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Statistical inference of stochastic differential equations driven by Gaussian noise

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Statistical inference of stochastic differential equations driven by Gaussian noise

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Content STATISTICAL INFERENCE OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY GAUSSIAN NOISE by Michael Moers 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) August 2012 Copyright 2012 Michael Moers Acknowledgements I would like to express sincere gratitude to my advisor Sergey Lototsky for his guidance and support during my graduate studies at USC. I feel very lucky to have an advisor who cares so much about the progress of his students and is extremely responsive to all questions and queries. This thesis could not have been written without his valuable suggestions and constructive feedback. I would also like to thank the members of my dissertation comittee Stephan Haas and Remigijus Mikulevicius for their helpful comments and assistance. ii Table of Contents Acknowledgements ii List of Figures v Abstract vi Chapter 1 Introduction 1 Chapter 2 Gaussian processes of Volterra type 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Gaussian processes of Volterra type . . . . . . . . . . . . . . . . . . 10 2.2.1 Hölder continuity of sample paths; Fundamental Martingale 15 2.2.2 Reproducing kernel Hilbert space . . . . . . . . . . . . . . . 18 2.2.3 Canonical representation . . . . . . . . . . . . . . . . . . . . 21 2.3 Stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.1 Malliavin Calculus; Derivative and adjoint operator . . . . . 24 2.3.2 Deterministic integrands . . . . . . . . . . . . . . . . . . . . 28 2.3.3 Stochastic integrands . . . . . . . . . . . . . . . . . . . . . . 32 Chapter 3 Statistical inference of Itô-processes of diffusion type with Volterra noise 37 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Maximum likelihood estimation of diffusion processes with Gaussian white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.1 Diffusion processes . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.3 Maximum likelihood estimation . . . . . . . . . . . . . . . . 49 3.3 Maximum likelihood estimation of Ornstein-Uhlenbeck process with Volterra noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . 58 3.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 Maximum likelihood estimation of harmonic oscillator driven by fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . 65 iii 3.4.1 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . 68 3.4.3 Properties of integrated fractional Brownian motion and strong consistency . . . . . . . . . . . . . . . . . . . . . . . 70 Chapter 4 Statistical inference of damped harmonic oscillator driven by Gaussian white noise 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Laplace transform of quadratic functionals . . . . . . . . . . . . . . 76 4.2.1 Properties of the solution . . . . . . . . . . . . . . . . . . . 76 4.2.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.3 Application to small deviations . . . . . . . . . . . . . . . . 90 4.3 ParameterestimationofharmonicoscillatordrivenbyGaussianwhite noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3.1 Convergence of moments . . . . . . . . . . . . . . . . . . . . 93 4.3.2 Asymptotics of the first and second moment . . . . . . . . . 98 Chapter 5 Hypothesis testing in a fractional Ornstein- Uhlenbeck model 101 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2 Strong consistency and large sample asymptotics . . . . . . . . . . . 107 5.3 Finite sample approximation and hypothesis testing . . . . . . . . . 116 5.3.1 Stochastic integration with respect to fBm . . . . . . . . . . 119 5.3.2 Asymptotic distribution of the statistics . . . . . . . . . . . 122 5.3.3 Alternative estimators . . . . . . . . . . . . . . . . . . . . . 125 5.3.4 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . 130 Chapter 6 Open questions 134 References 135 Index 140 iv List of Figures 5.1 Quantile-quantile plot of 1000 samples of √ T ( e θ T,H −θ) . . . . . . . 116 5.2 Quantile-quantile plot of 1000 samples of T e θ T,H . . . . . . . . . . . 124 v Abstract The objective of this thesis is to study statistical inference of first and second order ordinarydifferentialequations driven bycontinuous Gaussiannoise undercontinous time observations. The Gaussian process can be defined by a stochastic integral of a time-dependent triangular deterministic kernel with respect to standard Brow- nian motion. Especially, we do not assume the process to be Markovian nor a semimartingale. An important example for such a process is fractional Brownian motion. The thesis is focussed on (a) properties of these Gaussian processes (b) maximum likelihood estimation (c) asymptotic distribution of finite sample distri- bution of least squares type estimator. vi Chapter 1 Introduction Themainpurposeofthisthesisistopresentcertainaspectsofparameterestimation of stochastic differential equations. As in many areas in mathematics, the theory originated in a practical problem: Given a certain model and a finite amount of observations, it is required to estimate an unknown parameter. Stochastic differen- tial equations have been used to model various diverse phenomena, such as noisy electrical signals, tumor growth or interest rates. In many applications, the noise is assumed to be Gaussian, for instance white noise, fractional noise or Ornstein Uhlenbeck noise. A generalization of these noises are Gaussian noises of Volterra type, which is the type of noise considered in this thesis. They can be defined by a stochastic integral of a time-dependent triangular deterministic kernel with respect to standard Brownian motion. In [18] the name of this family of processes appeared the first time, it originated most likely from the similarity of the kernel to the Volterra kernel in the theory of integral equations. 1 The development of the theory of Volterra processes today moves in several di- rections: Stochastic integration, stochastic differential equations and statistical in- ference, the last being the main topic of this thesis. We consider for instance the fol- lowing Ornstein-Uhlenbeck process, which is driven by a Gaussian noise of Volterra type, where the drift depends on an unknown parameter θ. Let X ={X t ,t≥ 0} be a stochastic process which satisfies the stochastic differential equation dX t =θb(X t )dt +dB t , X 0 = 0, whereB ={B t ,t≥ 0} is a Gaussian process of Volterra type. That means that B t is a centered Gaussian process of the form B t = Z t 0 K(t,s)dW s , whereK(t,s) is a square integrable triangular kernel function andW ={W t ,t≥ 0} denotes a Brownian motion with respect to some filtration (F t ) t≥0 . An important problem is to find estimators for the parameter θ given observa- tions. Statistical inference using the Ornstein-Uhlenbeck process driven by various Gaussian noises of Volterra type has gained a lot of attention in the literature, see [37] for filtering, [30,36,68] for maximum likelihood estimation, [7,29] for least squares estimation, [6,41] for maximum likelihood estimation in higher dimensions. In the thesis we often address the following problems: 2 − For which b(x) exists a solution X t ? − Given continuous time data (X t , 0≤ t≤ T ) can we estimate the parameter θ? Especially, is there a maximum likelihood estimator (MLE) b θ T of θ? − Under which conditions is the MLE strongly consistent? That means