1 u k (t)h k (x): Denotedu(t;x)=v(t;x)dt, thenv(t;x) can also be written as v(t;x)= X k¸1 v k (t)h k (x) where v k (t;x) = du k (t;x)=dt: Substitution of this series in (1:1) suggests that each v k (t;x) should satisfy dv k (t)= ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ dt+dw k (t); 0<t·T; (1.1.2) with initial condition u k (0) = 0 and v k (0) = 0. If the trajectories of u k (t) and v k (t) are observed for a fixedk and all 0 < t < T , then the following scalar parameter estimation problems can be stated: 1. estimateµ 1 , assuming thatµ 2 is known; 2. estimateµ 2 , assuming thatµ 1 is known; 3. estimateµ 1 andµ 2 simultaneously. 2 The maximum likelihood estimator of µ (µ 1 or µ 2 subject to estimation) based on this observation for these three cases are: ^ µ (1) = R T 0 ¡k 2 u k dv k + R T 0 µ 2 k 2 v k u k dt R T 0 k 4 u 2 k dt ; (1.1.3) ^ µ (2) = R T 0 v k dv k + R T 0 µ 1 k 2 v k u k dt R T 0 v 2 k dt ; (1.1.4) ^ µ (3) 1 = R T 0 ¡k 2 u k dv k R T 0 v 2 k dt+ R T 0 v k dv k R T 0 k 2 v k u k dt R T 0 k 4 u 2 k dt R T 0 v 2 k dt¡ ¡R T 0 k 2 v k u k dt ¢ 2 ; (1.1.5) ^ µ (3) 2 = R T 0 ¡k 2 u k dv k R T 0 k 2 v k u k (t)dt+ R T 0 v k dv k R T 0 k 4 u 2 k dt R T 0 k 4 u 2 k dt R T 0 v 2 k dt¡ ¡R T 0 k 2 v k u k dt ¢ 2 ; (1.1.6) It was shown in [37, 38, 39] for parabolic equations that, for fixed T , the MLE for the parameter is super-efficient (i.e. µ can be reconstructed ”exactly” from measure- ments of u(t;x) on (0;T]£ [0;¼]). So we consider a two N-dimensional projection u N = (u 1 (t);¢¢¢ ;u N (t)), v N = (v 1 (t);¢¢¢ ;v N (t)) of the solution to (1.1.2). The number N is a finite number of Fourier coefficients of u(t) and v(t) which are used to approximate MLE ^ µ N . We can construct a sequence of maximum likelihood estimators ^ µ N based on the information about the fields of u(t) and v(t). It can be shown that this sequence converges to µ with probability one as N ! 1 or equivalently, as the amount of information increases under certain circumstances. To be more precise, the consistency is studied by investigating the asymptotic behavior of the Fisher’s information. 3 Let us now assume that the trajectories ofu k (t) andv k (t) are observed for all0<t< T and allk =1;:::;N, and combine the estimators (1.1.3), (1.1.4), (1.1.5) and (1.1.6) for differentk as follows: ^ µ (1) N = P N k=1 ¡R T 0 ¡k 2 u k dv k + R T 0 µ 2 k 2 v k u k dt ¢ P N k=1 R T 0 k 4 u 2 k dt ; (1.1.7) ^ µ (2) N = P N k=1 ¡R T 0 v k dv k + R T 0 µ 1 k 2 v k u k dt ¢ P N k=1 R T 0 v 2 k dt : (1.1.8) ^ µ (3) 1;N = ³ P N k=1 R T 0 ¡k 2 u k dv k ´³ P N k=1 R T 0 v 2 k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 + ³ P N k=1 R T 0 v k dv k ´³ P N k=1 R T 0 k 2 v k u k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 (1.1.9) ^ µ (3) 2;N = ³ P N k=1 R T 0 ¡k 2 u k dv k ´³ P N k=1 R T 0 k 2 v k u k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 + ³ P N k=1 R T 0 v k dv k ´³ P N k=1 R T 0 k 4 u 2 k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 (1.1.10) ^ µ (1) N , ^ µ (2) N , ^ µ (3) 1;N and ^ µ (3) 2;N are in fact the maximum likelihood estimators of µ 1 and µ 2 respectively based on the observationsv k (t),k =1;:::;N,0<t<T . (See, section 4.2) It then follows from (1.1.7)-(1.1.10) that 4 ^ µ (1) N ¡µ 1 =¡ P N k=1 k 2 R T 0 u k (t)dw k (t) P N k=1 k 4 R T 0 u 2 k (t)dt ; (1.1.11) ^ µ (2) N ¡µ 2 = P N k=1 R T 0 v k (t)dw k (t) P N k=1 R T 0 v 2 k (t)dt ; (1.1.12) and ^ µ (3) 1;N ¡µ 1 = ³ P N k=1 R T 0 k 2 u k dw k ´³ P N k=1 R T 0 v 2 k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 + ³ P N k=1 R T 0 v k dw k ´³ P N k=1 R T 0 k 2 v k u k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 ; (1.1.13) ^ µ (3) 2;N ¡µ 2 = ³ P N k=1 R T 0 k 2 u k dw k ´³ P N k=1 R T 0 k 2 v k u k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 + ³ P N k=1 R T 0 v k dw k ´³ P N k=1 R T 0 k 4 u 2 k dt ´ ³ P N k=1 R T 0 k 4 u 2 k dt ´³ P N k=1 R T 0 v 2 k dt ´ ¡ ¡P N k=1 R T 0 k 2 v k u k dt ¢ 2 : (1.1.14) Note that both the top and the bottom of the fraction on the right-hand side of (1.1.11)- (1.1.14) are sums of independent random variables, and the analysis of the properties of the estimators ^ µ (1) N , ^ µ (2) N , ^ µ (3) 1;N and ^ µ (3) 2;N is thus reduced to the study of these sums. 5 1. µ 2 is fixed and µ 1 is the parameter subject to estimation: by direct computation, if µ 2 =0, asN !1, E Z T 0 u 2 k (t)dt» T 2 4k 2 µ 1 ; Ek 4 Z T 0 u 2 k (t)dt» T 2 k 2 4µ 1 ; (1.1.15) where notationa N » b N meanlim N!1 (a N =b N ) = 1. Ifµ 2 6= 0, define2b =¡µ 2 , a 2 =¡µ 1 k 2 andl = p a 2 ¡b 2 . a andl are dependent onk, for simplicity, we omit k in these expressions. E Z T 0 u 2 k dt» e ¡2bT +2bT ¡1 8b 2 l 2 ; (1.1.16) Therefore, asN !1, Ek 4 Z T 0 u 2 k (t)dt» k 4 (e ¡2bT +2bT ¡1) 8b 2 l 2 ; (1.1.17) SinceE R T 0 k 2 u k (t)dw k (t)=0, it is reasonable to conjecture that, ² lim N!1 ( ^ µ (1) N ¡µ 1 )=0 with probability one by the law of large numbers; ² by the central limit theorem, the random variablesN 3=2 ( ^ µ (1) N ¡µ 1 ) converge in distribution to a zero-mean Gaussian random variable. 2. µ 1 is fixed andµ 2 is the parameter subject to estimation, asN !1, E Z T 0 v 2 k (t)dt» e ¡2bT +2bT ¡1 8b 2 : (1.1.18) We would be able to get similar results, 6 ² lim N!1 ( ^ µ (2) N ¡µ 2 )=0 with probability one; ² the random variables N 1=2 ( ^ µ (2) N ¡µ 2 ) converge in distribution to a zero-mean Gaussian random variable. 3. Estimation bothµ 1 andµ 2 , then we can show that lim N!1 ^ µ (3) 1;N =µ 1 ; lim N!1 ^ µ (3) 2;N =µ 2 (1.1.19) with probability one. ^ µ (3) 1;N converges toµ 1 at a rate ofN 3=2 and ^ µ (3) 2;N converges toµ 1 at a rate ofN 1=2 . The proof for the consistency depends on the properties of the second-order Ornstein- Uhlenbeck processes which will be discussed in section 1.3. In the rest of the introduction, we consider how (1.1.1) fits in the general framework of statistical estimation. 1.2 The Statistical Estimation Parameter estimation for ODEs has received a fair amount of attention in the last cen- tury with well developed methodology and substantial wealth of far reaching results, see e.g. [40, 55, 58]. Lately there has been a growing interest in extending the methods of this theory to statistical estimation for random fields, in particular, random fields driven by stochastic partial differential equations (SPDEs), see e.g. [2, 5, 41, 42, 78]. Parameter estimation is a particular case of inverse problem that arises when the solu- tion of a certain equation is observed and conclusions must be made about the coefficients 7 of the equation. In the deterministic setting, numerous examples of such problems in ecol- ogy, material sciences, biology, etc. are given in the book by Banks and Kunisch [7]. The stochastic term is usually introduced in the equation to take into account those components of the model that cannot be described exactly. (see e.g. [78]). To be more specific, stochas- tic hyperbolic equations can be used in physical and biological problems. Notable exam- ples are turbulent flow in fluid dynamics, diffusion and waves in random media [9, 13]. Besides, as a mathematical subject, they pose many interesting and challenging problems in stochastic analysis. In the classical statistical estimation problem, the starting point is a familyP µ of prob- ability measures depending on the parameter µ. Each P µ is the distribution of a random element. It is assumed that a realization of one random element corresponding to one value µ ¤ µ of the parameter is observed, and the objective is to estimate the value of this parameter from the observations. One approach is to select the valueµ corresponding to the random element that is most likely to produce observations. When observations are coming from an SODE, letB T as the Borel¾-algebra on the spaceC([0;T]). DenoteP µ to be the measure onB T generated by the process v , i.e., P µ := P(v 2 A) for all A 2 B T . Each P µ is the distribution of a random element. For a fixed number µ 0 2 £, which represents the true value of the parameterµ subject to estimation, it is assumed that the solution corresponding to one value µ = µ 0 of the parameter is observed and estimate the values of the parameter from the path of observations. We will consider the regular estimation problem whereP µ should satisfy the following two conditions: 8 ² there exists a new probability measure Q such that all measures P µ are absolutely continuous with Radon-Nikodym derivative. ² the ratiodP µ =dQ has the local asymptotic normality property. In the regular models, there exists a probability measureQ such that bothP µ andP µ 0 are absolutely continuous with respect to Q. The ratio or density dP µ =dQ or dP µ 0 =dQ can be derived from the Girsanov theorem and therefore dP µ =dP µ 0 which is called the likelihood ratio can be obtained by direct computation. An estimator ^ µ of the unknown parameterµ can be constructed by differentiating the density with respect to the parameter µ and it is called the maximum likelihood estimator (MLE). Since, as a rule, ^ µ 6= µ 0 , the consistency of the estimator is studied. That is, the convergency of ^ µ to µ 0 as more and more information becomes available. There are two ways to increase the amount of the information: (a) increasing the sample size, for example, the observation time interval (large sample asymptotic); (b) reducing the amplitude of noise (sample noise asymptotic). If the measureP µ are mutually singular for different values ofµ, the model is called sin- gular, and the value of the parameter can be determined ”exactly”. In reality, the singular model is often approximated by a sequence of regular models and the unknown parame- ter is usually calculated as the limit of the corresponding maximum likelihood estimators. For parabolic equations driven by additive space-time white noise, this approach was first suggested by Huebner et al. [38] and was further investigated by Huebner and Rozovskii [39], where a necessary and sufficient condition for the convergence of the estimators was stated in terms of the orders of the operators in the equation. When the observations are finite-dimensional diffusions, the necessary and sufficient conditions for absolute continuity of the corresponding measures are well-known (see, for 9 example, Lipster and Shiryaev [60, Chapter 7] ). Many of the results have been extended to infinite dimensions by Kozlov [50, 51], Loges [49, 61], Mikulevicius and Rozovskii [67, 68] and others. For linear equations, such as (1.1.1), whose solutions are Gaussian processes, there is another useful result, originally discovered independently by Feldman [23] and Hajek [28, 33]; Two Gaussian measures are either mutually absolutely continuous or mutually singular. In particular, the measures generated byv(t;x) = P k¸1 v k (t)h k (x) in a suitable Hilbert space are mutually singular for different values ofµ, which allows us to get the exact value of the parameterµ 1 orµ 2 as µ (1) = lim N!+1 P N k=1 ¡R T 0 ¡k 2 u k dv k + R T 0 µ 2 k 2 v k (t)u k (t)dt ¢ lim N!+1 P N k=1 R T 0 k 4 u 2 k dt ; and µ (2) = lim N!+1 P N k=1 ¡R T 0 v k dv k + R T 0 µ 1 k 2 v k (t)u k (t)dt ¢ lim N!+1 P N k=1 R T 0 v 2 k dt : And similarly for ^ µ (3) 1;N and ^ µ (3) 2;N . Since the limits in the top and the bottom of these two expressions are infinite, they must be approximated by ^ µ (1) N and ^ µ (2) N from (1.1.3) and (1.1.4). The situation is somewhat similar to the problem of estimating the diffusion coef- ficient in a finite-dimensional diffusion, where the exact value is known from the quadratic variation but is computed approximately using time discretization (see below). Our goal is to study the diagonalizable stochastic partial differential equations (SPDEs) depending on one or two parameters and investigate the statistical properties of the solu- tions. In particular, we introduce and investigate asymptotic properties of the maximum likelihood estimators for parameters in stochastic hyperbolic differential equations. 10 Assume that the measurements are given in the spectral form, i.e., the white noiseW can be written as a formal series W(t)= X k¸1 w k (t)h k : (1.2.1) wherefw k g 1 k=1 are independent 1-dimensional standard Brownian motions andfh k g 1 k=1 is an orthonormal basis in a Hilbert spaceH.fh k g 1 k=1 consists of a system of eigenfunction of the differential operatorsA,A 0 ,A 1 ,B,B 0 andB 1 . The existence and uniqueness of the solution to (1.1.1) is shown in Section 4.2. v(t;x) = P k¸1 v k (t)h k (x) where the series is convergent under some conditions in terms of the eigenvalues of A, A 0 , A 1 , B, B 0 and B 1 . An N-dimensional projection v N (t)= P N k=1 v k (t)h k (x) is then aN-dimensional process with independent components. The asymptotic properties of the MLE are studied as the dimension of those projections increases while the lengthT of the observation interval is fixed. Moreover, in some cases, the MLE is not a consistent estimator as the dimension of the projection increases for fixed T , it is a consistent estimator as the time intervalT !1 by Ergodic theorem which will be explained explicitly later. The following is the main result of this dissertation. Consider the following stochastic hyperbolic differential equations: 8 > > < > > : u tt +(A 0 +µ 1 A 1 )u(t)=(B 0 +µ 2 B 1 )u 0 (t)+ _ W;t2(0;T] u(0)=0; u t (0)=0: (1.2.2) 11 with zero initial and boundary conditions where µ 1 and µ 2 are parameters subject to esti- mation. DenoteA=A 0 +µ 1 A 1 andB =B 0 +µ 2 B 1 . Assume that this equation is diagonaliz- able, that is, the operatorsA 0 ,A 1 ,B 0 andB 1 have a common system of eigenfunctions: Ah k =¸ k h k ; B =¹ k h k ; A 0 h k =· k h k ; B 0 =½ k h k ; A 1 h k =¿ k h k ; B 1 =º k h k The sequencefa n ; n¸1g of positive numbers is calledslowly increasing if lim n!1 P n k=1 a 2 n ³ P n k=1 a k ´ 2 =0: (1.2.3) Define the functionsM andV are continuous and positive onR, and M(x)» 8 > > < > > : (2jxj) ¡1 ; x!