when is lim T→∞ b θ T =θ P-a.s.? Very often B t is assumed to be a Gaussian white noise, and in that case pa- rameter estimation has been studied in [43,44]. Recently, researchers worked on replacing the Gaussian white noise by fractional Gaussian noise. Fractional Brown- ian motion is a continuous Gaussian process that generalizes Brownian motion and has mean zero and covariance R(t,s) = V H 2 |t| 2H +|s| 2H −|t−s| 2H , ∀t,s≥ 0, whereV H is an appropriate constant to make the variance 1 att = 1. Here the pa- rameterH∈ (0, 1) is calledHurst-index, named after the biologist H. Hurst. Frac- tional Brownian motion was first described by Kolmogorov, who called it Wiener spiral, and then was further studied by Lévy and Mandelbrot [45], who gave the process its modern name. Fractional Brownian motion has become a popular model in research and applications: Due to its self-similarity and long-memory properties, 3 it has successfully been applied to diverse fields such as Local Area Network traffic in computer science, hydrology in biology or stock price models in finance, see [9]. GeneralizationsoffractionalBrownianmotionincludebifractionalmodels,which have been introduced in [27] and later studied in [63,69]. These models keep most of the properties of fractional Brownian motion (self-similarity, Gaussianity) but allows non-stationarity of large increments. Another natural generalization is multifractional Brownian motion, see [8], which allows multidimensional fractional Brownian motions. The second chapter of this thesis introduces certain aspects of the theory of Gaussian processes of Volterra type and Malliavin calculus. We collect and review recent results, which are essential mathematical tools for the consequent chapters of the thesis. The third chapter introduces statistical inference of stochastic differential equa- tions with Volterra noise. A contribution here is that some results which have been obtained for fractional Brownian motion are know generalized to the class of Gaus- sian processes of Volterra type. This is for instance a Girsanov theorem which is given in [22]. Moreover we find a maximum likelihood estimator for the parameter in the Ornstein Uhlenbeck process and show under which conditions it is strongly consistent. We show that this strong consistency assumptions hold in the fractional Brownian motion case and the case of Ornstein-Uhlenbeck process as driving force. 4 The fourth chapter contains a result on statistical inference of a second order diffusion process with Gaussian white noise. We contribute a Laplace transform of a quadratic functional of the diffusion process. We apply it then to small deviations of Gaussian processes and moreover apply it to obtain asymptotics of the moments of a maximum likelihood estimator. In the fifth chapter we approximate the finite sample distribution of an estima- tor using a newly developed approach. We illustrate through computer simulations and a Stein’s bound that asymptotic distributions found in the literature are inad- equate approximations of the finite sample distribution for moderate values of the parameter and the sample size. We show, that our approximation leads to better results. We moreover apply the asymptotics to regression analysis. 5 Chapter 2 Gaussian processes of Volterra type 2.1 Introduction In this chapter, we discuss several results and definitions on Gaussian processes of Volterra type (Volterra processes). The theory of stochastic integration with respect to Volterra processes has been introduced in [17,18,20]. Stochastic integration for Gaussian processes goes well beyondthestochasticintegrationwithrespecttosemimartingales, whereanalready well developed theory is available (see, for instance, [61]). Volterra processes are neither Markovian nor semimartingales in general, a fact that is well-known for a popular Volterra process, fractional Brownian motion (see [11, Section 1.8]). We distinguish between three different approaches for stochastic integration. 6 1. As for any Gaussian process, Malliavin calculus (see [50]) with respect to the Volterra process can be developed. In Malliavin calculus, the divergence op- erator can be interpreted as stochastic integral. This idea has been developed for fractional Brownian motion in [2,4,11,21–23]. The stochastic integral can be obtained as limit of Riemann sums where the products are defined as Wick products. 2. AdifferentconceptforintegrationandquadraticvariationistheRusso-Vallois calculus proposed in [64] and further developed in [65]. Russo-Vallois calculus was successfully applied to fractional Brownian motion with Hurst indexH≥ 1/4 to obtain Itô formula, see [26]. Then it was used to show that an Itô formula holds if and only if H > 1/6 in [25]. 3. Moreover we can define pathwise integrals with respect to Volterra processes. [72] proved that the Riemann-Stieltjes integral (then called Young integral) R f(s)dg(s) exists, whenever f and g are Hölder continuous of order α and β respectively, with α +β > 1. Therefore, assuming that the Volterra pro- cess has β-Hölder continuous sample paths and the integrator is α-Hölder continuous with α > 1−β, the stochastic integral can be interpreted as a Riemann-Stieltjes integral and coincides with the forward and Stratonovich integrals studied in [2]. This concept has been employed for fractional Brown- ian motion with parameter H > 1/2 in [15,42]. Using fractional integral and 7 derivative, [73] extended this integral to fractional Brownian motion with arbitrary Hurst index H∈ (0, 1). It turns out, that the only stochastic integral which is defined for any continu- ous Volterra process is the integral defined through the adjoint of the Malliavin derivative, which is the focus here. The connections to other stochastic integration techniques are not researched in such generality; They are known, in the case of fractional Brownian motion, which we discuss separately in Section Preliminaries in Chapter 4. Other aspects of Volterra processes studied in the literature are self- similarity [33], large deviations [52], equivalence [53], and canonical representations of Volterra processes [24]. Chapter 1 lays the groundwork for the mathematical treatment of Volterra processes for the remaining thesis. At first we introduce the general theory of Gaussian processes of Volterra type and give several examples. We address the problem, under which assumptions the process has a continuous representation and under which assumptions it can be transformed into a martingale. Then we show how to construct the reproducing kernel Hilbert space associated with the process. Every Volterra process can be written as integral over a time-dependent triangular kernel with respect to a standard Brownian motion. We are interested, under which condition the standard Brownian motion is unique in an almost sure sense, i.e. if there exists a canonical representation of the process. 