¡1; 2(2x) ¡2 e x ; x!+1; V(x)» 8 > > < > > : 4(2jxj) ¡3 ; x!¡1; 4(2x) ¡4 e 2x ; x!+1: Condition 1. The sequencef¿ 2 k M ¡ T¹ k (µ 2 ) ¢ =¸ k (µ 1 ); k¸1g is slowly increasing, and Condition 2. The sequencefº 2 k M ¡ T¹ k (µ 2 ) ¢ ; k¸1g is slowly increasing. 12 1.2.1 Theorem. Assume that equation u tt +(A 0 +µ 1 A 1 )u(t)=(B 0 +µ 2 B 1 )u t (t)+ _ W(t); 0<t·T; (1.2.4) is hyperbolic. 1. If Condition 1 holds, then lim N!1 ^ µ 1;N =µ 1 inprobability; (1.2.5) lim N!1 p ª 1;N ³ ^ µ 1;N ¡µ 1 ´ =» 1 indistribution; (1.2.6) where» 1 is a standard Gaussian random variable. 2. If Condition 2 holds then lim N!1 ^ µ 2;N =µ 2 inprobability; (1.2.7) lim N!1 p ª 2;N ³ ^ µ 2;N ¡µ 2 ´ =» 2 indistribution; (1.2.8) where» 2 is a standard Gaussian random variable. 3. If both Conditions 1 and 2 hold, then the random variables» 1 ;» 2 are independent. The object of first chapter is to study the asymptotic properties of the second order stochastic ordinary differential equation which is the prerequisite of those investigations of the SPDEs. To simplify our argument, we take the zero initial conditions. The fol- lowing sections of chapter one summarize the main technical tools necessary to study the estimation problems for stochastic hyperbolic equations: Ornstein-Uhlenbeck process 13 and its properties, Law of Large Numbers (LLN) and Central Limit Theorem (CLT) for independent but not identically distributed random variables, the Girsanov’s thoerem. The last section of Chapter one deals with stochastic ordinary differential equations and gives some significant theorems in the asymptotic properties for different situations which will be used to estimate the parameter. In Chapter two, we study the corresponding second order in time stochastic hyperbolic equations. In Chapter three, we discuss the estimation of stochastic hyperbolic equation depending on parameter(s) based on the information in the previous two chapters. In Chapter four, we first consider the equation is diagonalizable and hyperbolic and that the operators are positive-definite elliptic self-adjoint differential or pseudo-differential on a smooth bounded domain inR d with suitable boundary condi- tions or on a smooth compactd-dimensional manifold, we derive the conditions for which the estimators are consistent estimator. Also, we consider a more general case and we find out that an additional condition is needed for the consistency of the estimators. In Chapter five, we consider a multiplicative noise and consider the consistency while the time goes to infinity. 1.3 Definitions and Notations In this section, we introduce the notations and definitions which will be used through- out the presentation below. 1.3.1 Definition. A stochastic process is a collection of random variables fX t g t2T 14 defined on a probability space (;F;fF t g t¸0 ;P) and assuming values in an n- dimensional Euclidean spaceR n . 1.3.2 Definition. OrthonormalBasis. A subsetfh 1 ;h 2 ;:::;h k g of a vector spaceV , with the inner producth;i, is called orthonormal if 1. hh i ;h j i=0 wheni6=j. That is, the vectors are mutually perpendicular; 2. h k ;k¸1 are all required to have length one. That is,hh k ;h k i=1: For example, the setfe 1 = (1;0;0);e 2 = (0;1;0);e 3 = (0;0;1)g (the standard basis) forms an orthonormal basis ofR 3 . 1.3.3 Remark. 1. We assumeF = (;F;fF t g t¸0 ;P) is a complete probability basis andfF t g t¸0 are right-continuous. 2. We assume the stochastic basisF is large enough to support the independent stan- dard Brownian motions. 3. The standard basis of the n-dimensional Euclidean space R n is an example of orthonormal (and ordered) basis. 1.3.4 Definition. Let F = (;F;fF t g t¸0 ;P) be a complete probability basis. The Ornstein¡Uhlenbeck (OU)process X(t) is a stochastic process that satisfies the following stochastic differential equation dX(t)=¡¹X(t)dt+¾dW 15 with initial conditionX(0) = 0 whereW = W(t) is a standard Brownian motion on the given probability space(;F;fF t g t¸0 ;P) fort2[0;1). The Ornstein-Uhlenbeck processX is called stable is¹>0. The Ornstein-Uhlenbeck process is widely used for modeling biological processes such as neuronal response, and in mathematical finance, the modelling of the dynamics of interest rates and volatilities of asset prices. 1.3.5 Definition. Let (;F;P) be a probability space with a complete filtrationF = fF t g 0·t·T andG is a bounded domain inR d , for everyt2[0;T],W(t;x) is a Brownian motion or a Wiener process onG such that forÁ;Ã2C 1 0 (G),kÁk ¡1 L 2 (G) hW(t;¢);Á(¢)i is a one dimensional Winer process and E(hW(s;¢);Á(¢)ihW(t;¢);Ã(¢)i)=(s^t)hÁ;Ãi L 2 (G) : This process is called a cylindrical Brownian motion. Notation. 1. For two sequencesfa n ;n¸1g andfb n ;n¸1g,a n »b n means that lim n!1 a n b n =1 For example, P n k=1 n » n 2 =2 and P n k=1 n 2 » n 3 =3. Note that if a n » b n and P n a n diverges, then P n k=1 a k » P n k=1 b k ; 2. For two non-negative sequencesfa n ;n¸1g andfb n ;n¸1g, a n ./b n (1.3.1) 16 means that there exist positive numbers c 1 , c 2 such that c 1 · a n =b n · c 2 for all sufficiently largen; 3. For two non-negative sequencesfa n ;n¸1g andfb n ;n¸1g, a n ³b n (1.3.2) if lim k!1 a k b k =c for somec>0; 4. For a random variableX,EX andVarX denote the expectation and the variance ofX respectively. 5. f(n) = o(g(n)) means for allc > 0, there exists somek > 0 such that0· f(n) < cg(n) for alln¸k. Equivalently, lim n!1 f(n) g(n) =0: The value ofk must not depend onn, but may depend onc. As an example,3n+4 iso(n 2 ). 1.4 Theorems In the estimation analysis for the consistency, we need to use several theorems: Law of Large Numbers (LLN), the Central Limit Theorem for independent random variables and the absolute continuity of measures generated by processes of diffusion type. 17 1.4.1 Theorem. The Kolmogorov’s strong law of large numbers (LLN) LetÂ 1 ,Â 2 ,::: be a sequence of independent random variables withEjÂ i j<1 andc n , n¸ 1, an increasing sequence of positive real numbers such that lim n!1 c n = +1 . If for eachn, the variance ofÂ n is finite, and X n¸1 VarÂ n c 2 n <1; then the Kolmogorov’s strong law of large numbers holds. That is, lim n!1 P n k=1 (Â n ¡EÂ n ) c n =0: Proof. See, for example, Shiryaev [84, Theorem IV . 3.2] or Casella and L. Berger [11]. 1.4.2 Remark. The proof of the consistency relies on the LLN for independent but not identically distributed random variables. 1.4.3 Corollary. Let Â 1 , Â 2 ,::: be a positive sequence of independent random variables and X n¸1 VarÂ n ( P n k EÂ k ) 2 <1: Then lim n!1 P n k=1 Â k P n k=1 EÂ k =1 with probability one. Proof. Letc n = P n k=1 EÂ k , corollary follows from Theorem 1.4.1. 18 1.4.4 Theorem. Let Â k ; k ¸ 1; be independent random variables, each with zero mean and positive finite variance. If X k¸1 EÂ 2 k =+1; (1.4.1) then lim N!1 P N k=1 Â k P N k=1 EÂ 2 k =0withprobabilityone: (1.4.2) Proof. To prove (1.4.2), we take» n = Â n andb n = P n k=1 EÂ 2 k and apply Theorem 1.4.1; note that convergence of P n b ¡2 n EÂ 2 n follows from divergence of P k¸1 EÂ 2 k : X n EÂ 2 n b 2 n · X n µ 1 b n¡1 ¡ 1 b n ¶ : 1.4.5 Theorem. (A special case of Strong Law of Large Numbers) Let Â k , k ¸ 1, be independent random variables with the following properties: 1. EÂ k =0,EÂ 2 k >0, for allk¸1; 2. There exists real numbersc 2 >0,®¸¡1, such that lim k!1 k ¡® EÂ 2 k =c 2 : Then lim N!1 P N k=1 Â k P N k=1 EÂ 2 k =0 with probability one: 19 Proof. This is a particular case of Kolmogorove’s strong law of large numbers; see, for example, [84, Theorem IV .3.2]. 1.4.6 Corollary. If the above conditions are satisfied and additionally,EÂ 4 k ·C 1 ¡ EÂ 2 k ¢ 2 fork¸1 andC 1 ¸1 independent ofk, Then lim N!1 P N k=1 Â 2 k P N k=1 EÂ 2 k =1 with probability one: Proof. It follows from Theorem 1.4.5 and Theorem 1.4.1. 1.4.7 Theorem. A special case of Weak law of large numbers Let Â k , k ¸ 1, be independent random variables, each with zero mean and positive finite variance. If P k¸1 EÂ 2 k =+1 and in addition, assume that EÂ 4 k ·c 1 ¡ EÂ 2 k ¢ 2 for allk¸1, withc 1 >0 independent ofk. Then lim n!1 P n k=1 ¡ EÂ 2 k ¢ 2 ¡P n k=1 EÂ 2 k ¢ 2 =0 (1.4.3) implies lim N!1 P N k=1 Â 2 k P N k=1 EÂ 2 k =1 in probability: (1.4.4) Proof. follows from and Chebyshev’s inequality. 1.4.8 Theorem. Lindeberg’s Central Limit Theorem 20 LetÂ 1 ,Â 2 ,::: be a positive sequence of independent random variables withEÂ n =¹ n , VarÂ n = ¾ 2 n < 1 and distribution function F n = F n (x) for n ¸ 1. Denote S n = P n k=1 Â k andD 2 n = P n k=1 ¾ 2 k : Suppose that the Lindeberg condition is satisfied: for every²>0; (L) n X k=1 Z fx:jx¡¹ k j¸²Dng (x¡¹ k ) 2 dF k (x)!0; as n!1: Then the normalized partial sums(S n ¡ES n )=D n converges in distribution to a normal random variable with zero mean and unit variance: lim n!1 S n ¡ES n D n =N(0;1) in distribution. Proof. See, for example, Shiryaev [84, Theorem III.4.1]. 1.4.9 Corollary. Let Â 1 , Â 2 ,::: be a positive sequence of independent random variables withEÂ n = 0, VarÂ n = ¾ 2 n <1 and there exist postive constants C, C 1 and ® ·¡1, such that ¾ 2 n »Cn ® (1.4.5) EÂ 4 n ·C 1 ¾ 4 n : (1.4.6) Then the sequenceS n =D n converge in distribution to a standard normal random variable asn!1: lim n!1 S n D n =N(0;1): 21 Proof. We show the Lindeberg conditon is satisfied. That it, P n k=1 E ¡ Â 2 k I(jÂ k j>²D n ) ¢ D 2 n !0 as n!1 (1.4.7) for every²>0 whereI(A) is the indicator function for the setA. By Holder’s inequality, E ³ Â 2 k I(Â k )>²D n ´ · ³ EÂ 4 k ´ 1=2 ¢ ³ P(jÂ k j>²D n ) ´ 1=2 : (1.4.8) By the Chebyshev inequality, P(jÂ k j>²D n )· EÂ 4 k D 4 n ² 4 : (1.4.9) Therefore, condition (1.4.5) implies E ³ Â 2 k I(Â k )>²D n ´ · EÂ 4 k D 2 n ² 2 · C 1 ¾ 4 k ² 4 D 4 n : The Lindeberg condtion follows from lim n!1 P n k=1 k 2® ( P n k=1 k ® ) 2 =0: (1.4.10) 22 If¡1 · ® < ¡1=2, (1.4.10) is obvious because that the series on the top converges but the series on the bottom diverges. If®¸1=2, (1.4.10) follows from n X k=1 k ¯ » n 1+¯ 1+¯ for ¯ >¡1: 1.4.10 Theorem. Martingale Central Limit Theorem Let M n = M n (t), r ¸ 0, n ¸ 1 be a sequence of continuous square–integrable martingales with quadratic variationhM n i=hM n i(t): If, for someT >0, lim n!1 hM n i(T) EhM n i(T) =1 in probability; then lim n!1 hM n i(T) ¡ EhM n i(T) ¢ 1=2 d =N(0;1): Proof. IfX n andX are continuous square-integrable martingales such thatX is a Gaus- sian process andlim n!1 hX n i(T)=hXi(T) in probability, thenlim n!1 X n (T)=X(T) in distribution. It is the limit theorem for martingales. For the proof, see, for exam- ple, Jacod and Shiryave [43, Theorem VIII.4.17] or Lipster and Shiryaev [59, Theorem 5.5.4(II)]. The result follows if we take X n (t)= M n (t) ¡ EhM n i(T) ¢ 1=2 ; X(t)= w(t) p T ; wherew is a standard Brownian motion. 23 1.4.11 Corollary. Letf k =f k (t)t¸0,k¸1 be continuous, square-integrable processes andw k =w k (t) be independent standard Brownian motions such that P n k=1 R T 0 f 2 k (t)dt P n k=1 E R T 0 f 2 k (t)dt =1 in probability; then lim n!1 P n k=1 R T 0 f k (t)dw k (t) ³ P n k=1 E R T 0 f 2 k (t)dt ´ 1=2 d =N(0;1): Proof. This corollary follows from Theorem 1.4.10 by taking M n (t)= n X k=1 Z t 0 f k (s)dw k (s): 1.4.12 Corollary. Let M i;n = M i;n (t), t ¸ 0, n ¸ 1, i = 1;2, be two sequences of continuous square-integrable martingales. If, for someT >0, lim n!1 hM i;n i(T) EhM i;n i(T) =1; i=1;2; in probability; and lim n!1 hM 1;n ;M 2;n i(T) ³ EhM 1;n i ´ 1=2 ³ EhM 2;n i(T) ´ 1=2 =0; in probability; then lim n!1 0 B @ M 1;n (T) ¡ EhM 1;n i ¢ ¡1=2 M 2;n (T) ¡ EhM 2;n i ¢ ¡1=2 1 C A d =N(0;I); 24 (in distribution), whereN(0;T) is a two-dimensional vector whose components are inde- pendent standard Gaussian random variables. Proof. Take X n (t)= 0 B @ X 1;n X 2;n ; 1 C A X(t)= 0 B @ w 1 (t)= p T w 2 (t)= p T; 1 C A where X i;n = M i;n (t) ¡ EhM i;n i ¢ 1=2 ; i=1;2: andw 1 ,w 2 are one-dimensional independent Brownian motions. 1.4.13 Corollary. Letf k =f i;k (t),t¸0,i=1;2,k¸1 be continuous, square-integrable processes and w k = w k (t) be independent one-dimensional standard Brownian motions such that lim N!1 P N k=1 R T 0 f 2 i;k (t)dt P N k=1 Ef 2 i;k (t)dt =1 in probability; lim N!1 P N k=1 R T 0 f 1;k (t)f 2;k (t)dt P N k=1 Ef 1;k (t)f 2;k dt =1 in probability; lim N!1 P N k=1 E R T 0 f 1;k (t)f 2;k (t)dt ¡P N k=1 Ef 2 1;k (t)dt ¢ 1=2 ¡P N k=1 Ef 2 2;k dt ¢ 1=2 =0: Define ´ i;N = P N k=1 R T 0 f i;k (t)dw k (t) ¡P N k=1 E R T 0 f i;k (t)dt ¢ 1=2 ; i=1;2 Then lim N!1 0 B @ ´ 1;N ´ 2;N 1 C A d =N(0;I) 25 whereN(0;I) is a two-dimensional vector whose components are independent standard Gaussian random variables. Proof. This follows from Corollary 1.4.12 by taking M i;n (t)= P n k=1 R t 0 f i;k (s)dw k (s) ¡ E R T 0 f 2 i;k (t)dt ¢ 1=2 : becauseEhM i;n i(T)=1 and hM 1;n ;M 2;n i(T)= P N k=1 R T 0 f 1;k (t)f 2;k (t)dt ¡P N k=1 Ef 2 1;k (t)dt ¢ 1=2 ¡P N k=1 Ef 2 2;k dt ¢ 1=2 : 1.4.14 Theorem. [Absolute Continuity of Measures Generated by the process of diffusion type] Let w = (w t ;F t ) be a Wiener process on a probability space (;F;P), ¹ W is the measure generated byw andX(t)2R be an Ito’s process of the diffusion type: dX(t)=a t (X)dt+dw(t); t·T; X 0 =0; wherea t (X) is a non-anticipative process ofX,T ·1 is a given constant andX 0 = 0. IfP ¡R T 0 a 2 t (X)dt<1 ¢ =1, d¹ X d¹ W (t;X)=exp ³ ¡ Z t 0 a s (X)dw(s)¡ 1 2 Z t 0 a 2 s (X)ds ´ ; 26 andX(t) is a Brownian motion with respect toQ, where the ratio M t =exp ³ ¡ Z t 0 a s (X)dw(s)¡ 1 2 Z t 0 a 2 s (X)ds ´ is a martingale with respect toF t andP. Proof. See, for example, Lipster and Shiryaev [60, Theorem 7.6]. 