8 The last section introduces Malliavin calculus and its operators, the derivative, the adjoint, and the Ornstein-Uhlenbeck operator. We develop the theory in gen- eral, for every isonormal Gaussian process. Then we show the relationship between the Malliavin derivative and adjoint (so-called divergence integral) of the Volterra process and the underlying standard Brownian motion. Preliminary definitions Afiltered probability space (in the following also called stochastic basis), (Ω,F, (F t ) t∈T ,P) on some ordered set T consists of a probability space (Ω,F,P) and a filtration (F t ) = (F t ) t∈T ⊂F withF s ⊆F t for all s≤t, s,t∈T. A filtered probability space is said to satisfy the usual conditions if the fol- lowing conditions are met − (Ω,F,P) is complete. − The σ-fieldF t contain all null sets ofF for all t∈T. − The filtration (F t ) is right-continuous, i.e. for every t ∈ T excluding the maximal element, the σ-fields satisfy T s>t F s =F t . Astochastic processX ={X t ,t∈T} is a functionX :T×Ω→R such that X t (·) : Ω→R is a random variable for eacht∈T. We say thatX ={X t ,t∈T} is a stochastic process adapted to (F t ), ifX :T×Ω→R is such thatX t (·) : Ω→R 9 is anF t -random variable for each t∈T. X(·,ω) :T→R is called a sample path for each ω∈ Ω. A Gaussian process is a stochastic process for which any joint distribution is Gaussian. A martingale is a stochastic process withE|X t |<∞ for whichE(X t |X s ) =X s with probability 1 for any s≤t, s,t∈T. A Brownian motion (or Wiener process) with respect to a filtration (F t ) is a Gaussian process W ={W t ,t≥ 0}, which has almost surely continous sample paths,P (W 0 = 0) = 1,theincrementsW t −W s areindependentofF s andW t −W s ∼ N(0,t−s) for all 0≤s≤t. 2.2 Gaussian processes of Volterra type Let (Ω,F, (F t ),P) be a filtered probability space satisfying the usual hypotheses. We study Gaussian processes B ={B t ,t∈ [0,T ]} which are of the form B t = Z t 0 K(t,s)dW s , with a deterministic function K(t,s) and W ={W t ,t∈ [0,T ]} a Brownian motion with respect to (F t ). The conditions on K(t,s) are so that the following Gaussian processes are included: Ornstein-Uhlenbeck process B t = R t 0 e η(s−t) dW s , η> 0, see Example 2.2.4. 10 Fractional Brownian motion B t = R t 0 K H (t,s)dW s , where H∈ (0, 1) and K H is the kernel of fractional Brownian motion, see Example 2.2.5. Definition 2.2.1. A mean-zero Gaussian process B ={B t ,t∈ [0,T ]} is called Volterra process, if B t = Z t 0 K(t,s)dW s , t∈ [0,T ], with W ={W t ,t∈ [0,T ]} a Brownian motion with respect to (F t ) and K : [0,T ]× [0,T ]→ [0,∞) satisfying (H1) sup t∈[0,T ] kK(t,·)k L 2 ([0,T ]) <∞; (H2) K(t,s) = 0 for all s>t and K(0, 0) = 0. A function K(t,s) satisfying (H1) and (H2) is called Volterra kernel. Hypothesis (H1) implies that K(t,·)∈L 2 ([0,T ]) for each t∈ [0,T ] and there- fore the process{ R t 0 K(t,s)dW s ,t∈ [0,T ]} is in L 2 (Ω). Hypothesis (H2) implies that B t is adapted to the natural filtration of W t . A Volterra kernelK defines a linear operator onL 2 ([0,T ]), which is also denoted by the letter K Kf(t) = Z t 0 K(t,s)f(s)ds. (2.2.1) Lemma 2.2.2. 1. K is bounded. 11 2. K is a compact operator. Proof. 1. Boundedness follows since for all f∈L 2 ([0,T ]) kKfk 2 L 2 = Z T 0 (Kf) 2 (t)dt = Z T 0 Z t 0 K(t,s)f(s)ds 2 dt ≤ Z T 0 Z t 0 K 2 (t,s)ds Z t 0 f 2 (s)dsdt≤T sup t∈[0,T ] kK(t,·)k 2 L 2kfk 2 L 2. 2. The linear operator K can be approximated uniformly by finite rank oper- ators, and the uniform closure of the class of finite rank operators coincides with the class of compact operators. In fact since K is compact, it is also automatically a bounded operator. The following are examples of Volterra processes: Example 2.2.3 (Brownian motion). Let K(t,s) =1 [0,t] (s). Then B = W standard Brownian motion with covariance R(t,s) = t∧s. The mapping K on L 2 ([0,T ]) is Kf(t) = Z K(t,s)f(s)ds = Z t 0 f(s)ds. 12 Example 2.2.4 (Ornstein-Uhlenbeck process). Let K(t,s) =e μ(t−s) 1 [0,t] (s). Then B is Ornstein-Uhlenbeck process with covariance R(t,s) = 1 2μ (e −μ(t−s) − e −μ(t+s) ), t>s. It is the unique solution of the stochastic differential equation dB t =μB t dt +dW t , B 0 = 0. which can be verified with Itô’s formula. The integral equation with the associated linear operator K on L 2 ([0,T ]) is Kf(t) = Z e μ(t−s) f(s)ds =h(t), h(0) = 0, and the explicit solution is given by f(t) = d dt h(t)−μh(t). Example 2.2.5 (fractional Brownian motion). For H∈ (0, 1) let K(t,s) =K H (t,s) = 1 √ V H (H + 1/2) 2 F 1 (H− 1/2, 1/2−H,H + 1/2, 1−t/s)1 [0,t) (s) 13 where 2 F 1 is the Gauss hypergeometric function which has the following represen- tation for Re(c)> Re(b)> 0 2 F 1 (a,b,c,z) = Γ(c) Γ(b)Γ(c−b) Z 1 0 t b−1 (1−t) c−b−1 (1−tz) −a dt and V H = Γ(2− 2H) cos(πH) πH(1− 2H) a normalizing constant which makes EB 2 1 = 1. Then B t is fractional Brownian motion with Hurst index H∈ (0, 1), see [22]. The covariance is R(t,s) = V H 2 s 2H +t 2H −|t−s| 2H . The mapping K on L 2 ([0,T ]) is Kf(t) = 1 √ V H (H + 1/2) Z t 0 2 F 1 (H− 1/2, 1/2−H,H + 1/2, 1−t/s)f(s)ds. 14 2.2.1 Hölder continuity of sample paths; Fundamental Martingale LetB ={B t ,t∈ [0,T ]} be a Volterra process. Then the sample paths ofB t are not necessarily continuous. A simple example for a non-continuous Volterra process is the process which has the kernel K(t,s) =1 [0,S] (t)1 [0,t] (s) where S ∈ (0,T ). The process jumps from W S to 0 at time S. Consider the following hypothesis: (H3) There are constants C, α> 0 such that Z T 0 (K(t,r)−K(s,r)) 2 dr ! 1/2 ≤C|t−s| α for all s,t∈ [0,T ]. This guarantees that there exists a continuous modification of the process, since by Hölder’s inequality: E (|B t −B s | p )≤ E |B t −B s | 2 p/2 = Z T 0 (K(t,r)−K(s,r)) 2 dr ! p/2 ≤C|t−s| αp ≤C|t−s| β+1 (2.2.2) for p > 1/α, β := αp− 1. Therefore (H3) implies α-Hölder continuity: Due to Kolmogorov-Čentsov theorem (see [35, Prop. 2.8]), estimation (2.2.2) implies that 15 B t has a γ-Hölder modification for every 0 < γ < β/p = α− 1/p. Since p can be chosen arbitrarily large, B t has a γ-Hölder continuous modification for every 0<γ <α. Lemma 2.2.6. 1. Assume that the Volterra process B = {B t ,t ∈ [0,T ]} is almost surely continuous. Then Kf(·) is continuous for fixed f∈ L 2 ([0,T ]) and vanishes at zero. 2. If (H3) holds with some α∈ (0, 1], then Kf(·) is α-Hölder continuous for fixed f∈L 2 ([0,T ]). Proof. 1. Suppose 0≤s≤t≤T. By Cauchy-Schwarz inequality |Kf(t)−Kf(s)| = Z t 0 K(t,r)f(r)dr− Z s 0 K(s,r)f(r)dr ≤ Z t 0 |f(r)||K(t,r)−K(s,r)|dr ≤ Z t 0 f 2 (r)dr Z t 0 |K(t,r)−K(s,r)| 2 dr 1/2 . SinceE(B t −B s ) 2 = R t 0 |K(t,r)−K(s,r)| 2 dr, we have |Kf(t)−Kf(s)|≤kfk L 2 ([0,T ]) E(B t −B s ) 2 1/2 , which implies the continuity. 16 2. The α-Hölder continuity follows since |Kf(t)−Kf(s)|≤kfk L 2 ([0,T ]) Z t 0 |K(t,r)−K(s,r)| 2 dr 1/2 ≤C 0 |t−s| α . A Volterra process is in general not a semimartingale. For fractional Brownian motion, a proof that it is not a semimartingale unless H = 1/2 can be found in[11,Sect. 1.8]. Undercertainassumptionshoweverwecanconstructamartingale associated with a Volterra process. Assume there exist a Volterra kernelk(t,s) such that Z T 0 Z T 0 k(t,r)k(s,u)R(dr,du) = Z t∧s 0 k(t∧s,r)dr for all t,s∈ [0,T ]. Define the process M ={M t ,t∈ [0,T ]} by M t := Z t 0 k(t,s)dB s . (2.2.3) Theorem 2.2.7. Let the process M ={M t ,t∈ [0,T ]} be defined by (2.2.3), then M t has independent increments and variance EM 2 t = Z t 0 k(t,r)dr. 17 In fact, M t is a martingale. Proof. By assumption for 0≤s≤t≤T EM t M s = Z T 0 Z T 0 k(t,r)k(s,u)R(dr,du) = Z s 0 k(s,r)dr. Since the last term does not depend ont,M t has uncorrelated increments and since it is a Gaussian process, the increments are independent. IfM t generates the same filtration asB t , then it is also called the fundamental martingale associated with B t . The name fundamental martingale has been in- vented in [48], who found an explicit kernelk(t,s) in the case of fractional Brownian motion. 2.2.2 Reproducing kernel Hilbert space In the following we define reproducing kernel Hilbert space (RKHS) and its associ- ated reproducing kernel. Definition2.2.8. A functionR(t,s) defined on [0,T ]×[0,T ] is called reproducing kernel for a Hilbert spaceH of functions on [0,T ] if 1. R(t,·)∈H for each t∈ [0,T ]. 