27 Chapter 2 Second Order in Time Stochastic Ordinary Differential Equations 2.1 Introduction The theory of Stochastic Ordinary Differential Equations (SODEs) is an active field of mathematical research. It was first developed by mathematicians as a tool for the explicit construction of the paths of diffusion processes for a given coefficients of drift and dif- fusion. Rooted in probability and measure theory, beginning with the fundamental work of Wiener, Kolmogorov, Levy and Ito, stochastic differential equations have intrinsic and deep connections and many applications in modeling a variety of random dynamic phe- nomena in the physical, biology, as well as engineering and social science. The problems arise for a variety of reasons, e.g., when system parameters are fluctuating, when fluctuat- ing external forces exist, and when a small set of variables are extracted from a larger set of variables. 28 Stochastic differential equations are often used in the cases where it is not obvious that a simple systematic/stochastic split exists. For example, they offer a convenient and tractable way of modelling the dynamics of many stochastic systems including financial markets, DNA dynamics and cellular protein motor models. One of the most important steps in the modelling process is that of parameter estimation. However, most data is observed at a discrete time frequencies whereas SODEs are almost surely continuous pro- cesses. This introduces ”discretisation-bias” into estimates which is difficult to eliminate. Thus, the parameter estimation for SODEs problem is non-trivial and is generating a great deal of research effort. Tsay [93] outlines many recently developed methods of moments (also see Gallant and Tauchen [30] and Galland and Long [29]). A second approach applies Markov Chain Monte Carlo methods to estimate the diffusion equation (See Eraker [22] and Gray [31]). Much research focuses on a third approach, based on maximum likeli- hood estimation (MLE) or quasi-MLE (See Lo [62] and the references therein, and Kessler [46]). In this section, we consider the following second order in time stochastic ordinary differential equations: 8 > > > > > > < > > > > > > : u 00 (t)=¡a 2 u(t)¡2bu 0 (t)+ _ w u(0;t)=0 u t (0;t)=0 (2.1.1) where a > 0, b 2 R and _ w is a one-dimensional standard Brownian motion. We consider the statistical properties of the solution to (2.1.1). 29 2.2 Analysis of Ordinary Differential Equation Consider the stochastic ordinary equation 8 > > > > > > < > > > > > > : u 00 (t)=¡a 2 u(t)¡2bu 0 (t)+ _ w u(0;t)=0 u t (0;t)=0 (2.2.1) 2.2.1 b=0 Ifb=0, then stochastic differential equation (2.2.1) becomes: u 00 (t)+a 2 u(t)= _ w (2.2.2) with zero initial values u(0) = 0 and u 0 (0) = 0. The characteristic polynomial F(r) = r 2 +a 2 has two complex conjugate rootsai and¡ai. The solution to (2.2.1) is u(t;a)= 1 a Z t 0 sina(t¡s)dw(s): (2.2.3) Note thata6=0. 2.2.1 Theorem. Fort2[0;T] anddu(t;a)=v(t;a)dt, fixedT >0. 1. Define the random variable X(a)= Z T 0 u 2 (t;a)dt; 30 Y(a)= Z T 0 v 2 (t;a)dt; Then lim a!+1 a 2 EX(a)= T 2 4 ; (2.2.4) lim a!+1 a 4 VarX(a)= T 4 24 ; (2.2.5) EX n (a)·C(n;T)a ¡2n ; (2.2.6) and EY(a)» T 2 4 ; (2.2.7) VarY(a)» T 4 24 ; a!1: (2.2.8) 2. Ifdu(t;a) = v(t;a)dt, in the space of continuous functions on[0;T], denoteP a T as the measure generated by the processv(t;a),0·t·T . Then the measuresP a T are mutually absolutely continuous for alla and dP a 2 T dP a 1 T ¡ v(¢;a 1 ) ¢ =exp ³ ¡(a 2 2 ¡a 2 1 ) Z T 0 u(t;a 1 )dv(t;a 1 )¡ a 4 2 ¡a 4 1 2 Z T 0 u 2 (t;a 1 )dt ´ : (2.2.9) In particular,P 0 T is the measure generated by the standard Browninan motion which is also called the Winer measure, and dP a 1 T dP 0 T ¡ v(¢;a 1 ) ¢ =exp ³ ¡a 2 1 Z T 0 u(t;a 1 )dv(t;a 1 )¡ a 4 1 2 Z T 0 u 2 (t;a 1 )dt ´ : (2.2.10) 31 Proof. 1. By direct computation and the Ito’s Isometry: Eu 2 (t;a)= 1 a 2 Z a 0 sin 2 a(t¡s)ds= t 2a 2 ¡ sin2at 4a 3 ; (2.2.11) Asa!1, the main term that determines the asymptotic ofEu 2 (t;a) isg(t;a) = t=2a 2 . We write Eu 2 (t;a)»g(t;a)= t 2a 2 ; i.e., lim a!1 Eu 2 (t;a) g(t;a) =1: so EX(a)= Z T 0 Eu 2 (t;a)dt» Z T 0 g(t;a)dt= T 2 4a 2 : (2.2.12) Equality (2.2.4) then follows from (2.2.12), asa!+1, a 2 EX(a)! T 2 4 : For (2.2.6), by Jensen’s inequality, ifÁ(x) is a convex function, ³ Z T 0 Á(t)dt ´ 2 = ³ Z T 0 T ¢Á(t)d( t T ) ´ 2 ·T n Z T 0 Á n (t)d( t T ) (2.2.13) and ifu(t;a) is a Gaussian random variable with zero mean, by induction, Eu 2n (t;a)= ³ n Y k=1 (2k¡1) ´ ¡ Eu 2 (t;a) ¢ n : (2.2.14) 32 f(x)=x 2 is a convex function, therefore, EX n (a)=E ¡ Z T 0 u 2 (t;a)dt ¢ n ·T n¡1 Z T 0 Eu 2n (t;a)dt =T n¡1 Z T 0 ¡ n Y k=1 (2k¡1) ¢¡ Eu 2 (t;a) ¢ n dt =C(n)T n¡1 Z T 0 ¡ Eu 2 (t;a) ¢ n dt ·C(T;n)a ¡2n ; (2.2.15) The last inequality follows from (2.2.11). For (2.2.5),VarX(a)=EX 2 (a)¡(EX(a)) 2 and(EX(a)) 2 follows from (2.2.12) directly. It is necessary to computeEX 2 (a). By the definition ofX(a), EX 2 (a)= ³ Z T 0 Eu 2 (t;a)dt ´ 2 = Z T 0 Z T 0 E ¡ u 2 (t;a)u 2 (s;a) ¢ dsdt =2 Z T 0 Z t 0 E ¡ u 2 (t;a)u 2 (s;a) ¢ dsdt: (2.2.16) Fort>s,u(t;a) andu(s;a) are joint Gaussian with zero mean. By the property of Ito’s integral, the covariance ofu(t;a) andu(s;a) is Cov ¡ u(t;a);u(s;a) ¢ =E ¡ u(t;a)u(s;a) ¢ = ascosa(s¡t)¡cosatsinat 2a 3 » scosa(t¡s) 2a 2 a!1: (2.2.17) 33 So the correlation ½(t;s)» ascos[a(t¡s)] ³ (at)(as) ´ 1=2 a!1: If two random variables® and¯ are bivariate normal with mean zero and unit vari- ance, by direct computation, E(® 2 ¯ 2 )= 1 2¼ p 1¡½ 2 Z +1 ¡1 Z +1 ¡1 x 2 y 2 exp ³ ¡ 1 2(1¡½ 2 ) (x 2 +y 2 ¡2½xy) ´ dxdy =1+2½ 2 : (2.2.18) Therefore, E ¡ u 2 (t;a)u 2 (s;a) ¢ » ts 4a 4 ¡ 1+2½ 2 (t;s) ¢ =g(t)g(s)+2Cov 2 ¡ u(t;a);u(s;a) ¢ a!1: As a result, by (2.2.18), EX 2 (a)»2 Z T 0 Z t 0 g(t)g(s)dsdt+ 1 a 4 Z T 0 Z t 0 s 2 cos 2 a(t¡s)dsdt+o(a ¡4 ) = ¡ T 4 16a 4 ¢ 2 + T 4 24a 4 +o(a ¡4 )= ¡ EX(a) ¢ 2 + T 4 24a 4 +o(a ¡4 ) a!1: (2.2.19) So VarX(a)» T 4 24a 4 a!1: 34 Similarly,v(t;a)= R t 0 cosa(t¡r)dw(r). Asa!1, direct computation gives us Ev 2 (t;a)= 2at+sinat 4a » t 2 ; EY(a)» T 2 4 a!1; (2.2.20) andVarY(a)»T 4 =24 a!1. 2. We can write dv(t;a)=F(v)dt+dw(t): whereF(v) =¡a 2 R t 0 v(s)ds is a non-anticipating function ofv. Thus, the process v(t : a) is a process of diffusion type in the sense of Lipster and Shiryaev [60, Definiton 4.3.7]. AsEu 2 (t;a) andEv 2 (t;a) are both finite, the condition P n ¡ a 2 1 u(t;a 1 ) ¢ 2 dt<1 o =P n ¡ a 2 2 u(t;a 2 ) ¢ 2 dt<1 o =1 (2.2.21) is satisfied. By Theorem 7.19 in Lipster and Shiryaev [59], the measures generated byv(t;a 1 ) andv(t;a 2 ) are absolutely continuous with density dP a 2 T dP a 1 T ³ v(¢;a 1 ) ´ =exp Ã Z T 0 (a 2 1 ¡a 2 2 )u(t;a 1 )dv(t;a 1 ) ¡ 1 2 Z T 0 ³ (a 2 1 ¡a 2 2 )u(t;a 1 ) ´ 2 dt ! : (2.2.22) 35 It follows that dP a 2 T dP a 1 T ³ v(¢;a 1 ) ´ =exp Ã Z T 0 (a 2 1 ¡a 2 2 )u(t;a 1 )dw(t) ¡ 1 2 Z T 0 ³ (a 2 1 ¡a 2 2 )u(t;a 1 )dv(t;a 1 ) ´ 2 dt ! : In particular, E ³ dP a 2 T dP a 1 T ¡ v(¢;a 1 ) ¢ ´ =1: By the Girsanov Theorem (see, for example, Oksendal [74, Theorem 8.6.4]), there exists a probability measureQ which is generated by the standard Brownian motion such that ² the measureP a 1 T is absolutely continuous with respect toQ; ² the ratio dP a 1 T dQ ¡ v(¢;a 1 ) ¢ =exp Z T 0 ³ a 2 1 u(t;a 1 )dw(t)¡ 1 2 Z T 0 ³ a 2 1 u(t;a 1 ) ´ 2 dt =exp Z T 0 ³ a 2 1 u(t;a 1 ) ´ dv(t;a 1 )+ 1 2 Z T 0 ³ a 2 1 u(t;a 1 ) ´ 2 dt; whereQ is the probability measure on (;F T ) such that the ratio is a martingale with respect toF T and for0· t· T ,v(t;a) is a standard Browinan motion under the new probability measureQ. 36 Clearly, dQ dP a T ¡ v(¢;a) ¢ = ³ dP a T dQ ¡ v(¢;a) ¢ ´ ¡1 ; and E Q exp Ã ¡ Z T 0 ³ a 2 1 u(t;a 1 ) ´ dv(t;a 1 )¡ 1 2 Z T 0 ³ a 2 1 u(t;a 1 ) ´ 2 dt ! =E Q ³ dP a T dQ ¡ v(¢;a) ¢ ´ =1; (2.2.23) As a result, (2.2.23) is consistent with the Girsanov Theorem. 2.2.2 b 2 ·a 2 In this section, we discuss the case whenb 2 ·a 2 . The characteristic polynomialF(r) has two complex conjugate roots (a double root if b 2 = a 2 ). To simplify the notations, denotel = p a 2 ¡b 2 . The solution to the equation u 00 (t)+2bu 0 (t)+a 2 u(t)= _ w(t) (2.2.24) with zero initial values is u(t;a;b)= 1 l Z t 0 e ¡b(t¡r) sinl(t¡r)dw(r): (2.2.25) We have the following theorem. 2.2.2 Theorem. Fort2[0;T], fixedT >0. 37 1. Ifdu(t;a;b)=v(t;a;b)dt, define the random variable X(a;b)= Z T 0 u 2 (t;a;b)dt and Y(a;b)= Z T 0 v 2 (t;a;b)dt: Then lim a!+1 a 2 EX(a;b)= e ¡2bT ¡1+2bT 8b 2 ; (2.2.26) EY(a;b)» e ¡2bT ¡1+2bT 8b 2 ; a!1: (2.2.27) EX n (a;b)·C(n;T)( 1¡e ¡2bt 4bl 2 ) ¡n ; n=1;2;:::; (2.2.28) lim a!+1 a 4 VarX(a;b)= e ¡4bT +4e ¡2bT +8bTe ¡2bT +4bT ¡5 64b 4 : (2.2.29) VarY(a;b)» e ¡4bT +4e ¡2bT +8bTe ¡2bT +4bT ¡5 64b 4 ; a!1: (2.2.30) 2. In the space of continuous functions on[0;T], denoteP a T as the measure generated by the process v(t;a), 0 · t · T . Then the measuresP a T are mutually absolutely continuous for alla and dP a 2 T dP a 1 T ¡ v(¢;a 1 ;b 1 ) ¢ =exp Ã Z T 0 (a 2 1 ¡a 2 2 )u(t;a 1 )+2(b 1 ¡b 2 )v(t;a 1 )dv(t;a 1 ) ¡ 1 2 Z T 0 ³ (a 2 1 ¡a 2 2 )u(t;a 1 ) ´ 2 + ³ 2(b 1 ¡b 2 )v(t;a 1 ) ´ 2 dt ! (2.2.31) 38 In particular,P 0 T is the measure generated by the standard Browninan motion which is also called the Winer measure, and dP a 1 T dP 0 T ¡ v(¢;a 1 ;b 1 ) ¢ =exp Ã ¡ Z T 0 ³ a 2 1 u(t;a 1 )+2b 1 v(t;a 1 ) ´ dv(t;a 1 ) ¡ 1 2 Z T 0 ³ a 2 1 u(t;a 1 )+2b 1 v(t;a 1 ) ´ 2 dt ! (2.2.32) Proof. 1. Eu 2 (t;a;b)= 1 l 2 Z t 0 e ¡2b(t¡r) sin 2 l(t¡r)dr = e ¡2bt (¡a 2 +b 2 cos2lt¡blsin2lt)+l 2 4bl 2 a 2 ; (2.2.33) so Eu 2 (t;a;b)» 1¡e ¡2bt 4ba 2 =g(t); a!1 (2.2.34) and EX(a;b)» e ¡2bT ¡1+2bT 8b 2 a 2 ; a!1: (2.2.35) Asa!1, a 2 EX(a)! e ¡2bT ¡1+2bT 8b 2 : By taking the derivative ofu(t) with respect tot , we get v(t;a;b)= Z t 0 e ¡b(t¡r) ¡ ¡ b l sinl(t¡r)+cosl(t¡r) ¢ dw(r); 39 note that Ev 2 (t;a;b)= e ¡2bt ¡ ¡b 2 +l 2 (¡1+e 2bt )+b 2 cos2lt+blsin2lt ¢ 4bl 2 » 1¡e ¡2bt 4b ; and so EY(a;b)» e ¡2bT ¡1+2bT 8b 2 : (2.2.36) For (2.2.28), as before, EX n (a;b)=E ¡ Z T 0 u 2 a;b (t)dt ¢ n ·T n¡1 Z T 0 Eu 2n a;b (t)dt =T n¡1 Z T 0 ¡ n Y k=1 (2k¡1) ¢¡ Eu 2 a;b (t) ¢ n dt =C(n)T n¡1 Z T 0 ¡ Eu 2 a;b (t) ¢ n dt ·C(T;n)( 1¡e ¡2bt 4bl 2 ) ¡n ; (2.2.37) The last inequality follows from (2.2.33). Fors<t, Cov ¡ u(t);u(s) ¢ = 1 4bl 2 ( e ¡b(t+s) Ã ¡cosl(s¡t) ³ b 2 ¡l 2 (¡1+e 2bs ) ¡e ¡b(t+s) ´ +b ³ bcosl(s+t)+l ¡ e 2bs sinl(s¡t)+sinl(s+t) ¢ ´ !) » cosl(s¡t) ¡ e ¡b(t¡s) ¡e ¡b(t+s) ¢ 4b ; a!1 (2.2.38) 40 so Eu 2 (t)u 2 (s)»g(t)g(s)(1+2½ 2 (s;t))=g(t)g(s)+2Cov 2 ¡ u(s);u(t) ¢ ; a!1 (2.2.39) where ½ 2 (s;t)» Cov 2 ¡ u(t);u(s) ¢ g(s)g(t) ; a!1: EX 2 (a;b)» 1 8b 2 l 4 Z T 0 Z t 0 (1¡e ¡2bt )(1¡e ¡2bs )dsdt + 1 4b 2 l 4 Z T 0 Z t 0 cos 2 l(t¡s) £ e ¡b(t¡s) ¡e ¡b(t+s) dsdt ¤ = ³ EX(a;b) ´ 2 + e ¡4bT +4e ¡2bT +8bTe ¡2bT +4bT ¡5 64a 4 b 4 +o(a ¡4 ): (2.2.40) Asa!1, a 4 EX 2 (a)! e ¡4bT +4e ¡2bT +8bTe ¡2bT +4bT ¡5 64b 4 : (2.2.41) Similarly, VarY(a;b)» e ¡4bT +4e ¡2bT +8bTe ¡2bT +4bT ¡5 64b 4 : (2.2.42) 2. Forv(a 1 ) andv(a 2 ) be two processes with dv(t;a 1 )= ¡ ¡a 2 1 u(t;a 1 )¡2b 1 v(t;a 1 ) ¢ dt+dw(t) (2.2.43) 41 and dv(t;a 2 )= ¡ ¡a 2 2 u(t;a 2 )¡2b 2 v(t;a 2 ) ¢ dt+dw(t) (2.2.44) both ¡a 2 1 u(t;a 1 )¡ 2b 1 v(t;a 1 ) and ¡a 2 2 u(t;a 2 )¡ 2b 2 v(t;a 2 ) areF v s -measurable because dv = a t (v)dt + dw(t) where a t (x) = ¡a R t 0 x(s)ds ¡ 2bx(t) is F v s - measurable, so these two processes are of diffusion type by definition. AsEu 2 (t) andEv 2 (t) are both finite, the condition P n [a 2 1 u(t;a 1 )+2b 1 v(t;a 1 )] 2 dt<1 o = P n [a 2 2 u(t;a 2 )+2b 2 v(t;a 2 )] 2 dt<1 o =1 (2.2.45) is satisfied. By Theorem 7.19 in Lipster and Shiryaev [59], the measures generated by v(t;a 1 ) andv(t;a 2 ) are absolutely continuous with density dP a 2 T dP a 1 T ³ v(¢;a 1 ) ´ =exp Ã Z T 0 (a 2 1 ¡a 2 2 )u(t;a 1 )+2(b 1 ¡b 2 )v(t;a 1 )dv(t;a 1 ) ¡ 1 2 Z T 0 ³ (a 2 1 ¡a 2 2 )u(t;a 1 ) ´ 2 + ³ 2(b 1 ¡b 2 )v(t;a 1 ) ´ 2 dt ! (2.2.46) The remaining arguments are identical to the proof of Theorem 2.2.1 part 2. 42 2.2.3 b 2 >a 2 Now we considerb 2 >a 2 . Then the characteristic function has real roots. The solution is u(t)= 1 l Z t 0 e ¡b(t¡r) sinhl(t¡r)dw(r); with notationsl = p b 2 ¡a 2 . 2.2.3 Theorem. Fort2[0;T], fixedT >0. 1. Ifdu(t;a;b)=v(t;a;b)dt, define the random variable X(a;b)= Z T 0 u 2 (t;a;b)dt and Y(a;b)= Z T 0 v 2 (t;a;b)dt Then lim a!+1 a 2 EX(a;b)= T 4a 2 b ; (2.2.47) EY(a;b)» T 4b ; a!1: (2.2.48) EX n (a;b)·C(n;T)( 1¡e ¡2bt 4bl 2 ) ¡n ; n=1;2;:::; (2.2.49) lim a!+1 a 4 VarX(a;b)= T 16a 4 b 3 : (2.2.50) VarY(a;b)» T 16b 3 ; a!1: (2.2.51) 2. In the space of continuous functions on[0;T], denoteP a T as the measure generated by the process v(t;a), 0 · t · T . Then the measuresP a T are mutually absolutely continuous for alla and 43 dP a 2 T dP a 1 T ¡ v(¢;a 1 ;b 1 ) ¢ =exp Ã Z T 0 (a 2 1 ¡a 2 2 )u(t;a 1 )+2(b 1 ¡b 2 )v(t;a 1 )dv(t;a 1 ) ¡ 1 2 Z T 0 ³ (a 2 1 ¡a 2 2 )u(t;a 1 ) ´ 2 + ³ 2(b 1 ¡b 2 )v(t;a 1 ) ´ 2 dt ! (2.2.52) In particular,P 0 T is the measure generated by the standard Browninan motion which is also called the Winer measure, and dP a 1 T dP 0 T ¡ v(¢;a 1 ;b 1 ) ¢ =exp Ã ¡ Z T 0 ³ a 2 1 u(t;a 1 )+2b 1 v(t;a 1 ) ´ dv(t;a 1 ) ¡ 1 2 Z T 0 ³ a 2 1 u(t;a 1 )+2b 1 v(t;a 1 ) ´ 2 dt ! (2.2.53) Proof. Eu 2 (t)= e ¡2bt (a 2 ¡b 2 cosh2lt¡blsinh2lt)+l 2 4bl 2 a 2 ; and EX(a;b)=E Z T 0 u 2 (t)dt» T 4a 2 b : Ev 2 (t)= e ¡2bt (a 2 ¡b 2 cosh2lt+blsinh2lt)+l 2 4bl 2 ; and EY(a;b)=E Z T 0 v 2 (t)dt» T 4b ; a!1: 44 By a similar method, VarX(a;b)» T 16a 4 b 3 and VarY(a;b)» T 16b 3 ; a!1: The proof of the second part is identical as the casea 2 ¸b 2 : 45 Chapter 3 Stochastic Partial Differential Equations 3.1 Introduction The theory of stochastic equations in a Hilbert space has been well developed since Baklan [6]. With the aid of the associated Green’s function, he proved the existence the- orem for a stochastic parabolic equation or parabolic Ito equation by recasting it as an integral equation. The existence and uniqueness of strong solutions were first explored by Bensoussan and Teman [8, 9] and further investigated by Krylov, Rozovskii [53] and Paradoux [77], among many others. The estimation problems of stochastic partial differ- ential equation driven by additive space-time white noise were first suggested by Huebner et al. [38] and were further investigated by Huebner and Rozovskii [39]. While most of the existing papers concentrate on estimating either a single parameter or a function of time, estimation of several parameters in parabolic equations has also been studied [36, 63]. Partial differential equations are one of the most effective and useful tools in math- ematical modeling. Deterministic partial differential equations originated around 1740s 46 with the works of J.R. d’Alembert, D. Bernoulli and L. Euler. They were used in math- ematical models for physical problems, such as heat conduction and wave propagation in continuous media. However, the development of stochastic partial differential equations started to appear only recently in the mid-1960’s. Systematic research on stochastic partial differential equations has been going on only since about 1980’s. Stochastic partial differential equations (SPDEs) are known to be an effective tool in modeling complex physical and engineering phenomena subject to random influence, such as stochastic forcing, uncertain parameters, noisy sources, and random boundary condi- tions. Taking stochastic effects into account is of central importance for the development of mathematical modeling. Examples include wave propagation [75], diffusion through heterogeneous random media [76], randomly forced Burgers and Navier-Stokes equations (see e.g. [9, 18, 19, 69, 86, 87, 97, 98, 99] and the refernces therein.) Additional examples can be found in material science, chemistry, biology, and other areas. In these problems, the large structures and dominant dynamics are governed by deterministic physical laws, while the unresolved small scales, microscopic effects, and other uncertainties can be nat- urally modeled by stochastic processes. The resulting equations are usually PDEs with either random coefficients, random initial conditions, or random forcing. Unlike deter- ministic PDEs, solutions of stochastic PDEs are random fields. Hence, it is important to be able to study their statistical characteristics, e.g., mean, variance, and higher order moments. The study of stochastic PDEs has been a main research in probability theory in recent years and the trend is still increasing. In this chapter we introduce the stochastic hyperbolic differential equations depend- ing on one or two parameters. To be more specific, this chapter can be divided into two 47 sections. In the first part we begin with the basic setting for the stochastic partial differ- ential equations. We briefly review some basic definitions about stochastic processes and stochastic differential equations that will be needed in the subsequent sections. Then we will proceed to study some diagonalizable stochastic hyperbolic equations which will be estimated in the next chapter. 3.2 The setting Let H = L 2 (G) be a separable Hilbert space with the inner product (¢;¢) 0 and the corresponding normk¢k 0 and¤ is a densely-defined linear operator onH satisfying the following property: ² there exists a positive numberc, such thatk¤uk 0 ¸ckuk 0 for everyu in the domain of¤,fh i (µ)g 1 i=1 is a complete orthonormal system of eigenfunctions of¤ inH; This operator is assumed to be self-adjoint, that is, (¤u;v) 0 =(u;¤v) 0 : Consider the following stochastic hyperbolic equation : 8 > > < > > : u tt =(A 0 +µ 1 A 1 )u+(B 0 +µB 1 )u t + _ W(t;x);t2(0;T]; u(0)=0; u t (0)=0; (3.2.1) whereA 0 ,A 1 ,B 0 andB 1 are partial differential operators and µ 1 , µ 2 are parameters subject to estimation. 48 DenoteA = A µ = (A 0 + µA 1 ) andB = B µ = (µB 1 +B 0 ). We assume that the boundary@G ofG is aC 1 ¡manifold of dimension(d¡1) and locallyG is only on one side of@(G). For a multi-index° =(° 1 ;¢¢¢ ;° d ) we write D ° f(x):= @ j°j @x ° 1 1 ¢¢¢@x ° d d wherej°j=° 1 +¢¢¢+° d : Moreover, we assume there is a compact neighborhood£ ofµ so thatfA µ ;µ2£g (µ is the parameter eitherµ 1 orµ 2 subject to estimation ) is uniformly strongly elliptic of order 2m =max(m 1 ;m 0 ). For a partial differential operator and all other operators, this means that there exists a constant ± such that for all x 2 ¹ G;µ 2 £ which is a compact neighborhood ofµ 0 and³ 2R d , X j®j;j¯j=m a ®¯ (µ;x)³ ® ³ ¯ ¸±j³j 2m : The spectrum ofA is a discrete set¾(A) consisting of eigenvalues of finite multiplicity. We enumerate the eigenvalues ofA in order of magnitude: ¾(A)= © ¸ k (µ) ª 1 k=1 0<¸ 1 (µ)·¸ 2 (µ)·¢¢¢·¸ k (µ)·¢¢¢ ; where each one is counted as many times as its multiplicity. 3.2.1 Definition. Fors¸0, set: H s := 8 < : u2H:kuk s := Ã 1 X j=1 ¸ 2s j j(u;h j ) L 2 (G) j 2 ! 1=2 <1 9 = ; 49 For s < 0,H s is a closure ofH in the normkuk s defined for negative s by the same formula as for positive. The familyfH s g,s2R has the following properties: ² Fors 1 <s 2 , H s 2 ½H s 1 and the imbedding is dense and compact. ² For each pair(s 1 ;s 2 ),¤ s 1 :H s 2 !H s 2 ¡s 1 is a bounded operator. ² For anys2R; (¤ s u;¤ s v) 0 =(u;v) s for allu;v2H s Fors>0, denote the closure ofC 1 0 (G) in the Sobolev spaceW s;2 (G) byW s;2 0 . For everyk =1;2;:::;h k 2W m;2 0 (G)\C 1 ( ¹ G), Ah k =¸ k h k ; B =¹ k h k ; (3.2.2) A 0 h k =° k h k ; B 0 =½ k h k (3.2.3) A 1 h k =· k h k ; B 1 =º k h k (3.2.4) where¸ k ,° k ,· k ,º k ,¹ k and½ k are eigenvalues ofA,A 0 ,A 1 ,B,B 0 andB 1 respectively. 50 3.2.2 Definition. Let (;F;P) be a probability space with a complete filtrationF = fF t g 0·t·T and G is a bounded domain in R d , for every t 2 [0;T], W Ã =W Ã (t) is a process on this space such that for Á;Ã 2C 1 0 (G),kÁk ¡1 H(G) W Á (t) is a one dimensional Winer process and E ¡ W Á (t)W Ã (s) ¢ =(s^t)(Á;Ã) H(G) : This process is called a cylindrical Brownian motion. 3.2.3 Proposition. LetW be a cylindrical Brownian motion on a Hilbert spaceH. 1. The mapping Á ! W Á is linear and therefore, for every t i , i = 1;:::;m, j = 1;:::;m, fÁ j g n j=1 , the collection of random variablesfW Á j (t i )g is a Gaus- sian system. 2. Letfh j g n j=1 to be an orthnormal system inH such thatkh j k=1 and(h i ;h j ) H =0 for i 6= j. Then the processes defined by w i (t) = W h i (t) and w j (t) = W h j (t) are independent standard Brownian motions. Proof. 1. By the properties of inner product, fora;b2R andÁ;Ã2H, E ¡ W aÁ+bÃ (t)¡aW Á (t)¡bW Ã (t) ¢ 2 =0; cf. Nualart [73, Definition 1.1.1] 2. By definition, Ew i = Ew j = 0, and Ew i (t)w j (s) = (t^ s)(h i ;h j ) H for i 6= j. Therefore,w i (t) andw j (t) are independent standard Brownian motions. 51 If f = f(t) is an adapted, square-integrable process with values in H (that is, E R T 0 kf(t)k 2 H dt < 1), then we define the stochastic integral R T 0 hf(t);dW(t)i by the formula Z T 0 hf(t);dW(t)i= X k¸1 Z T 0 (f(t);h k ) H dw k (t); wherefh k , k ¸ 1g is an orthonormal basis in H and w k (t) = W h k (t). This definition does not depend on the choice of the basis inH (see, for example, Rozovskii [80, Chapter 2]) or Walsh [96, Chapter I]. Proposition 3.2.3 part 2 suggests that the space-time Brownian motionW is the formal series W(t)= X k¸1 w k (t)h k : (3.2.5) wherefh k g k¸1 and fw k g 1 k=1 = W h k are independent 1-dimensional standard Brownian motions. If the series does not converge inH, it is possible to embedH in a bigger Hilbert space so that the series converges in the new Hilbert space. As an example, letX to be the closure ofH in the norm kfk X = ³ X k¸1 k ¡2 (f;h k ) 2 H ´ 1=2 : It is easy to check thatW is a continuousX-valued square-integrable martingale, and EkW(t)k 2 X =t X k¸1 k ¡2 = ¼ 2 t 6 <1: To makeEkW(t)k 2 X finite,k ¡2 can be replaced byk ¡¯ with¯ >1. 52 IfW is a cylindrical Brownian motion onL 2 ((0;+1)), andÂ x is the indicator function of the interval[0;x], then, by direct computation, W(t;x) = W Âx (t) is a Brownian sheet and, for everyf 2L 2 ((0;+1)), W f (t)= Z t 0 Z +1 0 f(y)W(ds;dy); see, for example, Walsh [96, p. 284]. In general, definition 3.2.2 can be extended to allow spatial covariance: for a bounded, self-adjoint and non-negative operatorQ on the Hilbert spaceH, define theQ-cylindrical Brownian motionW Q onH by: E ¡ W Q Á (t)W Q Ã (t) ¢ =(QÁ;Ã) H (t^s) Iffh k ;k ¸ 1g is a complete orthnormal basis ofQ such thatQh k = q k h k with q k > 0, then (3.2.5) becomes W Q (t)= X k¸1 q k w k (t)h k ; wherew k ;k¸1 are independent standard Brownian motions by proposition (3.2.3)(b). 3.2.4 Definition. 1. An operatorB from a separable Hilbert spaceH to a separable Hilbert spaceX is called Hilbert-Schmidt if X k¸1 kBh k k 2 X <1 for one (hence all) orthornomal basisfh k g k¸1 inH; 53 2. A self-adjoint non-negative operatorQ on a separable Hilbert space H is called trace class if X k¸1 (Qh k ;h k ) H <1 for one (hence all) orthornomal basisfh k g k¸1 inH. 3.2.5 Proposition. 1. LetW be a cylindrical Brownian motion on a separable Hilbert spaceH andH is a dense sub-set of the Hilbert spaceX. ThenW is a continuous X-valued square-integrable martingale if and only if the embeddingB : H ! X is a Hilbert-Schmidt operator. In this case,W naturally extends to aQ-cylindrical Brownian motion onX withQ=BB > andQ is trace class; 2. LetW Q be aQ-cylindrical Brownian motion on a separable Hilbert spaceH. Then W Q is a continuous H-valued square-integrable if and only if the operator Q is trace class. Proof. 1. By direct computation, EkW(t)¡W(s)k 2 X =(t¡s) X k¸1 kh k k 2 X ; by the definition of the Hilbert-Schmidt operator, the series P k¸1 kh k k 2 X converges if and only if B is Hilbert-Schmidt. By the Kolmogorov criterion [96, Corollary 1.2],W has a continuous version. Let W Q Á =W B >; 54 it can be shown that the adjoint operatorB > is defined on all ofX (see, for example, Yosida [101, Theorem VII.2]). By direct computation, (Qh k ;h k ) H =kB > h k k 2 H =kh k k 2 X : As a result, P k¸1 kh k k 2 X converges if the series P k¸1 (Qh k ;h k ) H is finite. 2. By computation, EkW Q (t)¡W Q (t)k 2 H =(t¡s) 1 X k=1 (Qh k ;h k ) H ; and the series converges if and only if P 1 k=1 (Qh k ;h k ) H is finite. That is,Q is trace class. 55 3.3 Diagonalizable Stochastic Hyperbolic Equations The idea used to study the stochastic wave equation u tt =µ 1 u xx +µ 2 u t + _ W; 0<t·T; x2(0;¼); (3.3.1) on the interval can be extend with little or no modification to equations such as u tt =µ 1 ¢udt+ _ W; u tt =(¢u+µ 2 u t )+ _ W u tt =(¢u+µ 1 u)+ _ W; where jµ 1 j ¸ 1 and jµ 2 j · 1 are parameters subject to estimation, ¢ is the Laplace operator, and to abstract hyperbolic equations 8 > > < > > : u tt +(A 0 +µ 1 A 1 )u=(B 0 +µ 2 B 1 )u t + _ W(t;x);t2(0;T] u(0)=0: (3.3.2) with suitable assumptions about the operatorsA 0 ,A 1 ,B 0 andB 1 . In the wave equation u tt =µ 1 u xx +µ 2 u t + _ W; 0<t·T; x2(0;¼); (3.3.3) the key feature is the possibility to write the solution using separation of variables; in 56 what follows, we generalize this feature to (3.3.2) using the notation of a diagonalizable equation. In general, if W is a cylindrical Brownian motion on a Hilbert space H, the solution of (3.3.2) is not an element ofH for t > 0. There are two main approaches to circumvent this difficulty: ² To introduce spatial covariance in the noise and considerW Q instead ofW ; ² To consider the equation in a bigger Hilbert space. By Proposition 3.2.5, the two approaches are essentially equivalent, and we will use the second one. We will see that many equations driven byW Q can be reduced to equations driven byW later on. The following assumptions will be in force throughout the rest of the chapter. We will consider equation (3.3.2) from now on. 1. H, a separable Hilbert space with an orhthnomal basisfh k ; k¸1g; 2. W =W(t), a cylindrical Brownian motion onH; 3. A 0 ,A 1 ,B 0 andB 1 are linear operators onH; 4. £=[a;b], a closed bounded interval inR. Denote by¸ k ,¹ k ,· k ,½ k ,¿ k andº k the eigenvalues of the operatorsA,B,A 0 ,B 0 ,A 1 and B 1 . Ah k =¸ k h k ; B =¹ k h k ; A 0 h k =· k h k ; B 0 =½ k h k ; 57 A 1 h k =¿ k h k ; B 1 =º k h k : 3.3.1 Definition. 1. Equation (3.3.2) is called diagonalizable if the operatorsA,A 0 , A 1 , B, B 0 and B 1 have a common system of eigenfunctions h k ;k ¸ 1 such that fh k g k¸1 is an orthonormal basis inH. h k (µ) might depend onµ. However we omit µ for simplicity. 2. Equation (3.3.2) is called hyperbolic if ² there exist positive numbers C ¤ , c 1 and c 2 such thatf¸ k +C ¤ , k ¸ 1g is a positive, non-decreasing, and unbounded sequence for allµ2£ and c 1 · ¸ k (µ a )+C ¤ ¸ k (µ b )+C ¤ ·c 2 for allµ a ;µ b 2£. ² there exist positive numbers C, J such that, for all k ¸ J and all µ 1 2 £ 1 , µ 2 2£ 2 , T(½ k +µ 2 º k )·ln(· k +µ 1 ¿ k )+C: (3.3.4) Condition (3.3.4) implies that there is no restriction on the strength of dissipation, but amplification must be weak. For example, let ¢ be the Laplace operator in a smooth 58 bounded domainG2R d with zero boundary conditions, andH = L 2 (G). Then each of the following equations is diagonalizable and hyperbolic on[0;T] for allT >0: u tt =¢u+u t + _ W; u tt =¢u¡u t + _ W; u tt =¢(u+u t )+ _ w; u tt =¢u¡¢ 2 u t + _ W; (3.3.5) while equations u tt =¢(u=u t )+ _ W; u tt =¢u+¢ 2 u t + _ W (3.3.6) are diagonalizable but not hyperbolic on any[0;T]. In particular, if equation (3.3.2) is hyperbolic, thenlim k!1 ¸(µ 1 )=+1 uniformly in µ 1 2£, ¸ k (µ 1 )>0: (3.3.7) 3.3.2 Theorem. Assume that equation (3.3.2) is diagonalizable and hyperbolic. Then there exists a unique solutionu=u(t) ifj¹ k j is bounded. That is,sup k j¹ k j<1. Proof. See, for example [14, Theorem 8.4]. 59 We can easily verify that each of the following equations is diagonalizable and hyper- bolic: u tt =µu xx + _ W(t;x) (3.3.8) u tt =(u xx +µu t )+ _ W(t;x) (3.3.9) u tt =(u xx +µu)+ _ W(t;x) (3.3.10) 3.3.3 Example. 