2.hh,R(t,·)i H =h(t), for every h∈H, whereh·,·i H denotes the scalar product ofH. 18 If such an R exists, thenH is called reproducing kernel Hilbert space. The equivalent reverse direction to define a RKHS is to start with a positive definite kernel: Definition 2.2.9. A function R : [0,T ]× [0,T ]→ [0,∞) is called positive def- inite kernel if n X i,j=1 λ i λ j R(t i ,t j )≥ 0 holds for any n∈N, λ 1 ,...,λ n ∈R and t 1 ,...,t n ∈ [0,T ]. Theorem 2.2.10 (Moore-Aronszajn). SupposeR(t,s) is a symmetric, positive def- inite kernel on a set T. Then there is a unique Hilbert space of functions on T for which R(t,s) is a reproducing kernel. The RKHS associated with a Gaussian process B ={B t ,t∈ [0,T ]} can be obtained the following way (see [30, section 2.2]). − Denote by H the closure in L 2 (Ω) of the space spanned by{B t ,t∈ [0,T ]} equipped with the inner product hξ,ζi H =E(ξ,ζ) ξ,ζ∈H. − Define R(H) :={R(ζ),ζ∈ H}, where the linear operator R : H→ R(H) is defined by R(ζ)(t) =hζ,B t i H , t∈ [0,T ]. 19 − Then the covariance function R(t,s) =E(B t B s ) is a reproducing kernel with reproducing kernel Hilbert spaceH =R(H). The inner product inH is given byhF,Gi H =hR −1 F,R −1 Gi H , for all F,G∈H. Proof. We need to check the conditions in Definition 2.2.8. Fix a t∈ [0,T ]. Then R(t,s) =EB t B s =R(B t )(s) for all s∈ [0,T ], i.e. R(t,·)∈R(H). Secondly, let F ∈ R(H), F = R(ξ),ξ∈ H. ThenhF,R(t,·)i H =hξ,B t i H = E(ξB t ) =R(ξ)(t) =F (t). Let B ={B t ,t∈ [0,T ]} be a mean-zero Gaussian process with factorizable covariance function R(t,s) of the form R(t,s) = R T 0 K(t,r)K(s,r)dr with K satis- fying(H1). There exists an isometry betweenH and a closed subspace ofL 2 ([0,T ]) (see [24, Proposition 1]). Proposition 2.2.11. Let E be the closed subspace of L 2 ([0,T ]) spanned by the family{K(t,·),t∈ [0,T ]}. 1. A function h : [0,T ]→R is inH if and only if there exists a unique function f h ∈E such that h(t) = Z T 0 K(t,s)f h (s)ds. (2.2.4) 2. The scalar product onH is given by hh 1 ,h 2 i H =hf h 1 ,f h 2 i L 2 ([0,T ]) . 20 3. The representation (2.2.4) defines an isometry of Hilbert spaces Ψ :H→E, h→f h . Corollary 2.2.12. IfK is injective, then there exists a bijective isometry Ψ :H→ L 2 ([0,T ]). Proof. K is injective if and only if the family K(t,·) spans all of L 2 ([0,T ]), so E = L 2 ([0,T ]); Therefore by Proposition 2.2.11 there exists a bijective isometry Ψ :H→L 2 ([0,T ]). 2.2.3 Canonical representation LetB ={B t ,t∈ [0,T ]} be a mean-zero Gaussian process with covariance function R : [0,T ]× [0,T ]→ [0,∞), R(t,s) :=E [B t B s ]. Recall that the law of a mean-zero Gaussian process is uniquely determined by its covariance function. Assume that the covariance function is factorizable of the form R(t,s) = Z T 0 K(t,r)K(s,r)dr. (2.2.5) whereK : [0,T ]×[0,T ]→ [0,∞). In the following assume that (H1) to(H3) hold and that K is injective. 21 Since Gaussian random variables are completely determined by their first two moments, (H1) and (H2) imply that there exists the following representation of B (assuming (Ω,F,P) to be large enough): B t L = Z t 0 K(t,s)dW s , t∈ [0,T ], (2.2.6) where W ={W t ,t∈ [0,T ]} is a Brownian motion with respect to (F t ). Under the additional assumption (H3) and that K is injective, [24, Theorem 1] states that this representation is “canonical” , in the sense that (F B t ) = (F W t ) Following [30, Section 3.1] define the isometryI 1 :L 2 ([0,T ])→L 2 (Ω) by I 1 :=R −1 ◦K, wherethelinearoperatorR :H→R(H)isdefinedbyR(ζ)(t) =hζ,B t i H ,t∈ [0,T ]. Theorem 2.2.13 ( [24, Theorem 1]). Let B ={B t ,t∈ [0,T ]} be a mean-zero Gaussian process with factorizable covariance function R(t,s) as in (2.2.5), sat- isfying (H1) to (H3) and that the associated operator K is injective. Let W = 22 {W t ,t∈ [0,T ]} be given by W t :=I 1 (1 [0,t] ). Then W t is a (F B t )-Brownian motion and B t has representation B t = Z t 0 K(t,s)dW s P-a.s. with (F B t ) = (F W t ). We proof a Lemma which is an essential part of the proof of Theorem 2.2.13. Lemma 2.2.14. The stochastic process W ={W t ,t∈ [0,T ]} defined by W t :=I 1 (1 [0,t] ), t∈ [0,T ]. (2.2.7) is a (F B t )-Brownian motion. Proof. Denote by H t the closure of{B s ,s∈ [0,t]}. Then by Proposition 2.2.11I 1 is an isometry between L 2 [0,t] and H t . So W is (F B t )-adapted. Also for s,t∈ [0,T ] E(W t W s ) =E R −1 (K(1 [0,t] ))R −1 (K(1 [0,t] )) =hK(1 [0,t] )),K(1 [0,t] )i H =h1 [0,t] ,1 [0,s] i L 2 ([0,T ]) =t∧s. 23 Moreover, using that R(B t ) =R(t,·) =K(K(t,·)), for all 0≤r≤s≤t≤T E((W t −W s )B r ) =E R −1 (K(1 [s,t] ))R −1 (K(K(r,·))) =hK(1 [s,t] )),K(K(r,·))i H =h1 [s,t] ,K(r,·)i L 2 ([0,T ]) = Z t s K(r,u)du = 0. So W t −W s is independent ofF B s . 2.3 Stochastic calculus 2.3.1 Malliavin Calculus; Derivative and adjoint operator Malliavin Calculus consists of the study of three operators: the Malliavin derivative D, the divergence operator δ and the Ornstein-Uhlenbeck operator L. One impor- tant tool of Malliavin calculus is an integration by parts formula on the Wiener space. It turns out that Malliavin Calculus can be developed in much generality, for any isonormal Gaussian process. Consider a mean-zero real-valued Gaussian process X ={X(h) :h∈H} on a complete probability space (Ω,F,P), whereH is a real separable Hilbert space, and assume that E (X(h)X(g)) =hh,gi H , 24 whereh·,·i H denotes the inner product onH, and theσ-fieldF is generated byX. Denote byH q Wiener chaos of order q. This is the closed linear subspace of the Hilbert space L 2 (Ω,F,P) generated by the random variables {H q (X(h)) :h∈H,khk H = 1}. Here H q (x) denote the qth Hermite polynomial, which is defined by H q (x) = (−1) q q! e x 2 /2 d q dx q e −x 2 /2 , q≥ 1, and H 0 (x) = 1. Then H 1 (x) =x, H 2 (x) = 1 2 (x 2 − 1), H 3 (x) = 1 6 (x 3 − 3x) and so forth. Hermite polynomials form a basis of L 2 (R). One can show the following properties: For q≥ 1 H 0 q (x) =H q−1 (x), (2.3.1) (q + 1)H q+1 (x) =xH q (x)−H q−1 (x), (2.3.2) H q (−x) = (−1) q H q (x). (2.3.3) 25 One important result is that the space L 2 (Ω,F,P) can be written as the direct sum of the Wiener chaosH q , q≥ 0. Theorem 2.3.1 ( [50, Theorem 1.1.1] ). The space L 2 (Ω,F,P) can be decomposed into the direct sum of orthogonal subspacesH q : L 2 (Ω,F,P) = ∞ M q=0 H q . Denote by C ∞ p (R n ) the vector space of infinitely differentiable functions on f : R n →R, wheref and all its derivatives have at most polynomial growth. LetS be the set of random variables of the form F =f(X(h 1 ),...,X(h n )), h 1 ,...,h n ∈H, where f∈ C ∞ p (R n ) and n≥ 1. Here In the following the elements ofS are called smooth functionals. Define the derivative operator of such F as theH-valued random variable D X F = n X j=1 ∂f ∂x j (X(h 1 ),...,X(h n ))h j . Proposition 2.3.2 ( [50, Proposition 1.2.1]). The derivative operator D X is a closable unbounded operator from L p (Ω,F,P) to L p (Ω,F,P;H) for any p≥ 0. 26 Define asD 1,p X , p≥ 1, the closure ofS with respect to the norm kFk p X,1,p =kFk p L p (Ω) +kD X Fk p L p (Ω;H) . In the same fashion we can iteratively define the derivative operatorD X,k , which is a closable unbounded operator, mapping from L 2 (Ω,F,P) to L 2 (Ω,F,P;H ⊗k ). HereH ⊗k denotes the Hilbert spaceH ⊗k =H×···×H iterated k-times. Define asD k,p X (H),k,p≥ 1, the space ofH-valued random variables that are in the closure of smooth functionals F∈S with respect to the norm kFk p X,k,p =kFk p L p (Ω) + k X j=1 kD X,j Fk p L p (Ω;H ⊗j ) . Denote by δ X the adjoint of the operator D X . The domain dom(δ X ) are all u∈L 2 (Ω,F,P;H) such that there exists a constant c> 0 such that hD X F,ui L 2 (Ω;H) ≤ckFk L 2 (Ω) , for all F∈S. (2.3.4) For an element u∈ dom(δ X ) we can define δ X (u) through the relationship hF,δ X (u)i L 2 (Ω) =hD X F,ui L 2 (Ω;H) , for all F∈D 1,2 X (H). 