1. Consider the equation u tt = µu xx + _ W(t;x) for t 2 [0;1] and x 2 [0;1] with periodic boundary condition, whereu xx = @ 2 u=@x 2 : ThenH ° is the Sobolev space on the unit circle (see, for example, Shubin [85, Section 1.7]). In this case,A 0 = 0 and A µ = A 1 = ¢. We can show that H ° is the Sobolev space on the unit circel. For details, see Shubin [85]. It is a standard fact thatD(¡µ¢) =W 1;2 0 (0;1): Take h k (x) = p 2=¼sinkx. Obviously the sequence fh k g k¸1 forms a complete orthonormal system inL 2 (0;1). It is easily checked that the equation is diagonalizable and hyperbolic. It is 1- hyperbolic if and only if µ is positive. By the theorem above, the solution u(t) is uniquely determined by the uniqueness of u k (t)= 1 k p µ Z t 0 sin p µk(t¡s)dW k (s) fork¸1. 60 2. LetG be a smooth bounded domain inR d or a smooth compactd-dimensional man- ifold with a smooth measure. Let¢ be the Laplace operator onG with zero initial and boundary conditions ifG is a domain. It is easily checked that (see, for example, Safarov and Vassiliev [83] or Shubin [85]). (a) the eigenfunctions h k ;k ¸ 1 of¢ are smooth in G and form an orthonormal basis inH=L 2 (G); (b) the corresponding eigenvalues ¸ k ;k ¸ 1 are positive, can be arranged in ascending order so that0·¸ 1 ·¸ 2 ·:::; and there exists a positive number C such that j¸ k jsCk 2=d ; that is, lim k!1 j¸ k jk ¡2=d =C: Then, for everyµ positive, the stochastic equation u tt =(¢u+µu)+ _ W(t;x) (3.3.11) is diagonalizable and hyperbolic. We can verify that each of the following equations is diagonalizable and hyperbolic: u tt =µ¢udt+dW; 0<a·µ·b; u tt =¢udt+µu t +dW; 0<a·µ·b: (3.3.12) 61 From now on, we assume that equation (3.3.2) is diagonalizable and hyperbolic, and the eigenvalues of the operatorsA,A 0 , are enumerated so thatf¸ k (µ 1 ),k ¸ 1g is a non- decreasing sequence and (3.3.7) holds. 62 Chapter 4 The Estimation of Stochastic Hyperbolic Equations 4.1 Statistical models In mathematics, a statistical model (or experiment) generated by random elements X(µ) is frequently thought of as a parameterized set of probability distributions of the formP = fX;X;P µ ;µ 2 £g, where eachP µ is a probability measure on a measurable space (X;X) such that P µ (A) = P(X(µ) 2 A), A 2 X . In the parametric models, £ is a subset of a finite-dimensional Euclidean space. It is assumed that there is a distinct element in the setP which generates the observations. Statistical inference enables us to make statements about which element(s) of this set are likely to be the true one. An estimator ofµ is a random variableª, whereª is a measurable mapping fromX to £. The corresponding estimate ofµ is the numberª(X o (µ)), whereX o (µ) is the observed realization of the random elementX(µ ¤ ) corresponding toµ =µ ¤ . General speaking, an estimator ^ µ is a random variable, and is not equal to the true value µ ¤ . So we introduce a familyP N , N > 0 of statistical models, where N is the amount 63 of information about the parameter µ. The more information becomes available as the numberN increases. As an instance,P N can be a product ofN independent copies ofP, which corresponds to observingN independent realizations ofX. GivenP N , the corresponding family of estimators is constructed and studied in the limitN !1. One of the objectives is to establish consistency of the estimators (conver- gence to the true value of the parameter) asN !1. 4.1.1 Definition. A statistical model P is called absolutely continuous if there exists a probability measureQ such that everyP µ is absolutely continuous with respect toQ. The statistical modelP is called singular if the measuresP µa andP µ b are mutually singular forµ a 6=µ b . LetP be an absolutely continuous model and consider the density p(x;µ)= dP µ dQ (x;µ); x2X; µ2£: (4.1.1) The maximum likelihood estimator ^ µ ofµ is defined by ^ µ(X)=argmax µ2 ¹ £ p(x;µ)j x=X(µ) ; (4.1.2) where ¹ £ is the closure of £. A collection P N , N > 0 of absolutely continuous statistical models leads to a collection ^ µ N of maximum likelihood estimators. 4.1.2 Definition. An estimator ª N ofµ is said to converge toµ with the rate of converge 64 N ® ,® > 0, if the sequencefN ® (ª N ¡µ); N > 0g converges in distribution to a non- degenerate random vector³ (that isVar³ >0 ). If³ is a Gaussian random variable, then ª N is called asymptotically normal. 4.2 Analysis of The Stochastic Wave Equation on The Interval Wave motion and mechanical vibration are two of the most commonly observed physi- cal phenomena. As mathematical models, they are usually described by partial differential equations of hyperbolic type. The most well-known one is the wave equation. In contrast with heat equation, the first-order time derivative term is replaced by a second-order one. A typical example of a hyperbolic equation is the wave equation u tt =u xx : Naturally, we call the equation u 00 +Au=0 is a hyperbolic equation whereA is a linear elliptic operator in a more abstract setting. Damping in a hyperbolic equations is introduced via a term depending on the first time derivative of the solution. For example, a damped wave equation is u tt =u xx ¡au t ; a>0: 65 We can now define the total energyE(t) = R¡ u 2 t (t;x)+u 2 x (t;x) ¢ dx; then integration by parts gives that d dt =¡a Z u 2 t (t;x)dx; it also shows that negative damping (a<0) corresponds to amplification. A stochastic wave equation takes the form u tt =c 2 ¢u+¾ 0 _ W(t;x) in some domainG, where c is known as the wave speed and ¾ 0 is the noise intensity parameter. The formal derivative _ W(t;x) of the Brownian MotionW(t;x) is known as a white noise. LetW = W(t) be a cylindrical Brownian motion onL 2 ((0;¼)) andµ 1 > 0. Consider the following stochastic wave equation u tt =µ 1 u xx +µ 2 u t + _ W; 0<t·T; x2(0;¼): (4.2.1) To simplify the notations, we writeu(t) as the solution instead ofu(t;µ 1 ;µ 2 ). From now on, we assume 1. equation (4.2.1) is diagonalizable and hyperbolic; 2. µ 1 ¸1,jµ 2 j·1; 66 3. zero initial and boundary conditions: uj t=0 = @u @t j t=0 =uj x=0 =uj x=¼ =0: For ° 2R, define the spaceH ° as the closure of the set of smooth compactly supported function on(0;¼) with respect to the norm H ° = ( f : X k¸1 f 2 k k 2° <1 ) ; wheref k = p ¼=2 R ¼ 0 f(x)sinkxdx: 4.2.1 Definition. An adapted processu2L 2 (£(0;T)£(0;¼)) is called a solution of (4.2.1) if there exists an adapted processv such that 1. v2L 2 (;L 2 (0;T);H ¡1 ); 2. For every twice continuously-differentiable on(0;¼) functionf =f(x) withf(0)= f(¼)=0, the equalities (u;f)(t)= Z t 0 (v;f)(s)ds; (4.2.2) (v;f)(t)= Z t 0 ¡ µ 1 (u;f 00 )(s)+µ 2 (v;f)(s) ¢ ds+W f (t) (4.2.3) hold for allt2[0;T] on the same set of probability one. 67 4.2.2 Theorem. n equation (4.2.1), denote du(t) = v(t)dt, then there exists a unique solutionu=u(t) of (4.2.1) and for° <1=2, 1. u2L 2 (£(0;T);H ° )\L 2 (;C ¡ (0;T);H ° ¢ ; 2. v2L 2 (£(0;T);H °¡1 )\L 2 (;C ¡ (0;T);H °¡1 ¢ . Proof. This result can be derived from the general theory of stochastic hyperbolic equation (see, for example, Chow [15]). However, we present a different, and a more direct proof which will also help in the construction and analysis of the estimators. Denote a 2 = µ 1 k 2 and 2b = ¡µ 2 , then u(t) = P k¸1 u k (t)h k (x) where u k (t) is a solution of the equation u 00 k (t)+2bu 0 k (t)+a 2 = _ w(t): 1. By direct computation, ifµ 2 =0; u k (t)= 1 a Z t 0 sina(t¡r)dw(r); v k (t)= Z t 0 cosa(t¡r)dw(r): (4.2.4) The uniqueness of the solution then follows from the completeness of the system fh k = p ¼=2sinkxg,k¸1 inL 2 ((0;¼)). By (4.2.4) and Theorem 2.2.1, Ekuk 2 ° = X k¸1 t 2a 2 k 2° = X k¸1 t 2µ 1 k 2°¡2 and Ekvk 2 °¡1 = X k¸1 t 2 k 2(°¡1) both series converge because2°¡2<¡1 which impliesu(t)2L 2 (£(0;T);H ° ) andv2L 2 (£(0;T);H °¡1 ). 68 It remains to show that the processesu is a continuousH ° -valued process andv is a continuousH °¡1 -valued process. For0·s<t·T , E ³ u k (t)¡u k (s) ´ 2 = 1 a 2 Z s 0 ¡ sina(t¡r)¡sina(s¡r) ¢ 2 dr + 1 a 2 Z t s sin 2 a(t¡r)dr: Next, by the Lipschitz continuity ofsinx, there exists0 < ² < 1 and a constantC, such that E ³ u 2 k (t)¡u 2 k (s) ´ 2 · C(t¡s) ² a 2¡² ; so Eku(t)¡u(s)k 4 ° =E Ã X k¸1 k 2(°) µ u k (t)¡u k (s) ¶ 2 ! 2 = X k6=m E ¡ u 2 k (t)¡u 2 k (s) ¢ 2 E ¡ u 2 m (t)¡u 2 m (s) ¢ 2 k 2° m 2° +3 X k¸1 k 4° X k¸1 E ³ u k (t)¡u k (s) ´ 4 ; 69 and by induction, forn2N Eku(t)¡u(s)k 2n ° =E Ã X k¸1 k 2° µ u k (t)¡u k (s) ¶ 2 ! n =C(n) X k6=m E ¡ u 2 k (t)¡u 2 k (s) ¢ 2n E ¡ u 2 m (t)¡u 2 m (s) ¢ 2n k 2°n m 2°n +D(n) X k¸1 k 2n° X k¸1 E ³ u k (t)¡u k (s) ´ 2n ·C(n)C 2n =µ 2 1 X k6=m (t¡s) n² k 2°n¡2n+²n m 2°n +F(n) +D(n)C 2n X k¸1 k 2°n¡2n+²n (t¡s) n² +F(n); (4.2.5) whereF(n) is a function dependent onn. To makeEku(t)¡u(s)k 2n ° finite, we need n(2°¡2+²)<¡1; and also we want n²>1; that is, n>max ½ 1 ² ;¡ 1 2°¡2+² ¾ ; therefore, there existsn2N, such that Eku(t)¡u(s)k 2n ° <M(n)(t¡s) ® ; 70 where M(n) is a constant dependent on n and ® > 1, and the continuity of u(t) follows from the Kolmogorov’s continuity theorem (see, for example, Chow [14, Page 6]). Similarly, for0<²<1, Ekv k (t)¡v k (s)k 2 ·Ca ² (t¡s) ² ; (4.2.6) so Ekv(t)¡v(s)k 2n °¡1 ·C n (t¡s) n² X k¸1 k 2°n¡2n+²n : (4.2.7) and there existsn2N, such that Eku(t)¡u(s)k 2n °¡1 <M(n)(t¡s) ® : By the Kolmogorov criterion,v(t) is a continuousH °¡1 -valued process. 2. ifµ 2 6=0, u k (t)= 1 l Z t 0 e ¡b(t¡r) sinl(t¡r)dw(r); v k (t)= Z t 0 e ¡b(t¡r) ³ ¡ b l sinl(t¡r)+cosl(t¡r) ´ dw(r); (4.2.8) 71 wherel = p a 2 ¡b 2 . By Theorem 2.2.2, Ekuk 2 ° = X k¸1 4k 2° e µ 2 t ¡1 µ 2 (4µ 1 k 2 ¡µ 2 ) <1; (4.2.9) and Ekvk 2 °¡1 = X k¸1 k 2° e µ 2 t ¡1 2µ 2 <1: (4.2.10) So E ³ u k (t)¡u k (s) ´ 2 = 1 l 2 Z t s e ¡2b(t¡s) sin 2 l(t¡r)dr + 1 l 2 Z s 0 ³ e ¡b(t¡r) sinl(t¡r)¡e ¡b(s¡r) sinl(s¡r) ´ 2 dr: (4.2.11) Similarly, there exists 0 < ² < 1 and a constant C, such that there exists n 2 N, such that Eku(t)¡u(s)k 2n ° <M(n)(t¡s) ® ; where M(n) is a constant dependent on n and ® > 1, and then the continuity of u(t) follows from the Kolmogorov’s continuity theorem. By a similar argument, there exists0<²<1, Ekv k (t)¡v k (s)k 2 ·Ca ² (t¡s) ² ; (4.2.12) 72 so Ekv(t)¡v(s)k 2n °¡1 ·C n (t¡s) n² X k¸1 k 2°n¡2n+²n : (4.2.13) and there existsn2N, such that Eku(t)¡u(s)k 2n °¡1 <M(n)(t¡s) ® : By the Kolmogorov criterion,v(t) is a continuousH °¡1 -valued process. 4.2.3 Remark. 1. Since the solution is defined by its Fourier coefficients, it is only necessary to represent the derivative ofu as a process. 2. Forµ 2 = 0 andµ 2 6= 0, thek-th Fourier coefficients of the solutionu(t) have differ- ent expressions. But for the second case, asb approaches 0, the solution approach the solution when µ 2 = 0. As a result, for these two cases, the spaces of solution u(t) and its derivativev(t) are consistent. 3. Note that the processes u k (t) and v k (t) are independent because the Brownian motionsw k ,k¸1 are independent. 4. We can now comment on the significance of the second assumption jµ 1 j ¸ 1, jµ 2 j· 1 and the third assumption (zero initial and boundary conditions). The sec- ond assumption can be relaxed to µ 1 > 0, because we will still have a 2 > b 2 for sufficiently large k. In other words, if µ 1 > 0, then the free motion (any solution 73 of the homogeneous version of (4.2.1)) oscillatory for all sufficiently large k ¸ 1; the oscillations are damped if b > 0, harmonic if b = 0, and amplified if b < 0. This is also a reason to call b the damping coefficient, with an understanding that negative damping means amplification. Thus, the second assumption is only needed to simplify the computations by ensuring that equalities (4.2.8) hold for all k ¸ 1: Non-zero initial conditions, if sufficiently regular, will not affect the regularity of the solution. Similarly, the analysis will not change much for zero Neumann or other homogeneous boundary conditions. Let us now consider the problem of estimating the parameters µ 1 and µ 2 from the observations ofu(t) andv(t). If the trajectories ofu(t) andv(t),0·t·T , are observed, then the following scalar parameter estimation problems can be stated: 1. estimateµ 1 , assuming thatµ 2 is known; 2. estimateµ 2 , assuming thatµ 1 is known; 3. estimateµ 1 andµ 2 simultaneously. It can be shown that the solution generates a Gaussian measure in space of continu- ous processes with values in a suitable Hilbert space, and the measures are singular for different values ofµ 1 andµ 2 . It is a fact that the observed fieldv =v(t) satisfies dv k (t)=F k (v)dt+dw(t); 74 whereF k (v) = (¡µ 1 k 2 R t 0 u k (t)dt+µ 2 v k (t)) is a non-anticipating function ofv for some unknown but fixed valueµ 0 of the parametersµ 1 andµ 2 . Depending on the circumstances, µ 0 can correspond to either µ 1 or µ 2 in (4.2.1). Even though the whole random field is observed, the estimate ofµ 0 will be computed using only finite dimensional processes. By Theorem 4.2.2, the solution of the equationu tt =µ 1 u xx +µ 2 u t + _ W ,0<t·T ,x2(0;¼) has a Fourier series expansion. We will construct the maximum likelihood estimators ofµ 1 andµ 2 using the observations of the2N-dimensional processfu k (t),v k (t),k =1;:::;N, t 2 [0;T]g and study the asymptotic properties of the estimators in the limit N ! 1. Note that both the amplitude of noise and the observations are fixed. By u k (t)= Z t 0 v k (s)ds; v k (t)=¡µ 1 k 2 Z t 0 u k (s)ds+µ 2 Z t 0 v k (s)ds+w k (t); (4.2.14) for each k ¸ 1, the processes u k , v k and w k generates P u k , P v k , P w k , in the space C((0;T);R) of continuous, real-valued functions on [0;T]. Since u k is a continuously- differentiable function, the measures P u k and P w k are mutually singular. For each k, the Ornstein-Uhlenbeck process v k generates the measures P v;µ k (T) in the space of the continuous real-valued functions on [0;T], and by theorem 2.2.1 and 2.2.2, the mea- sures are equivalent for different valued of the estimated parameter. Similarly, the vector v N (µ)=(v 1 ;:::;v N ) generates a probability measureP v;µ N (T) on the space of continuous real-valued functions on [0;T]. P µ N (T) is a product measure P v;µ N (T) = Q N k=1 P v;µ k (T) because the random processesv k (t) are independent for differentk and thus the measures P v;µ N (T) are equivalent for different values ofµ. To prove the consistency of the estimators µ 1 and µ 2 , we need to use the Theorem (strong law of large numbers)1.4.5 and its Corollary 1.4.6. 75 4.2.1 Estimateµ 1 , assuming thatµ 2 is known For eachk, the processv k (t) satisfies dv k (t)=F k (v)dt+dw(t); where F k (v) = (¡µ 1 k 2 R t 0 u k (t)dt+µ 2 v k (t)) is a non-anticipating function of v. Thus, the process v is a process of diffusion type in the sense of Lipster and Shiryaev. Further analysis shows that the measuresP v k is absolutely continuous with respect to the measure P w k (see, Theorem 2.2.1 and 2.2.2), and dP v k;T dP w k;T ³ v k (µ 1 ) ´ =exp Ã ³ Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ dv k (t) ¡ 1 2 Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ 2 dt ´ ! : (4.2.15) Since the processesw k are independent for differentk, so are the processesv k (t). There- fore, the measureP v N;T generated by the vector processfv k ,k = 1;:::;Ng is absolutely continuous with respect to the measure P w N;T generated in C((0;T);R N ) by the vector processfw k ,k =1;:::;Ng, and the density is dP v;µ 1 N;T dP w N;T ³ v N (µ 1 ) ´ =exp Ã N X k=1 ³ Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ dv k (t) ¡ 1 2 Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ 2 dt ´ ! : (4.2.16) 76 Let us now introduce the following notations: J 1;N = N X k=1 k 4 Z T 0 u 2 k (t)dt; J 2;N = N X k=1 Z T 0 v 2 k (t)dt; J 12;N = N X k=1 k 2 Z T 0 u k (t)v k (t)dt; B 1;N =¡ N X k=1 k 2 Z T 0 u k (t)dv k (t); » 1;N = N X k=1 k 2 Z T 0 u k (t)dw k (t); B 2;N = N X k=1 Z T 0 v k (t)dv k (t); » 2;N = N X k=1 Z T 0 v k (t)dw k (t): (4.2.17) Ifµ 2 =0, the maximum likelihood ratio dP v;µ 1 N;T dP w N;T ³ v N (µ 1 ) ´ =exp Ã N X k=1 ³ ¡µ 1 k 2 Z T 0 u k (t)dv k (t)¡ µ 2 1 k 4 2 Z T 0 u 2 k (t)dt ´ ! : (4.2.18) By definition, maximizing the expression on the right-hand side of (1.7) with respect to µ 1 , the maximum likelihood estimate (MLE) is then equal to argmax µ 1 ³ dP v;µ 1 N;T dP 0 N;T ´³ v N (µ 1 ) ´ ; (4.2.19) we can get the following expression for the MLE ^ µ 1 N ofµ 1 based on the observationsv k (t), k =1;:::;N,t2[0;T]: ^ µ (1) N = B 1;N J 1;N : (4.2.20) 77 4.2.4 Theorem. Estimator (4.2.20) is strongly consistent and asymptotically normal in the limitN !1: lim N!1 ^ µ (1) N =µ 1 with probability one; (4.2.21) lim N!1 N 3=2 ( ^ µ (1) N ¡µ 1 )=³ in distribution; (4.2.22) where³ is a Gaussian random variable with zero mean and variance12µ 1 =T 2 . Proof. It follows from (4.2.20) that ^ µ (1) N ¡µ 1 =¡ » 1;N J 1;N : (4.2.23) To get a consistent estimator, it is intuitively clear that P N k=1 k 4 R T 0 u 2 k (t)dt should tend to infinity asN !1. Since eachv k (t) is a stable Ornstein-Uhlenbeck process with a 2 =k 2 µ 1 , by Theorem 2.2.1, E Z T 0 u 2 k (t)dt» T 2 4k 2 µ 1 ; Var Z T 0 u 2 k (t)dt» T 4 24k 4 µ 2 1 : (4.2.24) and so Ek 4 Z T 0 u 2 (t)dt» T 2 k 2 4µ 1 ; Vark 4 Z T 0 u 2 (t)dt» T 4 k 4 24µ 2 1 : (4.2.25) 78 Therefore, Vark 4 Z T 0 u 2 (t)dt< µ Ek 4 Z T 0 u 2 (t)dt ¶ 2 ; (4.2.26) and lim k!1 k ¡2 Ek 4 Z T 0 u 2 (t)dt= T 2 4µ 1 : (4.2.27) As a result, EJ 1;N » N X k=1 T 2 k 2 4µ 1 = T 2 N 3 12µ 1 : (4.2.28) Let us now apply Theorem 1.4.5 and Corollary 1.4.6 to prove the consistency. First apply Corollary 1.4.6 withÂ 2 k =k 4 R T 0 u 2 k (t)dt and by (4.3.2), (4.2.27), lim N!1 J 1;N EJ 1;N = P N k=1 Â 2 k P N k=1 EÂ 2 k =1: (4.2.29) Next apply Theorem 1.4.5 withÂ k =k 2 R T 0 u k dw k (t) to conclude that lim N!1 » 1;N EJ 1;N = P N k=1 Â k P N k=1 EÂ 2 k =0; (4.2.30) with probability one and then (4.2.21) follows from (4.2.29) and (4.2.30). 79 Asymptotic normality (4.2.22) now follows from (4.2.39) and Corollary 1.4.9. Since u k (t) is a Gaussian random variable, let ¾ 2 k =Ek 4 Z T 0 u 2 (t)dt» T 2 k 2 4µ 1 : Therefore, P n k=1 ¾ 2 k »Tn 3 =12µ 1 : The rate of this convergence isN 3=2 : Ifµ 2 6=0, the maximum likelihood ratio dP v;µ 1 N;T dP w N;T ³ v N (µ 1 ) ´ =exp Ã N X k=1 ³ Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ dv k (t) ¡ 1 2 Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ 2 dt ´ ! ; (4.2.31) and the maximum likelihood estimator ^ µ (1) N = B 1;N+ µ 2 J 12;N J 1;N : (4.2.32) The results in Theorem 4.2.4 also hold in this case except that the variance of the Gaussian variable ³ is 3µ=C(µ 2 ;T) which is defined on the next page by applying Theorem 2.2.2. The following is an outline for the proof. ^ µ (1) N ¡µ 1 =¡ » 1;N J 1;N : (4.2.33) 80 Define2b=¡µ 2 ,a 2 =µ 1 k 2 andl = p a 2 ¡b 2 , C(µ 2 ;T)= e µ 2 T ¡µ 2 T ¡1 2µ 2 2 ; M(µ 2 ;T)= e 2µ 2 T +4e µ 2 T ¡4µ 2 Te µ 2 T ¡2µ 2 T ¡5 4µ 2 2 : (4.2.34) Note thatlim µ 2 !0 C(µ 2 ;T)=T 2 =4 andlim µ 2 !0 M(µ 2 ;T)=T 4 =24. By Theorem 2.2.2, Ek 4 Z T 0 u 2 k (t)dt» k 2 C(µ 2 ;T) µ 1 ; Vark 4 Z T 0 u 2 k (t)dt» k 4 M(µ 2 ;T) µ 2 1 : (4.2.35) Therefore, Vark 4 Z T 0 u 2 k (t)dt< µ Ek 4 Z T 0 u 2 k (t)dt ¶ 2 ; (4.2.36) and lim k!1 k ¡2 Ek 4 Z T 0 u 2 k (t)dt= C(µ 2 ;T) µ 1 ; (4.2.37) EJ 1;N = N 3 C(µ 2 ;T) 3µ 1 ; (4.2.38) Apply Corollary 1.4.6 withÂ 2 k = R T 0 k 4 u 2 k (t)dt, lim N!1 J 1;N EJ 1;N = P N k=1 Â 2 k P N k=1 EÂ 2 k =1: (4.2.39) Next apply Theorem 1.4.5 withÂ k =k 2 R T 0 u k dw k (t) to conclude that 81 lim N!1 » 1;N EJ 1;N = P N k=1 Â k P N k=1 EÂ 2 k =0; (4.2.40) with probability one and then (4.2.21) follows from (4.2.39) and (4.2.40). 4.2.2 Estimateµ 2 , assuming thatµ 1 is known In this section we assume that µ 2 is not zero because µ 2 is the parameter subject to estimation. Similar to the previous section, the maximum likelihood ratio is dP v;µ 2 N;T dP w N;T ³ v N (µ 2 ) ´ =exp Ã N X k=1 ³ Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ dv k (t) ¡ 1 2 Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ 2 dt ´ ! ; (4.2.41) and the maximum likelihood estimator is ^ µ (2) N = B 2;N +µ 1 J 12;N J 2;N (4.2.42) 4.2.5 Theorem. Estimator (4.2.42) is strongly consistent and asymptotically normal in the limitN !1: lim N!1 ^ µ (2) N =µ 2 with probability one; (4.2.43) lim N!1 N 1=2 ( ^ µ (2) N ¡µ 2 )=³ in distribution (4.2.44) where³ is a Gaussian random variable with zero mean and variance1=C(µ 2 ;T). 82 Proof. It follows from (4.2.42) that ^ µ (2) N ¡µ 2 = » 2;N J 2;N : (4.2.45) To prove consistency, we apply Theorem 1.4.5 and Corollary 1.4.6. Define 2b = ¡µ 2 , a 2 =µ 1 k 2 andl = p a 2 ¡b 2 . By Theorem 2.2.2, E Z T 0 v 2 k (t)dt»C(µ 2 ;T) Var Z T 0 v 2 k (t)dt»D(µ 2 ;T); (4.2.46) therefore, Var Z T 0 v 2 k (t)dt< µ E Z T 0 v 2 k (t)dt ¶ 2 ; (4.2.47) and lim k!1 E Z T 0 v 2 k (t)dt=C(µ 2 ;T); (4.2.48) EJ 2;N »NC(µ 2 ;T): (4.2.49) Apply Corollary 1.4.6 withÂ 2 k = R T 0 v 2 k (t)dt, by the previous argument, lim N!1 J 2;N EJ 2;N = P N k=1 Â 2 k P N k=1 EÂ 2 k =1: (4.2.50) 83 and by Theorem 1.4.5 withÂ k = R T 0 v 2 k (t)dt, lim N!1 » 2;N E» 2;N = lim N!1 P N k=1 Â k EÂ 2 k =0: (4.2.51) Therefore, ^ µ (2) N is a consistent estimator of µ 2 . The Asymptotic normality follows from (1.32) and Corollary 1.4.9. (4.2.48) shows that the rate of convergence isN 1=2 . 4.2.3 Estimateµ 1 andµ 2 simultaneously. The maximum likelihood ratio dP v;µ 1 ;µ 2 N;T dP w N;T ³ v N (µ 1 ;µ 2 ) ´ =exp Ã N X k=1 ³ Z T 0 ¡ ¡µ 1 k 2 u k (t)+µ 2 v k (t) ¢ dv k (t) ¡ 1 2 Z T 0 ¡ µ 1 k 2 u k (t)+µ 2 v k (t) ¢ 2 dt ´ ! : (4.2.52) We differentiate the expression on the right-hand side with respect toµ 1 andµ 2 and get the following expression for the maximum likelihood estimators ^ µ (1) N ofµ 1 and ^ µ (2) N ofµ 2 based on the observationsu k (t);k =1;:::;N,t2[0;T]: ^ µ (3) 1;N = B 1;N J 2;N +B 2;N J 12;N J 1;N J 2;N ¡J 2 12;N (4.2.53) ^ µ (3) 2;N = B 1;N J 12;N +B 2;N J 1;N J 1;N J 2;N ¡J 2 12;N : (4.2.54) 84 4.2.6 Theorem. If bothµ 1 andµ 2 are unknown, then lim N!1 ^ µ (3) 1;N =µ 1 ; lim N!1 ^ µ (3) 2;N =µ 2 (4.2.55) with probability one and lim N!1 N 3=2 ( ^ µ (3) 1;N ¡µ 1 ) d =³; (4.2.56) where³ is a Gaussian random variable with zero mean and variance3µ 1 =C(µ 2 ;T), lim N!1 N 1=2 ( ^ µ (3) 2;N ¡µ 2 ) d =³ (4.2.57) where³ is a Gaussian random variable with zero mean and variance1=C(µ 2 ;T). Proof. Define D N = J 2 12;N J 1;N J 2;N : (4.2.58) Sinceu k (T) is a Gaussian process and by (2.2.33), k 2 Eu 2 k (T)» e µ 2 T ¡1 2µ 1 µ 2 ; Vark 2 u 2 k (T)» (e µ 2 T ¡1) 2 2µ 2 1 µ 2 2 <C µ k 2 Eu 2 k (T) ¶ 2 ; whereC is a constant and lim k!1 k 2 Eu 2 k (T)= e µ 2 T ¡1 2µ 1 µ 2 : Note that EJ 12;N =1=2 N X k=1 k 2 Eu 2 k (T)» N(e µ 2 T ¡1) 4µ 1 µ 2 : 85 Then by the strong law of large numbers (Corollary 1.4.6)withÂ 2 k =k 2 u 2 k (T), lim N!1 J 12;N EJ 12;N = lim N!1 P N k=1 k 2 u 2 k (T) P N k=1 Ek 2 u 2 k (T) =1: As a result,lim N!1 D N =0 with probability one. Also B 1;N =µ 1 J 1;N ¡µ 2 J 12;N ¡» 1;N ; B 2;N =¡µ 1 J 12;N +µ 2 J 2;N +» 2;N : (4.2.59) It follows from (4.2.53) and (4.2.54) that ^ µ (3) 1;N ¡µ 1 = 1 1¡D N ³ » 1;N J 1;N + » 2;N J 12;N J 1;N J 2;N ´ ; ^ µ (3) 2;N ¡µ 2 = 1 1¡D N ³ » 2;N J 2;N + » 1;N J 12;N J 1;N J 2;N ´ : (4.2.60) By the argument in the previous sections, lim N!1 » 1;N EJ 1;N =0; lim N!1 » 2;N EJ 2;N =0; lim N!1 J 1;N EJ 1;N =1; lim N!1 J 2;N EJ 2;N =1 (4.2.61) And sincelim N!1 J 12;N =EJ 12;N =1; lim N!1 » 2;N J 12;N J 1;N J 2;N =0; lim N!1 » 1;N J 12;N J 1;N J 2;N =0: (4.2.62) and so lim N!1 ^ µ (3) 1;3 =µ 1 ; lim N!1 ^ µ (3) 2;3 =µ 2 : (4.2.63) 86 There are some pictures of the trajectories and solution: 0 5 10 15 20 −5 0 5 10 Fourier coefficient θ 1 Estimates True value 0 5 10 15 20 −5 0 5 10 Fourier coefficient θ 2 Estimates True value Figure 4.1: Estimates vs true values whenµ 1 =1 andµ 2 =¡1. 87 0 5 10 15 20 −10 −8 −6 −4 −2 0 2 4 6 8 10 Fourier coefficient θ 1 Estimates True value 0 5 10 15 20 −15 −10 −5 0 5 10 15 Fourier coefficient θ 2 Estimates True value Figure 4.2: Estimates vs true values whenµ 1 =1 andµ 2 =1. 4.3 Parameter Estimation in General Stochastic Hyperbolic Equations 4.3.1 Analysis of Estimators: Algebraic Case Asymptotic problems for stochastic differential equations arose and were solved simultaneously with the very beginnings of the theory of such equations, because one of the founders of this theory, I.I. Gikhman, was considering first and foremost problems on asymptotic behavior, and he constructed the equations themselves partly in order to be able to pose and solve these problems rigorously. A cycle of papers by Krylov and Bogolyubov [10, 54, 52] were devoted to these investigations. 88 0 5 10 15 20 −2 0 2 4 6 8 10 12 Fourier coefficient θ 1 Estimates True value 0 5 10 15 20 −2 0 2 4 6 8 10 Fourier coefficient θ 2 Estimates True value Figure 4.3: Estimates vs true values whenµ 1 =1 andµ 2 =0. In this section, we consider the diagonalizable hyperbolic equation u tt +(A 0 +µ 1 A 1 )u=(B 0 +µ 2 B 1 )u t + _ W; 0<t·T; x2(0;¼); (4.3.1) driven by a cylindrical Brownian motion on a Hilbert spaceH with zero initial and bound- ary conditions. LetX be a Hilbert space such thatH is a dense subset ofX andW =W(t) is anX-valued continuous square-integrable martingale. The objectives of this chapter are: ² to determine the conditions on the operators so that the equation has a generalized solution that is a square-integrable random element with values in a suitable Hilbert space; 89 ² to construct a maximum likelihood estimator of the unknown parametersµ 1 ,µ 2 using a finite-dimensional projection of the solution, and to study the asymptotic proper- ties of the estimator as the dimension of the projection grows. The solutionu=u(t) of the equation is a continuousX-valued process u(t)= X k¸1 u k (t)h k (x) whereu k (t) is the solution to the second order stochastic ordinary differential euqation u 00 k (t)¡¸ k (µ 1 )u 0 k (t)¡¹ k (µ 2 )k 2 = _ w k ; u k (0)=0; u 0 k (t)=0; fh k , k ¸ 1g is an orthonormal basis in H, w k = W h k , A 0 h k = · k h k , A 1 h k = ¿ k h k , ¡¸ k (µ 1 )=· k +µ 1 ¿ k ,B 0 h k =½ k h k ,B 1 h k =º k h k and¹ k (µ 2 )=½ k +µ 2 º k . Assume that the observations ofv k (t) are available fort2 [0;T] andk = 1;:::;N. For each µ (the parameter subject to estimation, either µ 1 or µ 2 ) and each k, the Ornstein- Uhlenbeck process v k (t) generates the measure P v;µ;k T in the space of continuous real- valued functions on [0;T], and, by Theorem 1:4:1(a), the measures are equivalent for different valued of µ. Similarly, the vector v N;µ = (v 1 ;:::;v N ) generates a proba- bility measure P µ N;T on the space of continuousR N -valued functions on [0;T]. Since the random processes u k are independent for different k, P v;µ N;T is a product measure: P v;µ N;T = Q N k=1 P v;µ;k T , and thus the measures P v;µ N;T are equivalent for different values of µ. 90 Substitution of the Fourier series v(t;x)= X k¸1 v k (t)h k (x); u(t;x)= X k¸1 u k (t)h k (x) suggests that eachv k (t) should satisfy dv k (t)= ¡ ¡¸ k (µ 1 )u k +¹ k (µ 2 )v k ¢ dt+dw k (t): (4.3.2) The density generated by the processv N (µ 1 ) can be derived directly from (4.3.2): Z N (µ 1 ;µ 2 )=exp Ã N X k=1 ³ Z T 0 ¡ ¡¸ k (µ 1 )u k (t)+¹ k (µ 2 )v k (t) ¢ dv k (t) ¡ 1 2 Z T 0 ¡ ¡¸ k (µ 1 )u k (t)+¹ k (µ 2 )v k (t) ¢ 2 dt ´ ! : (4.3.3) Introduce the following notations: A 1;N =¡ N X k=1 Z T 0 ¿ k u k (t)dv k (t); A 2;N = N X k=1 Z T 0 º k v k (t)dv k (t); F 1;N =¡ N X k=1 Z T 0 · k ¿ k u 2 k (t)dt; F 2;N =¡ N X k=1 Z T 0 ½ k º k v 2 k (t)dt; K 1;N = N X k=1 Z T 0 ¿ 2 k u 2 k (t)dt; K 2;N = N X k=1 Z T 0 º 2 k v 2 k (t)dt; ¶ 1;N = N X k=1 Z T 0 ¿ k u k (t)dw k (t); ¶ 2;N = N X k=1 Z T 0 º k v k (t)dw k (t); 91 L 1;N =¡ N X k=1 Z T 0 ½ k ¿ k u k (t)v k (t)dt; L 2;N =¡ N X k=1 Z T 0 · k º k u k (t)v k (t)dt; K 12;N =¡ N X k=1 Z T 0 º k ¿ k u k (t)v k (t)dt; D N = K 2 12;N K 1;N K 2;N : Note that the numbersA,F ,L andK are computable from the observations ofu k and v k ,k =1;:::;N, and also A 1;N =¡F 1;N ¡µ 1 K 1;N ¡L 1;N ¡µ 2 K 12;N +¶ 1;N ; A 2;N =¡F 2;N ¡µ 2 K 12;N ¡L 2;N ¡µ 1 K 12;N +¶ 2;N ; L 1;N =¡ 1 2 N X k=1 ½ k ¿ k u 2 k (T); L 2;N =¡ 1 2 N X k=1 · k º k u 2 k (T); K 12;N =¡ 1 2 N X k=1 ¿ k º k u 2 k (T); (4.3.4) because, by assumption,u k (0)=0 and thus Z T 0 u k v k (t)dt= Z T 0 u k (t)du k (t)= 1 2 u 2 k (T): We consider three estimation problems: 4.3.1 Problem. Estimateµ 1 ifµ 2 is known. Maximizing the expression on the right-hand side of (4.3.2) with respect toµ 1 , we get the following expression for the maximum likeli- hood estimator ^ µ (1) N ofµ 1 based on the observationsu k (t),k = 1;:::;N,t2 [0;T]. Then the maximum likelihood estimator ^ µ (1) 1;N ofµ 1 satisfies(@Z N =@µ 1 )( ^ µ (1) 1;N ;µ 2 ) or 92 ^ µ (1) 1;N = A 1;N +F 1;N +L 1;N +µ 2 K 12;N K 1;N (4.3.5) 4.3.2 Problem. Estimate µ 2 if µ 1 is known. Then the maximum likelihood estimator ^ µ (2) 2;N ofµ 2 satisfies(@Z N =@µ 2 )(µ 1 ; ^ µ (2) 2;N ) or ^ µ (2) 2;N = A 2;N +F 2;N +L 2;N +µ 1 K 12;N K 2;N (4.3.6) 4.3.3 Problem. Estimate both µ 1 and µ 2 . Then the maximum likelihood estimator ^ µ (3) 1;N , ^ µ (3) 2;N satisfies(@Z N =@µ 1 )( ^ µ (3) 1;N ; ^ µ (3) 2;N )=0 and(@Z N =@µ 2 )( ^ µ (3) 1;N ; ^ µ (3) 2;N )=0, or ^ µ (3) 1;N = A 1;N K 2;N ¡F 1;N K 2;N ¡L 1;N K 2;N ¡A 2;N K 12;N +L 2;N K 12;N +F 2;N K 12;N K 1;N K 2;N ¡K 2 12;N ; (4.3.7) ^ µ (3) 2;N = A 1;N K 12;N ¡F 1;N K 12;N ¡L 1;N K 12;N ¡A 2;N K 1;N +L 2;N K 1;N +F 2;N K 1;N K 1;N K 2;N ¡K 2 12;N : (4.3.8) If the equation (4.3.1) is hyperbolic, thenlim k!1 ¸ k (µ 1 ) = +1 uniformly inµ 1 2£ and there exists an idexJ ¸1 such that, for allk¸J andµ2£, ¸ k (µ 1 )>0: (4.3.9) Obviously, as N ! 