27 In [50, Proposition 1.3.1] it is shown that for any u∈D 1,2 X (H) kδ X (u)k L 2 (Ω;H) ≤kuk X,1,2 , and hence the space D 1,2 X (H) is contained in dom(δ X ). In fact, for any simple function F∈D 1,2 X (H) of the form F = n X i=1 F i h i , F i ∈S, h i ∈H, i = 1,...,n we have δ X (F ) = n X i=1 F i X(h i )− n X i=1 hD X F j ,h j i L 2 (Ω;H) . Denote by J q F the projection of F ∈ L 2 (Ω) onto the qth Wiener chaos. We define the operator L X as follows: L X F = ∞ X q=0 −qJ q F, (2.3.5) whenever F∈ dom(L) :={F∈L 2 (Ω) : P ∞ q=0 q 2 kJ q Fk 2 2 <∞}. 2.3.2 Deterministic integrands This section generalizes some of the integrals defined for fractional Brownian mo- tion, see [19] for a review. 28 At first we consider the class of stochastic integrals with respect to deterministic functions. Stochastic integrals of deterministic functions with respect to Gaussian processes are also called Wiener integrals. Afterwards we generalize the obtained integrals to stochastic integrands and obtain the divergence integral. Assume in the following B ={B t ,t∈ [0,T ]} is a mean-zero Gaussian process withfactorizablecovariancefunctionR(t,s)oftheformR(t,s) = R T 0 K(t,r)K(s,r)dr andK satisfying (H1), i.e. sup t∈[0,T ] kK(t,·)k L 2 ([0,T ]) <∞. LetH the RKHS asso- ciated with B t . Definition 2.3.3. The abstract Wiener integral is defined as the linear exten- sion in L 2 (Ω) of the map (WIENER)- Z :H→L 2 (Ω) R(t,·)→B t . (2.3.6) That means, if f(s) = P i a i R(t i ,s), then (WIENER)- R f(s)dB s = P i a i B t i . More- over for anyf∈H, there exists a sequence of (f n ) n , where eachf n is a finite linear combination of R(t,·), which converges to f inH. Then by definition (WIENER)- Z f(s)dB s = lim n→∞ (WIENER)- Z f n (s)dB s in L 2 (Ω). 29 A characterization of elements in the RKHSH is given in Proposition 2.2.11. It is however in general difficult to say which functions are inH. Often it is convenient to find an isometrically isomorphic Hilbert space and replaceH with it. Definition 2.3.4. LetH be a Hilbert space. A representation of H is a pair (E,i) consisting of a functional space E and a bijective isometry i between E and H. Note that RKHSH and a subspace ofL 2 ([0,T ]) are isometrically isomorphic by Proposition 2.2.11. This motivates the definition of the following Wiener integral. Definition2.3.5. LetE be the closed subspace ofL 2 ([0,T ]) spanned by{K(t,·),t∈ [0,T ]}. Then the (W1)- R -integral is a Wiener integral defined as the extension of the map (W1)- Z :E→L 2 (Ω) K(t,·)→B t . (2.3.7) Assume here that 1 [0,t] ∈ E. With Wiener integral (W1)- R , the function 1 [0,t] is no longer the preimage of B t , i.e. (W)- R t 0 1 [0,t] (s)dB s =B t does not hold, since E (W1)- Z 1 [0,t] (s)dB s 2 =k1 [0,t] k L 2 ([0,T ]) =t6=EB 2 t =R(t,t). In fact, (W1)- R 1 [0,t] (s)dB s L = f W t , where f W t is a standard Brownian motion. 30 Wiener integral (W1)- R is the deterministic integral usually found in the literature in the case thatB is a Brownian motion. In that case, the RKHS is the space{f∈ C([0,T ]) : ˙ f∈L 2 ([0,T ]),f(0) = 0} endowed with the scalar producthf,gi = R ˙ f ˙ g. This RKHS is being replaced byL 2 ([0,T ]). A representation of the RKHS space is here given by (L 2 ([0,T ]),d/dt). In the case of Brownian motion, the extension of integral (W1)- R to stochastic integrands coincides with the Itô stochastic integral. ConsiderthespaceofstepfunctionsE on [0,T ]withscalarproducth1 [0,t] ,1 [0,s] i 1 = R(t,s). Consider the linear map given by (E,h·,·i 1 )→H, 1 [0,t] →R(t,·). (2.3.8) The extension of this linear map to the closure of (E,h·,·i 1 ) is a representation of H. It is essential to take the closure of (E,h·,·i 1 ): In the case of fractional Brownian motion with H > 1/2 the space (E,h·,·i 1 ) is not complete, as proofed in [59]. Definition 2.3.6. Let B ={B t ,t∈ [0,T ]} be Gaussian process with covariance function R(t,s). Define Wiener integral as the linear extension of the map Z : (E,h·,·i 1 )→L 2 (Ω) 1 [0,t] →B t . (2.3.9) 31 From now on identify (H,h·,·i H ) = (E,h·,·i 1 ). (2.3.10) The elements ofH may not be function but distributions. This is in particular the case for fractional Brownian motion for H > 1/2, as is shown in [58,59]. For H < 1/2 however, the space (E,h·,·i 1 ) is a function space. In most cases integral (W1)- R is different from integral R . For fractional Brownian motion this can be seen by computing the second moments E (W1)- Z 1 [0,t] (s)dB s 2 =k1 [0,t] k L 2 ([0,T ]) =t6=t 2H =E Z 1 [0,t] (s)dB s 2 . In the case of a Brownian motion, integral (W1)- R is the same as integral R , and they have the following relation to integral (WIENER)- R : (W1)- Z f(s)dW s = Z f(s)dW s = (WIENER)- Z Z s 0 f(u)du dW s , f∈L 2 ([0,T ]). 2.3.3 Stochastic integrands Inthefollowingwewillonlyworkwiththeintegral R fromDefinition2.3.6. Thenext step is to generalize integral R in Definition 2.3.6 to non-deterministic integrands. Note at first thatH⊂ dom(δ B ). Moreover for any ϕ∈H, δ B (φ) = R T 0 ϕ(t)dB t . 32 This follows by the linearity of the operator and by observing that δ B (1 [0,t] ) =B t . Thus the Malliavin adjoint is a generalization of the integral in Definition 2.3.6. In the following denote by D, δ,D k,p (L 2 [0,T ]) the operators and spaces associ- ated with the Wiener process W t . For any u∈ dom(δ) we call δ(u) the divergence integral and write δ(u) = Z T 0 u t dW t , and similarly for any u∈ dom(δ B ) we write δ B (u) = R T 0 u t dB t . It turns out that the operator and spaces D B , δ B ,D k,p B (H) can be expressed in terms ofD,δ,D k,p . The following Lemmas and the concluding Theorem show their relationship. Lemma 2.3.7 ( [3, Lemma 1]). Denote byE the set of step functions on [0,T ]. DenoteK ∗ the adjoint of the operator K in the following sense: Z T 0 φ(t)(Kf)(dt) = Z T 0 (K ∗ φ)(t)f(t)dt for all φ∈E and f∈L 2 ([0,T ]). For any function φ∈E we have (K ∗ φ)(s) =φ(s)K(T,s) + Z T s [φ(t)−φ(s)]K(dt,s). (2.3.11) 33 If for example φ = P n 1 a i 1 (s i ,s i+1 ] , where a i ∈ R and 0 = s 1 < ... < s n+1 = T, then (K ∗ φ)(s) = n X i=1 a i 1 (s i ,s i+1 ] (s)K(T,s) + n X i=1 a i 1 (s i ,s i+1 ] (s) n X j=i+1 (a j −a i )(K(s j+1 ,s)−K(s j ,s)). The proof for this Lemma is a straightforward calculation and given in [3, Lemma 1]. The next claim is a consequence of Lemma 2.3.7 if one formally replacesf with ˙ W. We give a proof, since the proof is omitted in [3]. Lemma 2.3.8. For all φ∈E Z T 0 φ(t)dB t = Z T 0 (K ∗ φ)(t)dW t , (2.3.12) where the operatorK ∗ is defined as in (2.3.11). 34 Proof. Let φ = P n 1 a i 1 (s i ,s i+1 ] , where a i ∈R and 0 =s 1 <...