1, asymptotic behavior of the estimators is determined by 93 ¶ i;N =K i;N , i = 1;2, and K 12;N =(K 1;N K 2;N ). Note that each of ¶ i;k , K i;N , K 12;N is a sum of independent random variables. Moreover, by Ito’s Isometry, E¶ 2 i;N =EK i;N ; i=1,2. Iff k is the function satisfying Ä f k (t)¡¹ k (µ 2 ) _ f k (t)+¸ k (µ 1 )f k (t)=0; f k (0)=0; _ f k (0)=1; (4.3.10) then, by direct computation,u k (t)= R t 0 f k (t¡s)dw k (s), so that Eu 2 k (t)= Z t 0 jf k (s)j 2 ds; Ev 2 k (t)= Z t 0 j _ f k (s)j 2 ds; and we define © 1;N :=EK 1;N = N X k=1 ¿ 2 k Z T 0 Z t 0 jf k (s)j 2 dsdt; (4.3.11) © 2;N :=EK 2;N = N X k=1 º 2 k Z T 0 Z t 0 j _ f k (s)j 2 dsdt; © 12;N :=EK 12;N =¡ 1 2 N X k=1 ¿ k º k Z T 0 Z t 0 jf k (s)j 2 dsdt: (4.3.12) The following is a necessary conditions for the consistency of the estimators. 94 4.3.4 Theorem. Iflim N!1 ^ µ (1) 1;N =µ 1 in probability, thenlim N!1 © 1;N =+1. Similarly, iflim N!1 ^ µ (2) 2;N =µ 2 in probability, thenlim N!1 © 2;N =+1. Proof. Each of the sequencesf© i;N , N ¸ 1g is monotonically increasing and thus has a limit, finite or infinite. If lim N!1 © i;N <1, then lim N!1 ¶ i;N =K i;N exists with proba- bility one and is a non-degenerate random variable. Define ¹ D N = K 2 12;N K 1;N K 2;N : (4.3.13) ¹ D N ·1 by Cauchy-Swartz inequality and in fact ¹ D N <1. ^ µ (3) 1;N ¡µ 1 = ¶ 1;N K 2;N ¡¶ 2;N K 12;N K 1;N K 2;N ¡K 2 12;N = 1 1¡ ¹ D N ³ ¶ 1;N K 1;N ¡¶ 2;N p ¹ D N p K 1;N K 2;N ´ ; ^ µ (3) 2;N ¡µ 2 = ¶ 2;N K 1;N ¡¶ 1;N K 12;N K 1;N K 2;N ¡K 2 12;N = 1 1¡ ¹ D N ³ ¶ 2;N K 2;N ¡¶ 1;N p ¹ D N p K 1;N K 2;N ´ (4.3.14) implies that the estimators cannot converge toµ i . 4.3.5 Proposition. If equation (4.3.1) is diagonalizable and hyperbolic, then lim k!1 ¸ k =+1 (4.3.15) uniformly in µ 2 £ 1 , and there exists an index J ¸ 1 and a number c 0 such that, for all k¸J andµ2£ 1 , ¸ k >1; (4.3.16) 95 j¿ k j ¸ k ·c 0 : (4.3.17) Proof. Since equation (4.3.1) is hyperbolic and the condition C 1 · ¸ k (µ)+C ¤ ¸ k (µ 0 ) ·C 2 ; (4.3.18) for all µ, µ 0 2 £ 1 , C 1 , C 2 , C ¤ > 0, we have f¸ k (µ) + C ¤ ,k ¸ 1g is a positive, non- decreasing, and unbounded sequence for all µ 2 £, so that lim k!1 ¸ k = +1, and then (4.3.16) follows. To prove (4.3.17), assume that the sequence fj¿ k j¸ ¡1 k , k ¸ 1g is not uniformly bounded, then there exists a subsequencefj¿ k j j¸ ¡1 k j ,j¸1g, such that lim j!1 j¿ k j j ¸ k j =1: (4.3.19) Since£ 1 is compact, assume that lim j!1 µ j =µ 0 2£; Without loss of generality, we also assume that¿ k j >0; then (4.3.19) implies lim j!1 · k j ¿ k j =¡µ 0 : Becauselim j!1 (µ 0 ¿ k j +· k j )=+1,lim j!1 j¿ k j j=+1. Consequently, lim j!1 ¸ k j (µ)+C ¤ ¸ k j (µ 0 )+C ¤ = µ¡µ 0 µ 0 +lim j!1 (· k j =¿ · k j ) =+1; µ6=µ 0 : 96 As a result, if (4.3.17) fails, so does (4.3.18) forµ6=µ 0 ,µ 0 =µ 0 . In the usual setting to define a variational solution, (see, for example, Chow [13]), we have the assumption that the operators are self-adjoint or act in a normal triple of Hilbert space. However, in our setting, we do not have enough information about the operatorsA i andB i to define the traditional variational solution. To state the existence and uniqueness of solution for (4.3.1), we use the following assumptions: assuming that the equation (4.3.1) is diagonalizable, that is, the operators A i and B i have a common system of eigenfunctions such that it is an orthonomal basis in the Hilbert space. The solutionu has the following expansion in the basisfh k ,k¸1g: u(t)= X k¸1 u k (t)h k (x); (4.3.20) u 00 k (t)=¡¸ k (µ 1 )u k (t)+¹ k (µ 2 )u 0 k (t)+ _ w k (t) (4.3.21) with conditionsu k (0)=u 0 k (0)=0: We define a generalized solution of equation (4.3.1) under more general assumptions. 4.3.6 Theorem. Assume that equation (4.3.1) is diagonalizable and hyperbolic, and there exist positive constantsC andJ, such that, T(½ k +µ 2 º k )·ln(· k +µ 1 ¿ k )+C; (4.3.22) for allk¸ J. Then there exists a unique adaptedX-valued processu = u(t) with repre- sentation (4.3.20) and (4.3.21). This process is called a generalized solution of (4.3.1). In 97 addition, if there exists a real numberC 0 such that¹ k · C 0 for allk, thenv(t) = _ u(t) is also anX-valued process. Proof. Consider the solutionu k (t) to the equation u 00 k (t)=¡¸ k (µ 1 )u k (t)+¹ k (µ 2 )u 0 k (t)+ _ w k (t) (4.3.23) with conditionsu k (0) = u 0 k (0) = 0: By the computations in Chapter Two, It has a unique solution andu k (t) satisfies u k (t)= Z t 0 f k (t¡s)dw k (s); where the fundamental solutionf k satisfies f 00 k (t)¡¹ k f 0 k (t)+¸ k f k (t)=0; f k (0)=0; f 0 k (0)=1: (4.3.24) Since the processu k is a Gaussian process,Eu k (t)=0 andu k are independent for differ- entk, the seriesfu k (t)g,k¸1 defines anX-valued process if and only if sup k¸1 sup t2[0;T] Eju k (t)j 2 <1: By Ito’s isometry, Eju k (t)j 2 = Z t 0 f 2 (t¡s)ds= Z t 0 f 2 (s)ds: 98 Therefore, we need sup k¸1 sup t2[0;T] Z t 0 f 2 (s)ds<1: To proof the theorem, consider the corresponding homogeneous characteristic equation of the equation (4.3.24): r 2 ¡¹ k r+¸ k =0: By hyperbolicity, lim k!1 ¸ k = +1. In particular, ¸ k > 0 for all k sufficiently large. Condition (4.3.22) implies that ¹ k can be very negative, but if ¹ k is positive, it must be much smaller than¸ k . Consider the following two cases: ¡2 p ¸ k · ¹ k · 2 p ¸ k and¹ k <¡2 p ¸ k . By the computations in Chapter two, if¡2 p ¸ k · ¹ k · 2 p ¸ k , the characteristic function has two complex conjugate roots (a double root if¹ k =4¸ k . Denotel k = p ¸ k ¡(¹ 2 k =4), f 2 k (t)=te ¹ k t ¡ sin(l k t) l k t ¢ 2 : Obviously,f 2 k (t) is bounded byT if¹ k ·0 for allt2[0;T]. If¹ k >0, condition (4.3.22) implies thate ¹ k t ·¸ k e C and¸ k =l 2 k <2. Therefore, sup t2[0;T] sup k¸1 f 2 k (t)<1: 99 If ¹ k < ¡ p ¸ k , then the characteristic equation has two real roots. Denote l k = p (¹ 2 k )=4¡¸ k , f 2 k (t)=4t 2 e ¹ k +l k =2 ¡ 1¡e ¡l k t=2 l k t ¢ 2 ·T 2 : This completes the proof the the Theorem. Let us make a few comments about the result. 1. By proposition (4.3.5), we know that¸ k > 1 for all sufficiently largek. Condition (4.3.22) means certain subordination of the dissipation operatorB 0 +µ 2 B 1 to the evo- lution operatorA 0 +µ 1 A 1 . In particular, any dissipation (negative¹ k = ½ k +µ 2 º k ) is admissible, as well as certain unbounded amplification (positive and bounded¹ k ), as long as the sequencef¹ k ,k¸ 1g does not grow too fast; the critical growth rate depends on the length of the time interval. This possibility to have an unbounded dissipation operator makes the result different from those considered in the litera- ture. 2. The resulting generalized solution is in the PDE sense, but it is strong in the proba- bility sense, being constructed on a given stochastic basis. 3. The solution is defined by its Fourier coefficients and therefore does not depend on the choice of the spaceX. The role ofX is to ensure that the equation is well-posed in the sense that the output process (the solution u) takes values in the same same as the ”input” processW . Given the special form ofW , we are not discussing any continuous dependence ofu onW . 100 Under the assumption of Theorem (4.3.6), we derived a boundjf k (t)j 2 ·const¢T 2 , which can be used to establish the existence and uniqueness condition of (4.2.8). The expected value and variance of u 2 k (t) and v 2 k (t) over the interval [0;T] can be derived from the computations in Chapter three. Eu 2 k (T)» e ¹ k (µ 2 )T ¡1 2¹ k (µ 2 )¸ k (µ 1 ) ; Varu 2 k (T)»3 ³ e ¹ k (µ 2 )T ¡1 2¹ k (µ 2 )¸ k (µ 1 ) ´ 2 ; (4.3.25) E Z T 0 u 2 k (t)dt» T 2 M ¡ T¹ k (µ 2 ) ¢ ¸ k (µ 1 ) ; Var Z T 0 u 2 k (t)dt» T 4 V ¡ T¹ k (µ 2 ) ¢ ¸ 2 k (µ 1 ) ; (4.3.26) E Z T 0 v 2 k (t)dt»T 2 M ¡ T¹ k (µ 2 ) ¢ ; Var Z T 0 v 2 k (t)dt»T 4 ¡ T¹ k (µ 2 ) ¢ ; (4.3.27) where M(x)= 8 > > < > > : e x ¡x¡1 2x 2 ; ifx6=0; 1 4 ; ifx=0; (4.3.28) v(x)= 8 > > < > > : e 2x +4e x ¡4xe x ¡2x¡5 4x 4 ; ifx6=0; 1 24 ; ifx=0: (4.3.29) 101 Note that the functionsM andV are continuous and positive onR, and M(x)» 8 > > < > > : (2jxj) ¡1 ; x!¡1; 2(2x) ¡2 e x ; x!+1; V(x)» 8 > > < > > : 4(2jxj) ¡3 ; x!¡1; 4(2x) ¡4 e 2x ; x!+1: These computations are derived on the fact that u k andv k are Gaussian processes, so that, for example, Var Z T 0 u 2 k (t)dt=4 Z T 0 Z t 0 ³ E ¡ u(t)u(s) ¢ ´ 2 dsdt: It follows from (4.3.26) and (4.3.27) that iflim N!1 © i;N =+1, then © 1;N » T 2 M ¡ T¹ k (µ 2 ) ¢ ¸ k (µ 1 ) ; © 2;N »T 2 M ¡ T¹ k (µ 2 ) ¢ : These relations show that conditions for consistency and asymptotic normality of the estimators require additional assumptions on the asymptotical behavior of the eigenvalues of the operatorsA i andB i . The asymptotic behavior of the eigenvalues of an operator is well-known when the operator is elliptic and self-adjoint. For example, letD be an operator defined on smooth functions by Df(x)=¡ d X i;j=1 @ @x i ³ a ij (x) @f(x) @x j ´ ; in a smooth bounded domainG2R d , with zero Dirichlet boundary conditions. Assume 102 that the functionsa ij are all finitely differentiable inG and are bounded with all the deriva- tives, and the matrix(a ij (x),i;j =1;:::;d) is symmetric and uniformly positive-definite for allx2G. Then the eigenvaluesd k ofD can be enumerated so that d k ³k 2=d (4.3.30) in the sense of notation (1.3.2). More generally, for a positive-definite, elliptic self-adjoint differential operatorD of orderm on a smooth bounded domain inR d with suitable bound- ary conditions or on a smooth compact d-dimensional manifold, the asymptotic of the eigenvaluesd k ;k¸1, is d k ³k m=d ; (4.3.31) Note thatm can be an arbitrary positive number. This result is well-known; for exam- ple, Safarov and Vassiliev [83]. An example ofD is(1¡¢) m=2 ,m > 0, where¢ is the laplace operator; note that, for this operator, relation (4.3.30) holds even whenm·0. In our setting, when the operators are defined by their eigenvalues and eigenfunctions, more exotic eigenvalues are possible, for example,¿ k =e k orº k =(¡1) k =k. On the other hand, it is clear that the analysis of the estimators should be easier when all the eigenvalues are of the type (4.3.30). Accordingly, we make the following definition. 4.3.7 Definition. Equation (4.2.1) is called algebraically hyperbolic if it is diagonaliz- able, hyperbolic, and the eigenvalues ¸ k (µ) = · k , ¹ k (µ) = ½ k +µº k have the following properties: 103 1. There exist real numbers®,® 1 such that, for allµ2£ 1 , ¸ k ³k ® ; j¿ k j³k ® 1 ; (4.3.32) 2. Eitherj¹ k (µ)j· C for allµ 2 £ 2 or there exist numbers¯ > 0,¯ 2R such that, for allµ2£ 2 , ¡¹ k (µ)³k ¯ ; jº k j³k ¯ 1 (4.3.33) To emphasis the importance of the numbers ® and ¯, we will sometimes say that the equation is(®;¯)-algebraically hyperbolic;¯ =0 includes the case of uniformly bounded ¹ k (µ). It can be easily verified that ² under hyperbolic assumption,®>0 and no unbounded amplification is possible; ² each of the equation in (3.3.5) is algebraically hyperbolic. We derive the following theorem: 4.3.8 Theorem. Assume that equation (4.2.1) is diagonalizable and hyperbolic and that A i ,B i are positive-definite elliptic self-adjoint differential or pseudo-differential operators on a smooth bounded domain inR d with suitable boundary conditions or on a smooth compactd-dimensional manifold. Then 1. the maximum likelihood estimator of µ 1 is consistent and asymptotically normal in the limitN !1 if and only if order(A 1 )¸ order(A 0 +µ 1 A 1 )+ order(B 0 +µ 2 B 1 )¡d 2 ; (4.3.34) 104 2. the maximum likelihood estimator of µ 2 is consistent and asymptotically normal in the limitN !1 if and only if order(B 1 )¸ order(B 0 +µ 2 B 1 ) 2 : (4.3.35) Similar to the parabolic case (Huebner [36]), the results extend to a more general esti- mation problem Ä u+ n X i=0 µ 1i A i u= m X j=0 µ 2j B j _ u+ _ W; as long as all the operatorsA i ,B j have a common system of eigenfunctions. For example, in the setting similar to Theorem 4.3.8, the coefficientµ 1p can be consistently estimated if and only if order(A p )¸ order( P n i=0 µ 1i A i )+ ³ P m j=0 µ 2j B j ´ ¡d 2 : 4.3.9 Theorem. Assume that equation (4.2.1) is (®;¯)-algebraically hyperbolic in the sense of Definition 4.3.7. 1. If ® 1 ¸ ®+¯¡1 2 ; (4.3.36) then the estimator ^ µ (3) 1;N is a strongly consistent and asymptotically normal with rate p © 1;N asN !1: lim N!1 ^ µ (3) 1;N =µ 1 with probability one; (4.3.37) lim N!1 p © 1;N ¡ ^ µ (3) 1;N ¡µ 1 ¢ =» 1 in distribution; (4.3.38) 105 where» 1 is a standard Gaussian random variable. 