<s n+1 =T, then Z T 0 φ(t)dB t = n X i=1 a i (B s i+1 −B s i ) =a n B T − n−1 X i=1 (a i+1 −a i )B s i = Z T 0 a n K(T,s)dW s − Z T 0 n−1 X i=1 (a i+1 −a i )1 (0,s i+1 ] K(s i+1 ,s) ! dW s = Z T 0 n X i=1 a i 1 (s i ,s i+1 ] (s)K(T,s)dW s + Z T 0 n−1 X i=1 (a i+1 −a i )1 (0,s i+1 ] (s)(K(T,s)−K(s i+1 ,s)) ! dW s = Z T 0 φ(s)K(T,s)dW s + Z T 0   n X i=1 1 (s i ,s i+1 ] (s) n X j=i+1 (a j −a i )(K(s j+1 ,s)−K(s j ,s))   dW s = Z T 0 (K ∗ φ)(t)dW t . Corollary 2.3.9. Assume there is an element h∈H such thatK ∗ h = 1 [0,t] then W t = R t 0 h(s)dB s , which implies W t isF B t -adapted. Taking squares and expectation in Lemma 2.3.7 yields the isometry kφk H =kK ∗ φk L 2 ([0,T ]) 35 for all φ∈E. Taking the closure ofE with respect to the normk·k H yields that K ∗ is an isometry betweenH and a closed subspace of L 2 ([0,T ])., i.e. H = (K ∗ ) −1 (L 2 ([0,T ])). (2.3.13) Theorem 2.3.10. The following relationships hold D 1,2 B (H) = (K ∗ ) −1 (D 1,2 (L 2 [0,T ])); (2.3.14) dom(δ B ) = (K ∗ ) −1 (dom(δ)); (2.3.15) δ B (u) = Z T 0 (K ∗ u) t dW t , ∀u∈ dom(δ B ). (2.3.16) Proof of Theorem 2.3.10. The proof is essentially given in [3, Section 2]. From relation (2.3.13) we can deduce that (2.3.14) holds. Moreover if F = f(B t ) is a smooth functional andu is anH-valued square integrable random variable, we get: Ehu,D B Fi H =Ehu,f 0 (B t )1 [0,t] i H =EhK ∗ u,f 0 (B t )K ∗ 1 [0,t] i L 2 ([0,T ]) =EhK ∗ u,f 0 (B t )K(t,·)i L 2 ([0,T ]) =EhK ∗ u,DFi L 2 ([0,T ]) and therefore (2.3.15) and (2.3.16) follow. 36 Chapter 3 Statistical inference of Itô-processes of diffusion type driven by Volterra noise 3.1 Introduction Diffusion processes are used for mathematical modeling in such diverse fields as economics [10], random media [54], financial mathematics [34] and others, see [38] for additional fields. We define diffusion processes as Itô-processes ξ ={ξ t ,t∈ [0,∞)} of the form (see [44, Definition 7]) ξ t = Z t 0 α s (ξ)ds + Z t 0 β s (ξ)dW s , 0≤t<∞, where α s and β s areB(C[0,s]) + -measurable for every 0≤s<∞. 37 The definition of diffusion processes varies in the literature. In the traditional approach the term diffusion process is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal gen- erator, see [35, Ch. 5]. An alternative, probabilistic approach to diffusion processes has been initiated by Lévy and developed further by Itô. They consider continuous processes X ={X t , 0≤ t <∞}, which satisfy for every x∈ R the stochastic integral equation X t =x + Z t 0 a(X s )ds + Z t 0 b(X s )dW s , 0≤t<∞. This stochastic differential equations approach provides a useful alternative and is used in [38] for statistic parameter estimation. As is claimed in [35], the only important Markov processes with continuous paths not included in this approach arethosewithspecialboundarybehavior,suchasreflectionorkillingonaboundary. Note that our definition of diffusion processes is more general, since the coefficients may depend on the whole path of ξ t up to time t. Solutions to diffusion equations are distinguished between strong and weak so- lutions. Strong solutions, which will be considered in this chapter, are solutions with respect to a given filtration and Brownian motionW t . For weak solutions, the idea is that the probability space, the filtration and the Brownian motion are part of the solution rather than part of the problem. 38 At first we obtain a maximum likelihood estimator for diffusion processes. Sta- tistical inference of diffusion processes has been studied in [38,43] amongst others. We consider the problem of estimating the parameter θ which is multiplied to the coefficient α s (ξ). The coefficient β s (ξ) is assumed to be 1. If it were another con- stant β s (ξ) =σ for s∈ [0,T ], the quadratic variation process of ξ t is given almost surely by hξi t =σ 2 t, whereh·i t denotes the quadratic variation process. Hence the parameter σ 2 is obtained by the following limit: 1 T lim n→∞ n X i=1 x t n i+1 −x t n i 2 =σ 2 a.s., where 0 = t n 0 < ... < t n n = T is a partition of [0,T ] with max i |t n i+1 −t n i |→ 0 as n→∞. Moreover we analyze the relation between strong consistency of the maximum likelihood estimator and the singularity between the measures of the diffusion pro- cess and the Brownian motion. In some applications, Gaussian white noise is an unrealistic assumption. In the second section, we consider stochastic processes of the form X t =θ Z t 0 b(X s )ds + Z t 0 K(t,s)dW s , t≥ 0, X 0 = 0, 39 where b(x) is a Lipschitz continuous function and K(t,s) is a square-integrable kernel satisfying assumptions that imply a.s. Hölder continuity of the sample paths ofB t = R t 0 K(t,s)dW s (see Subsection 2.2.1 in Chapter 2 for sufficient condition for Hölder continuity). In [66] existence and uniqueness of differential equations driven by Hölder con- tinuous functions are studied. Furthermore the results are applied to fractional Brownian motion. Several existence and uniqueness results of differential equations driven by fractional noise can be found in [51]. Parameter estimation problems with an integrated kernel including fractional Brownian are discussed in [22,30,36,68]. In [36] the authors study maximum like- lihood estimator of Ornstein Uhlenbeck processes driven by fractional Brownian motion with H≥ 1/2. They show strong consistency and asymptotic behavior of the bias and the mean square of the estimator. [68] use a different maximum like- lihood estimator for a non-linear Ornstein Uhlenbeck process driven by fractional Brownian motion withH∈ (0, 1) and proof its strong consistency using techniques from Malliavin calculus. An estimator for Ornstein Uhlenbeck motivated by least squares is proposed in [29], and its strong consistency and asymptotic distribution is shown. We consider the problem of estimating the parameter θ through a maximum likelihood type estimator. Our strategy is to reduce the problem to estimating the parameter of an Itô-process of diffusion type. We develop a suitable version of the 40 Girsanov theorem, and use it to find maximum likelihood estimator. We illustrate thetheoryontwoexamples: IncasethatthekernelK(t,s)isthekerneloffractional Brownian motion, and then when it is the kernel of an Ornstein-Uhlenbeck process. In both examples, we proof strong consistency of the estimator. In the third section, we consider the stochastic process ˙ X t =θ Z t 0 b(X s )ds + Z t 0 K(t,s)dW s , t≥ 0, X 0 = ˙ X 0 = 0, where K(t,s) is the kernel of fractional Brownian motion. We show, that under a suitable Girsanov transform the process X t is an inte- grated fractional Brownian motion. We develop properties of integrated fractional Brownianmotionandusethepropertiestoproofstrongconsistencyoftheestimator in the case that b(x) =x. There is a decent amount of literature available covering topics related to frac- tional Brownian motion. In [48] the authors show that integration over a certain kernel with respect to fractional Brownian motion leads to a martingale with the same filtration, which they call fundamental Martingale. Stochastic analysis with respect to fractional Brownian motion is discussed in [19,22,23,50]. Especially [11] is a useful reference for stochastic calculus and applications of fractional Brownian motion and includes useful references to recent literature. Ergodicity of fractional Ornstein Uhlenbeck processes is discussed in [14]. 41 3.2 Maximum likelihood estimation of diffusion processes with Gaussian white noise Denote by (C([0,∞)),B(C[0,∞))) the measure space of continuous functions x = {x s ,s∈ [0,∞)} and byB(C[0,t]) the Borel σ-algebra generated by open sets of C[0,t]. For any filtration (F t ) write (F t ) + for it to be right-continuous such thatF + t = ∩ ε>0 F t+ε . Moreover denoteF ∞ =σ (∪ t≥0 F t ). Denote by P Z the measure generated by a continuous stochastic process Z = {Z t ,t∈ [0,∞)}inthemeasurespaceofcontinuousfunctions (C[0,∞),B(C[0,∞))). In the following let (Ω,F, (F t ),P) be a stochastic basis satisfying the usual hypotheses and W ={W t ,t∈ [0,∞)} a Brownian motion. Denote by (F W t ) the filtration withF W t :=σ (W s ,s∈ [0,t]). 