2. If ¯ 1 ¸ ¯¡1 2 ; (4.3.39) then the estimator ^ µ (3) 2;N is a strongly consistent and asymptotically normal with rate p © 2;N asN !1: lim N!1 ^ µ (3) 2;N =µ 2 with probability one; (4.3.40) lim N!1 p © 2;N ¡ ^ µ (3) 2;N ¡µ 2 ¢ =» 2 in distribution; (4.3.41) where» 2 is a standard Gaussian random variable. 3. if both (4.3.36) and (4.3.39) hold, then the random variables» 1 and» 2 are indepen- dent. Proof. Note that ifbeta>0, thenlim k!1 ¹ k (µ)=¡1, and therefore, by (4.3.30), M k ¡ T¹ k (µ) ¢ » 1 2Tj¹ k (µ)j ³k ¯ ; V k ¡ T¹ k (µ) ¢ » 1 2jT¹ k (µ)j 3 ³k ¡3¯ : (4.3.42) Let° 1 =2® 1 ¡®¡¯,° 2 =2¯ 1 =¯,° 12 =®¡®+¯ 1 ¡¯. We have ³ see (1.3.1) for the definition of./ ´ ¿ 2 k E Z T 0 u 2 k (t)dt./k ° 1 ; ¿ 4 k Var Z T 0 u 2 k (t)dt./k 2° 1 ¡¯ ; (4.3.43) 106 º 2 k E Z T 0 v 2 k (t)dt./k ° 2 ; º 4 k Var Z T 0 v 2 k (t)dt./k 2° 2 ¡¯ ; (4.3.44) jº k ¿ k jEu 2 k (T)./k ° 12 ; º 2 k ¿ 2 k Varu 2 k (T)./k 2° 12 ; (4.3.45) and therefore © 1;N ./ 8 > > > > > > < > > > > > > : const; if° 1 <¡1; lnN; if° 1 =¡1; N ° 1 +1 ; if° 1 >¡1; (4.3.46) © 2;N ./ 8 > > > > > > < > > > > > > : const; if° 2 <¡1; lnN; if° 2 =¡1; N ° 2 +1 ; if° 2 >¡1; (4.3.47) j© 12;N j./ 8 > > > > > > < > > > > > > : const; if° 12 <¡1; lnN; if° 12 =¡1; N ° 12 +1 ; if° 12 >¡1; (4.3.48) 107 Next, we show that condition (4.3.36) implies lim N!1 K 1;N © 1;N =1 with probability one; (4.3.49) condition (4.3.39) implies lim N!1 K 2;N © 2;N =1 with probability one; (4.3.50) and either condition (4.3.36) or condition (4.3.39) implies lim N!1 D N =0 with probability one: (4.3.51) Convergence of (4.3.49) follows from (4.3.26) and Corollary 1.4.6, because (4.3.36) implies X n n 2° 1 © 2 1;N <1: Similarly, (4.3.50) follows from (4.3.27) and Corollary 1.4.6, because (4.3.39) implies X n n 2° 2 © 2 2;N <1: For (4.3.1), we first observe that lim B!1 K 12;N =© 12;N exists with probability one. If ° 12 <¡1, then the limit is aP-a.s. finite random variable. If° 12 ¸¡1, then (4.3.25) and Corollary 1.4.6 imply that this limit is1. Then direct analysis shows that © 2 12;N © 1;N © 2;N =0 108 if at least one of© 1;N and© 2;N is unbounded. (4.3.1) then follows. Next, we show that (4.3.36) implies lim N!1 ¶ 1;N © 1;N =0 with probability one; (4.3.52) and lim N!1 ¶ 1;N p © 1;N =» 1 in distribution; (4.3.53) whereas (4.3.39) implies lim N!1 ¶ 2;N © 2;N =0 with probability one; (4.3.54) and lim N!1 ¶ 2;N p © 2;N =» 1 in distribution: (4.3.55) In fact, (4.3.52) follows from (4.3.26) and (1.4.2), because (4.3.36) implies that P k k ° 1 =+1: Similarly, (4.3.49) follows from (4.3.27) and (1.4.2). Both (4.3.53) and (4.3.55) follow from Corollary 1.4.13. Together with (4.3.1), the same Corollary also implies independence of» 1 and» 2 if both (4.3.36) and (4.3.39) hold. It is easy to verify that lim N!1 ¶ i;N K 12;N K 1;N K 2;N =0; i=1;2; with probability one; (4.3.56) follows from (1.4.2) and lim N!1 p © i;N ¶ i;N K 12;N K 1;N K 2;N =0; i=1;2; with probability one: (4.3.57) 109 follows from (4.3.1), (4.3.53) and (4.3.55). Indeed, if one of© i;N ,i=1;2 diverges, lim N!1 ¶ i;N K 12;N K 1;N K 2;N = lim N!1 ¶ i;N p D N p K 1;N K 2;N =0; i=1;2; with probability one: And lim N!1 p © i;N ¶ i;N K 12;N K 1;N K 2;N = lim N!1 p © i;N ¶ i;N p D N p K 1;N K 2;N =0; i=1;2; in probability: This completes the proof of this theorem. 4.3.10 Remark. From Theorem 4.3.4, we can see that (4.3.36) is a both necessary and sufficient condition for consistency and asymptotic normality of estimator ^ µ (3) 1;N . Similarly, condition (4.3.39) is necessary and sufficient for consistency and asymptotic normality of estimator ^ µ (3) 2;N . Since in the algebraic case the sum P N k=1 k ° appears frequently, we introduce a special notation to describe the asymptotic of this sum asN !1 for°¸¡1: ¨ N (°)= 8 > > < > > : N °+1 ; if° >¡1; lnN; if° =¡1: (4.3.58) With this notation, P N k=1 k ° ³¨ N (°),°¸1. Let us consider the following examples where¢ is the Laplace operator in a smooth bounded domainG inR d with zero boundary conditions: H =L 2 G. u tt =µ 1 ¢u+µ 2 u t + _ W; µ 1 >0; µ 2 2R; (4.3.59) 110 Asymptotic Eq. (4.3.59) Eq. (4.3.60) Eq. (4.3.61) ª 1;N N 2 d +1 N ¨ N (¡2=d); d¸2 ª 2;N N N 2 d +1 N 4 d +1 Table 4.1: Asymptotes for equations (4.3.59), (4.3.60) and (4.3.61). u tt =¢(µ 1 u+µ 2 u t )+ _ W; µ 1 >0; µ 2 2R; (4.3.60) u tt =µ 1 ¢u¡µ 2 ¢ 2 u t + _ W; µ 1 >0; µ 2 2R; (4.3.61) The following table summarizes the results: In equations (4.3.59)–(4.3.61),A 1 andB 1 are the leading operators, that is, ® = ® 1 and¯ =¯ 1 . This, in particular, ensures that ^ µ 2;N is always consistent. Let us now consider examples whenA 1 andB 1 are not the leading operators: u tt = ¡ ¢u+µ 1 u ¢ + ¡ ¢u t +µ 2 u t ¢ + _ W; µ 1 2R; µ 2 2R; (4.3.62) u tt + ¡ ¢ 2 u+µ 1 u ¢ = ¡ µ 2 ¢u t ¡¢ 2 u t ¢ + _ W;µ 1 2R; µ 2 2R; (4.3.63) u tt + ¡ ¢ 2 u+µ 1 ¢u ¢ = ¡ µ 2 u t ¡¢ 2 u t ¢ + _ W; µ 1 2R; µ 2 2R: (4.3.64) The following table summarizes the results: A multi-parameter estimation problem, such as u tt + ¡ µ 11 ¢ 2 u+µ 12 ¢u ¢ = ¡ µ 21 u t ¡µ 22 ¢ 2 u t ¢ + _ W; 111 Asymptotic Eq. (4.3.62) Eq. (4.3.63) Eq. (4.3.64) ª 1;N ¨ N (¡4=d); d¸4 ¨ N (¡8=d); d¸8 ¨ N (¡4=d); d¸4 ª 2;N ¨ N (¡2=d); d¸2 N ¨ N (¡4=d); d¸4 Table 4.2: Asymptotes for equations (4.3.62), (4.3.63) and (4.3.64). can be studied in the same way. 4.3.2 Analysis of Estimators: General Case The algebraic case which corresponds to elliptic partial differential operators seems to be the most natural, there is also a more general case, where the eigenvalues such as ¸ k » e k or ¹ k » lnk is also worth considering. It serves as an example of a model with observations coming from independent but not identical channels (see, for example, Korostelev and Yin ). As in the proof of Theorem 4.3.9, the arguments rely heavily on a suitable law of large numbers. Verification of the corresponding conditions is straightforward in the algebraic case, but is impossible in the general case unless we make additional assumptions about the eigenvalues of the operators. In fact, as we work with weighted sums of independent random variables, we need some conditions on the weights for a law of large numbers to hold. In particular, the weights should not grow too fast: If » 2 k , k ¸ 1 are iid stan- dard Gaussian random variables, then the sequencefn ¡2 P n k=1 n» 2 k ; n ¸ 1g converges with probability one to 1=2, butfe ¡n P n k=1 e k » 2 k ; n ¸ 1g does not have a limit, even in probability. 112 Theorem 1.4.5 in Chapter Two summarizes some of the laws of large numbers, and leads to the following 4.3.11 Definition. The sequence fa n ; n ¸ 1g of positive numbers is called slowly increasing if lim n!1 P n k=1 a 2 n ³ P n k=1 a k ´ 2 =0: (4.3.65) The purpose of this definition is to simplify the statement of the main theorem (The- orem 4.3.12 below). It was not necessary in the algebraic case because the sequence fn ° ; n¸ 1g is a slowly increasing sequence if and only if°¸¡1. The reason for intro- ducing the terminology is that the sequencefe n r ; n¸1g has property (4.3.65) if and only if r < 1. Further discussion of (4.3.65), including the connections with the weak law of large numbers, is after the proof of Theorem 1.4.5 in Chapter 2. In general, we have to replace (4.3.36) with Condition 1. The sequencef¿ 2 k M ¡ T¹ k (µ 2 ) ¢ =¸ k (µ 1 ); k¸1g is slowly increasing, and (4.3.39), with Condition 2. The sequencefº 2 k M ¡ T¹ k (µ 2 ) ¢ ; k ¸ 1g is slowly increasing, whereM andV are defined in the previous section. 4.3.12 Theorem. Assume that equation u tt +(A 0 +µ 1 A 1 )u(t)=(B 0 +µ 2 B 1 )u t (t)+ _ W(t); 0<t·T; (4.3.66) is hyperbolic. 113 1. If Condition 1 holds, then lim N!1 ^ µ (3) 1;N =µ 1 inprobability; (4.3.67) lim N!1 p ª 1;N ³ ^ µ (3) 1;N ¡µ 1 ´ =» 1 indistribution; (4.3.68) where» 1 is a standard Gaussian random variable. 2. If Condition 2 holds then lim N!1 ^ µ (3) 2;N =µ 2 inprobability; (4.3.69) lim N!1 p ª 2;N ³ ^ µ (3) 2;N ¡µ 2 ´ =» 2 indistribution; (4.3.70) where» 2 is a standard Gaussian random variable. 3. If both Conditions 1 and 2 hold, then the random variables» 1 ;» 2 are independent. Proof. The main steps are the same as in the algebraic case (Theorem 4.3.9). In particular, (4.3.52) and (4.3.54) continue to hold as long as© 1;N !1 and© 2;N !1, respectively. The only difference is that Conditions 1 and 2 do not provide enough information about the almost sure behavior ofK 12;N =EK 12;N , and, in this general setting, there is no natural condition that would do that. As a result, in (4.3.1), the convergence is in probability rather than with probability one, and then, in both (4.3.49) and (4.3.50), convergence in probability will suffice. Conditions 1 and 2 ensure (4.3.49) and (4.3.50), respectively, but with convergence in probability rather than almost sure. This is a direct consequence of the weak law of large numbers. 114 We will mainly show that under condition 1 or condition 2, lim N!1 D N = lim N!1 K 12;N K 1;N K 2;N =0 (4.3.71) In the case of (4.3.1), we have EjK 12;N j· N X k=1 j¿ k º k jEu 2 k (T) and, for all sufficiently largek, Eu 2 k (T)· 4T ¸ k (µ 1 ) M ¡ T¹ k (µ 2 ) ¢ ³ 1+max ¡ 0;T¹ k (µ 2 ) ¢ ´ ; becausexe x ¡x·4(e x ¡x¡1)(1+max(0;x)) for allx2R. Then lim N!1 EjK 12;N j p © 1;N © 2;N =0: (4.3.72) 115 Indeed, under Condition 1, (4.3.72) follows from ¡ EjK 12;N j ¢ 2 © 1;N © 2;N ·· ³ P N k=1 j¿ k º k j¢(4=T¸ k )M ¡ T¹ k (µ 2 ) ¢¡ 1+max(0;T¹ k (µ 2 )) ¢ ´ 2 © 1;N © 2;N · 16 P N k=1 ¿ 2 k M ¡ T¹ k (µ 2 ) ¢ ¸ k (µ 1 ) ¡ 1+max ¡ 0;T¹ k (µ 2 ) ¢¢ 2 ¸ k (µ 1 ) ¢ P N k=1 º 2 k M ¡ T¹ k (µ 2 ) ¢ T 2 P N k=1 ¿ 2 k M ¡ T¹ k (µ 2 ) ¢ ¸ k (µ 1 ) ¢ P N k=1 º 2 k M ¡ T¹ k (µ 2 ) ¢ · 16 P N k=1 ¿ 2 k M ¡ T¹ k (µ 2 ) ¢ ¸ k (µ 1 ) ¡ 1+max ¡ 0;T¹ k (µ 2 ) ¢¢ 2 ¸ k (µ 1 ) T 2 P N k=1 ¿ 2 k M ¡ T¹ k (µ 2 ) ¢ ¸ k (µ 1 ) (4.3.73) (Cauchy-Schwartz inequality) and lim k!1 ¡ 1+max ¡ 0;T¹ k (µ 2 ) ¢¢ 2 ¸ k (µ 1 ) =0 (4.3.74) (hyperbolicity condition), therefore, the denominator of the last expression of (4.3.73) diverges faster than the numerator and then (4.3.71) follows. While, under Condition 2, (4.3.72) follows from ¡ EjK 12;N j ¢ 2 © 1;N © 2;N · 16 P N k=1 º 2 k M ¡ T¹ k (µ 2 ) ¢ ¡ 1+max ¡ 0;T¹ k (µ 2 ) ¢¢ 2 ¸ k (µ 1 ) T 2 P N k=1 º 2 k M ¡ T¹ k (µ 2 ) ¢ (Cauchy-Schwartz inequality with a different arrangement of terms) and (4.3.74). 116 As an example, consider the model with· k = e 2k ,¿ k = e k ,½ k = 0,º k = lnln(k+3) and assume thatµ 1 >0,µ 2 >0. Then ¸ k =e 2k +µ 1 e k ; ¹ k =µ 2 lnln(k+3); so that¿ 2 k =¸ k »1. Next, ¡ ln(k+3) ¢ Tµ 2 =2 <M(T¹ k )< ¡ ln(k+3) ¢ Tµ 2 for all sufficiently largek, and alsoº 2 k M(T¹ k )³ ¡ ln(k+3) ¢ Tµ 2 . Using integral compar- ison, we conclude that, for allr >0, N X k=1 ¡ lnk ¢ r »N ¡ lnN ¢ r : Thus, both Condition 1 and Condition 2 hold. By Theorem 4.3.12, both ^ µ (3) 1;N and ^ µ (3) 2;N are consistent and asymptotically normal. Further computations show that ª 1;N ³ N ¡ lnN ¢ Tµ 2 ¡ lnlnN ¢ 2 ; ª 2;N ³N ¡ lnN ¢ Tµ 2 : 4.3.13 Remark. Here we only proved the conditions for consistency of ^ µ (3) 1;N and ^ µ (3) 2;N when bothµ 1 andµ 2 are unknown parameters. If onlyµ 1 is unknown and condition 1 holds, then ^ µ (1) 1;N is a consistent estimator ofµ 1 . Similarly, If onlyµ 2 is unknown and condition 2 holds, then ^ µ (2) 2;N is a consistent estimator ofµ 2 . 117 Chapter 5 Other Statistical Inference for Stochastic Hyperbolic Equations In the previous chapters, we consider the estimation of the parameters in the sense that as the fourier coefficientN goes to infinity. In this chapter, we present large sample theory of statistical inference for the stochastic hyperbolic equation u tt =(A 0 +A 1 µ 1 )u+(B 0 +B 1 µ 2 )u t +² _ W; (5.0.1) where _ W is a standard white noise and ² > 0 is called the small noise. The asymptotic in such problems of estimation of observations are investigated in the limit as T ! 1 or ² ! 0 or both. As in before, the equation above can be written as two first-order Ito equations 8 > > < > > : du(t)=v(t)dt; dv(t)=F(v)dt+²dW; (5.0.2) 118 where F(v) = (A 0 +A 1 µ 1 ) R t 0 v(t)dt+(B 0 +B 1 µ 2 )v is a non-anticipating function of v(t). Therefore, the processv(t) is a diffusion process in the sense of Lipster and Shiryaev [60, Definition 4.3.7]. Diffusion processes are widely used in applied problems. (Biomedical sciences [27, 44, 57, 71, 88, 90, 94, 103], economics [4, 17, 24, 25, 34, 70, 79, 89], mechanics [1, 20, 21, 26, 32, 66, 91, 102], physics [16, 35, 47, 64, 95, 100], and especially in financial mathematics [3, 12, 45, 48, 56, 65, 72, 81, 82, 92]). From what we have discovered, the estimation problems are typical for classical statistical inference, i.e., statistics of independent identically distributed (i.i.d) trajectories u N = fu 1 ;:::;u N g in the limit N ! 1. 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Abstract (if available)

Abstract A parameter estimation problem is considered for a stochastic wave equation and a linear stochastic hyperbolic driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimators are studied as the number of the Fourier coefficients increases, while the observation time and noise intensity are fixed.

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Creator Liu, Wei (author)

Core Title Statistical inference for stochastic hyperbolic equations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 11/16/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag diffusion process,maximum likelihood estimators,OAI-PMH Harvest,ordinary differential equation,partial differential equation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lototsky, Sergey V. (committee chair), Cheng, Hsiao (committee member), Kukavica, Igor (committee member)

Creator Email liu5@usc.edu,wlcrystal@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3530

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