3.2.1 Diffusion processes The following is as in [44, Definition 6]. Definition 3.2.1. A continuous random process ξ ={ξ t ,t∈ [0,∞)} is called an Itô-process if there exists two (F t )-adapted processes a ={a t ,t∈ [0,∞)} and b ={b t ,t∈ [0,∞)} such that for 0≤t<∞ P Z t 0 |a s |ds<∞ = 1, P Z t 0 b 2 s ds<∞ = 1 (3.2.1) 42 and with probability 1 ξ t =ξ 0 + Z t 0 a s ds + Z t 0 b s dW s . (3.2.2) A shorthand notation for (3.2.2) is dξ t =a t dt +b t dW t . (3.2.3) The following is as in [44, Definition 7]. Definition 3.2.2. The Itô-process ξ ={ξ t ,t∈ [0,T ]} is called a process of the diffusion type if the functionalsa t andb t in Definition 3.2.1 are (F ξ t )-measurable for almost all t, 0≤t<∞. Remark 3.2.3. Every process ξ ={ξ t ,t∈ [0,∞)} satsifying definition 3.2.2 with ξ 0 = 0 and b s = 1 for all s≥ 0 is of the form ξ t = Z t 0 α s (ξ)ds +W t . (3.2.4) whereα s isB(C[0,s]) + -measurableforevery 0≤s<∞andP R t 0 |α s (ξ)|ds<∞ = 1 for any t> 0. 43 3.2.2 Girsanov theorem Consider a process κ(W ) ={κ t (W ),t∈ [0,T ]} (or κ(ξ) respectively) given by κ t (W ) = dP ξ dP W (t,W ) and κ t (ξ) = dP ξ dP W (t,ξ). Thefollowingtwotheoremsarewell-known, buttheyhaveanimportantfunctionfor therestofthepresentation. That’swhywerecallthematthisplaceforconvenience. The following is as in [44, Theorem 7.6]. Theorem 3.2.4. Let ξ ={ξ t ,t∈ [0,T ]} be a process of diffusion type with the differential dξ t =α t (ξ)dt +dW t , ξ 0 = 0, 0≤t≤T. If P R T 0 α 2 t (ξ)dt<∞ = 1, then the process κ(W ) ={κ t (W ),t∈ [0,T ]} is the unique solution to the equation κ t (W ) = 1 + Z t 0 κ s (W )α s (W )dW s ; dP ξ dP W (t,ξ) = exp Z t 0 α s (ξ)dξ s − 1 2 Z t 0 α 2 s (ξ)ds (P-a.s.), 44 P Z T 0 α 2 t (W )dt<∞ ! =E exp − Z t 0 α s (ξ)dW s − 1 2 Z t 0 α 2 s (ξ)ds =E exp − Z t 0 α s (ξ)dξ s + 1 2 Z t 0 α 2 s (ξ)ds . The following is as in [44, Theorem 7.7]. Theorem 3.2.5. Let ξ ={ξ t ,t∈ [0,T ]} be a process of diffusion type with the differential dξ t =α t (ξ)dt +dW t , ξ 0 = 0, 0≤t≤T. Then P ξ ∼P W ⇔              P R T 0 α 2 t (ξ)dt<∞ = 1 P R T 0 α 2 t (W )dt<∞ = 1 Here dP ξ dP W (t,W ) = exp Z t 0 α s (W )dW s − 1 2 Z t 0 α 2 s (W )ds (P-a.s.), dP W dP ξ (t,ξ) = exp − Z t 0 α s (ξ)dξ s + 1 2 Z t 0 α 2 s (ξ)ds (P-a.s.). Definition 3.2.6. Let P and Q be two probability measures on a filtered measure space (Ω,F, (F t )). Denote byP t andQ t the restriction ofP andQ toF t respectively. 1. Q is called absolutely continuous with respect toP, writtenQP, if for every set A∈F it holds that Q(A) = 0 whenever P(A) = 0. 45 2. Q and P are called equivalent, written Q∼P if QP as well as PQ hold. 3. Q andP are called singular, writtenQ⊥P, if there exists a setE∈F such that Q(Ω/E) = 1 and P(E) = 1. 4. Q is called locally absolutely continuous with respect toP, writtenQ loc P, if Q t P t for all 0≤t<∞. 5. Q and P are called locally equivalent, written Q loc ∼P, whenever Q t ∼P t for all 0≤t<∞. In the following the measure space (C([0,∞)),B(C[0,∞))) is equipped with filtrationF t =B(C[0,t]). Theorem 3.2.4 and 3.2.5 imply the following Corollary. Corollary 3.2.7. Let ξ be given as in (3.2.4). 1. P ξ loc P W on (F t )⇔P R t 0 α 2 s (ξ)ds<∞ = 1 for all 0≤t<∞. 2. P W loc ∼P ξ on (F t )⇔              P R t 0 α 2 s (ξ)ds<∞ = 1, P R t 0 α 2 s (W )ds<∞ = 1, for all 0≤t<∞. Example3.2.8. LetP andQ betwomeasuresonafilteredmeasurespace (Ω, (F t ),F) satisfying P loc ∼Q. Then in general it is false thatPQ orQP onF ∞ . Here is a counterexample (adapted from [35, after corollary 5.2]). Consider ξ t = μt +W t , μ < 0. Then for 46 any 0≤ t <∞ the two integrals in the conditions in corollary 3.2.7[2.] are finite a.s., soP ξ loc ∼P W .Consider the set n x∈C([0,∞)) : lim T→∞ x T T =μ o ∈B(C[0,∞)). Then P W x : lim T→∞ x T T =μ =P lim T→∞ W T T =μ = 0, but P ξ x : lim T→∞ x T T =μ =P lim T→∞ ξ T T =μ ! = 1 by the law of large numbers for martingales, so P ξ ⊥P W onB(C[0,∞)). Theorem 3.2.9. Let ξ be given as in (3.2.4). Assume P W loc ∼P ξ .Then P W ⊥P ξ ⇔P Z ∞ 0 α 2 s (W )ds =∞ = 1 (3.2.5) ⇔P Z ∞ 0 α 2 s (ξ)ds =∞ = 1. (3.2.6) Proof. First construct a measure Q on (Ω, (F W t ),F W ∞ ) as in [35, discussion after theorem 5.1] such that ξ is a Wiener process underQ. Define Z t := exp − Z t 0 α s (ξ)dW s − 1 2 Z t 0 α 2 s (ξ)ds 0≤t<∞. By theorem 3.2.5 it follows thatE (Z t ) = 1 for all 0≤t<∞. Lett 1 ,...,t n ∈ (0,∞) be distinct numbers and set T := maxt i . Define a measureQ T on (Ω, (F W t ),F W T ) by dQ T (ω) =Z T (ω)dP(ω). 47 Then by [44, theorem 6.3] the processξ is a Wiener process on (Ω, (F W t ),F W T ,Q T ). Set μ t 1 ,...,tn (A) :=Q T (ω∈ Ω : (W t 1 (ω),...,W tn (ω))∈A), A∈B(R n ). (3.2.7) Then μ t 1 ,...,tn is a consistent family of finite dimensional distributions, so by Kol- mogorov extension theorem exists a measure μ on (R [0,∞) ,B(R [0,∞) )) such that μ t 1 ,...,tn (A) =μ w∈R [0,∞) : (w t 1 ,...,w tn )∈A , A∈B(R n ). (3.2.8) Since every set inF W ∞ can be written as{ω :W (ω)∈A} for some A∈B(R [0,∞) ), there exists aQ defined onF W ∞ defined as Q (ω :W (ω)∈A) :=μ(A). Secondly, one can use a theorem from [31]. The conditionP R t 0 α s (ξ)ds<∞ = 1 for all 0≤t<∞ is satisfied by corollary 3.2.7, so it follows by [31, theorem IV.4.23, c)] that Q⊥P⇔Q Z ∞ 0 α s (ξ)ds =∞ = 1. SinceQ ξ =P W , Q⊥P⇔P Z ∞ 0 α s (W )ds =∞ = 1. 48 For (3.2.5) to hold it remains to show that Q⊥P if and only if P W ⊥P ξ . Assume Q⊥P. Since Q ξ =P W this implies P W =Q ξ ⊥P ξ . To show the reverse implication, note that since P W ⊥P ξ there exists a set E∈B(C[0,∞)) such that P W (E) = 1 and P ξ (Ω/E) = 0. Set B :={ω : ξ(ω)∈ E}∈F W ∞ . Then Q(B) = Q (ω :ξ(ω)∈E) = Q ξ (E) = P W (E) = 1 and P(Ω/B) = P (ω :ξ(ω)∈ Ω/E) = P ξ (Ω/E) = 0. Therefore Q ⊥ P, so (3.2.5) holds. By applying [31, theorem IV.4.23, c)] again with the measureQ andP interchanged using that also Q R t 0 α s (ξ)ds<∞ = 1 for all 0≤ t <∞, it follows similarly that (3.2.6) holds. 3.2.3 Maximum likelihood estimation In the following let θ∈R. An estimator b θ t of the parameter θ is called strongly consistent if lim t→∞ b θ t =θ, P-a.s. Theorem 3.2.10. Let a diffusion process ξ ={ξ t ,t∈ [0,∞)} be given by ξ t =θ Z t 0 α s (ξ)ds +W s , (3.2.9) 49 where P R t 0 α 2 s (ξ)ds<∞ = 1 for all t> 0. Given observations (ξ s , 0≤ s≤ t), the maximum likelihood estimator (MLE) is given by b θ t = R t 0 α s (ξ)dξ s R t 0 α 2 s (ξ)ds . (3.2.10) provided that P R t 0 α 2 s (ξ)ds> 0 = 1. Proof. IfP R t 0 α 2 s (ξ)ds> 0 = 1, the representation of the MLE follows after max- imising the density given by Theorem 3.2.4 dP ξ dP W (ξ,t) =E exp ( θ Z t 0 α s (ξ)dξ s − θ 2 2 Z t 0 α 2 s (ξ)ds )! . with respect to the parameter θ. Theorem 3.2.11. Let ξ ={ξ t ,t∈ [0,∞)} be given by (3.2.9). Then the following conditions imply that the maximum likelihood estimator b θ t in (3.2.10) is strongly consistent 1. P ( R ∞ 0 α 2 s (ξ)ds =∞) = 1. 2. P R t 0 α 2 s (W )ds<∞ = 1 for all t> 0 and P ( R ∞ 0 α 2 s (W )ds =∞) = 1. 50 Proof. Assume that (1) holds. By an application of the strong law of large numbers for local martingales applied to the local martingaleM t := R t 0 α s (ξ)dW s , (1) implies that b θ t = R t 0 α s (ξ)dξ s R t 0 α 2 s (ξ)ds = R t 0 α s (ξ)dW s R t 0 α 2 s (ξ)ds +θ = M t hMi t +θ→θ P-a.s. Moreover by Theorem 3.2.9, (2) implies (1). Example 3.2.12. Consider an Ornstein-Uhlenbeck process X t =θ Z t 0 X s ds +W t with θ∈ R. Since X t is continuous R t 0 X 2 s ds≤ t sup s∈[0,t] X 2 s <∞ P-a.s. for all 0≤t<∞. The MLE for θ is given by b θ t = R t 0 X s dX s R t 0 X 2 s ds . Futhermore R t 0 W 2 s ds <∞ P-a.s. for all 0≤ t <∞. Therefore P W loc ∼ P X so for strong consistency it is enough to show that P ( R ∞ 0 W 2 s ds =∞) = 1. By scale invariance of Brownian motion Z t 0 W 2 s ds =t Z 1 0 W 2 ts ds L =t Z 1 0 tW 2 s ds =t 2 Z 1 0 W 2 s ds→∞ (P-a.s.). 51 SinceforanyC > 0andanyε> 0thereexistsat> 0suchthatP R t 0 W 2 s ds>C > 1−ε and R t 0 W 2 s ds is increasing, it must converge, and the limit is infinityP-a.s.. Define the Brownian local time at time t and level a by L a (t) = lim ε&0 1 2ε |{s∈ [0,t] :W s ∈ (a−ε,a +ε)}| (3.2.11) see [61, IV. Corollary 1, page 217]. Using the occupation times formula for Brownian local time one can generalise Example 3.2.12: The formula says that for every positive Borel-function g :R→R Z t 0 g(W s )ds = Z ∞ −∞ g(a)L a (t)da P-a.s., (3.2.12) see [62, VI Corollary 1.6]. The following Lemma is based on the main result in [74]. Theorem 3.2.13. Let ξ ={ξ t ,t∈ [0,∞)} be as in (3.2.9) and assume that P Z t 0 α 2 s (W )ds<∞ = 1, t≥ 0, and α s (ξ) is a function of ξ s , i.e. α s (ξ) =b(ξ s ), 0≤s<∞, 52 where b(x) is a Borel function onR. Then the maximum likelihood estimator b θ t in (3.2.10) is a strongly consistent estimator of θ. Proof. By Theorem 3.2.11 it is enough to proof thatP ( R ∞ 0 b 2 (W s )ds =∞). By the occupation times formula (3.2.12) Z ∞ 0 b 2 (W s )ds = lim t→∞ Z t 0 b 2 (W s )ds = lim t→∞ Z ∞ 0 b 2 (a)L a (t)da = Z ∞ 0 b 2 (a)L a (∞)da. Since b6= 0, the last expression is infinite P-a.s., so by Theorem 3.2.11, b θ t is a strongly consistent estimator of θ. Corollary 3.2.14. Let ξ ={ξ t ,t≥ 0} be a diffusion process satisfying ξ t =θ Z t 0 b(ξ s )ds +W t , with b(x)6= 0 and b(x) is either Lipschitz or b(x) is continuous and bounded. Then the MLE for θ given by b θ t = R t 0 b(ξ)dξ s R t 0 b 2 (ξ)ds (3.2.13) is strongly consistent. We can also analyze how largely the MLE b θ t fluctuates aroundθ, which is based on a standard result for local martingales. 53 Theorem 3.2.15. Let ξ ={ξ θ t ,t∈ [0,∞)} be as in (3.2.9), moreover assume P Z ∞ 0 α 2 s (ξ)ds =∞ = 1, then lim sup t→∞   b θ t −θ q R t 0 α 2 s (ξ θ )ds q 2 log(log( R t 0 α 2 s (ξ θ )ds)   = 1 P-a.s. For the proof of Theorem 3.2.15 we use the following well-known law of it- erated logarithm. Lemma 3.2.16. If M ={M t ,t≥ 0} is a continuous local martingale andhMi t → ∞ P-a.s., then lim sup t→∞ M t q 2hMi t log(log(hMi t )) = 1 P-a.s. Proof of Theorem 3.2.15. By an application of the law of iterated logarithm for local martingales applied to M t := R t 0 α s (ξ θ )dW s , it follows that lim sup t→∞ ( b θ t −θ)hMi t q 2hMi t log(log(hMi t )) = lim sup t→∞ M t q 2hMi t log(log(hMi t )) = 1 P-a.s. 54 3.3 Maximum likelihood estimation of Ornstein- Uhlenbeck process with Volterra noise Let (Ω,F, (F t ),P) be a filtered probability space satisfying the usual hypotheses. Suppose B ={B t ,t≥ 0} is a continuous Volterra process (see Definition 2.2.1) such that B t = R t 0 K(t,s)dW s , where W ={W t ,t≥ 0} is a Wiener process with respect to (F t ). Consider a stochastic process X ={X t ,t≥ 0} which is of the form X t =θ Z t 0 b(X s )ds +B t , X 0 = 0. (3.3.1) We investigate the following questions: 1. Solving the equation: For whichb(x) exists a solution of equation (3.3.1) and what are properties of the solution? 2. Parameter estimation: Given there exists continuous time data (X t , 0≤t≤ T ), is it possible to estimate θ from it? 3. Under which conditions is the maximum likelihood estimator (MLE) for θ consistent? The firstquestion canbeansweredthe followingway; sincethere is nostochastic multiplicative term for the noise, the answer is rather simple. For the solution in the multiplicative noise case, see [66, Thm 3]. 55 Theorem 3.3.1. Suppose that B ={B t ,t∈ [0,T ]} is a Volterra process (see Def- inition 2.2.1) which is continuous a.s. Assume b :R→R is Lipschitz-continuous. Then the stochastic integral equation X t = Z t 0 b(X s )ds +B t , X 0 = 0, (3.3.2) has a unique, strong solution. Proof. Since B t is continuous a.s., the proof is the same as the standard proof in the deterministic case, by using Picard iteration for a.e. ω. Indeed, the unique strong solution is given by X t = Z t 0 e R t s b(u)du dB s :=B t + Z t 0 b(s)e R t s b(u)du B s ds, where the last Riemann-Stieltjes integral exists because of the a.s. continuity of the sample paths. 3.3.1 Girsanov theorem Note that X t can be written as X t =θ Z t 0 b(X s )ds +B t =θ Z t 0 b(X s )ds + Z t 0 K(t,s)dW s = Z t 0 K(t,s)d f W s , (3.3.3) 56 assuming the following is an Itô-process f W t =θ Z t 0 K −1 Z · 0 b(X r )dr (s) ds +W t . (3.3.4) Moreover since X has representation (3.3.3), X is (F e W t ) t≥0 -adapted, so (3.3.4) is a process of the diffusion type, see Definition 3.2.2. To simplify notation define a process Q(X) ={Q t (X),t∈ [0,T ]} as Q t (X) :=K −1 Z · 0 b(X r )dr (t), so that f W t =θ Z t 0 Q s (X)ds +W t . (3.3.5) This observation leads to the following Corollary, which is a suitable adaption of Theorem 3.2.4. Corollary 3.3.2. Assume that P R T 0 Q 2 s (X)ds<∞ = 1. Then the measure P e W in the space of continuous functions C([0,T ]) generated by f W ={ f W t ,t∈ [0,T ]} is absolutely continuous to the measure P W generated by W ={W t ,t∈ [0,T ]} P e W P W . 57 The Radon-Nikodym derivative is given by dP e W dP W (T, f W ) = exp ( θ Z T 0 Q s (X)d f W s − θ 2 2 Z T 0 Q 2 s (X)ds ) . The stochastic process X is a Volterra process with kernel K under the measure e P(ω) = dP e W dP W (T,ω)P(ω). 3.3.2 Maximum likelihood estimation Theorem 3.3.3. Assume that t→ Q t ∈ L 2 ([0,T ]) almost surely. Then the MLE is given by b θ T = R T 0 Q t d f W t R T 0 Q 2 t dt . (3.3.6) Proof. According to Corollary 3.3.2 the Log-likelihood function is given by log dP e W dP W (T, f W ) =θ Z T 0 Q s d f W s − θ 2 2 Z T 0 Q 2 s ds (3.3.7) For optimising the Log-likelihood function with respect to θ it is necessary that 0 ! = Z T 0 Q s d f W s −θ Z T 0 Q 2 s dr. (3.3.8) And since R T 0 Q 2 r dr> 0 almost surely, the result follows. 58 LetH be defined as in (2.3.10) in Subsection 2.3.2 andK ∗ be defined as in Lemma 2.3.7. If there exist elements h t ∈H such that (K ∗ h t )(s) = 1 [0,t] (s) for all t∈ [0,T ], then the stochastic process f W ={ f W t ,t∈ [0,T ]} is given by f W t = Z t 0 h t (s)dX s . That implies that the MLE (3.3.6) is observable, given the knowledge of a sample path (X t , 0≤t≤T ). The bias of the MLE (3.3.6) is given by E b θ T −θ =E (R T 0 Q t d f W t R T 0 Q 2 t dt ) −θ =E (R T 0 Q t dW t R T 0 Q 2 t dt ) , so in general b θ T is a biased estimator of the parameter θ. Theorem3.3.4. The MLE is strongly consistent, that means b θ T →θ almost surely, if the following condition is satisfied: lim T→∞ Z T 0 Q 2 t dt =∞ P-a.s. (3.3.9) Proof. The strong consistency follows by an application of the strong law of large numbers for martingales. Indeed, plugging (3.3.5) in (3.3.6) yields b θ T −θ = R T 0 Q t d f W t R T 0 Q 2 t dt −θ = R T 0 Q t dW t R T 0 Q 2 t dt . 59 DefineN ={N t ,t∈ [0,T ]} byN t := R t 0 Q s dW s . ThenN t is locally square integrable martingale with brackethNi T = R T 0 Q 2 t dt. So by an application of the strong law of large numbers, the condition lim T→∞ Z T 0 Q 2 t dt =∞ P-a.s. (3.3.10) implies b θ T →θ P-almost surely. 3.3.3 Examples Example 3.3.5 (Ornstein Uhlenbeck process driven by fractional Brownian mo- tion). See Example 2.2.5 for the definition of fractional Brownian motion. To show that MLE is given by (3.3.6), one needs to verify Q t (X) = K −1 ( R · 0 X s ds)(t) ∈ L 2 ([0,T ]). The following identities hold (see [67, page 186]): K −1 h(s) =s H−1/2 I 1/2−H 0+ (s 1/2−H h 0 (s))(s), H≤ 1/2 (3.3.11) K −1 h(s) =s H−1/2 D H−1/2 0+ (s 1/2−H h 0 (s))(s), H≥ 1/2 (3.3.12) 60 where the fractional Riemann-Liouville integral and derivative are given for a func- tion f∈L 1 ([0,t]) and α> 0 by I α 0+ f(t) = 1 Γ(α) Z t 0 f(s) (t−s) 1−α ds D α 0+ f(t) = 1 Γ(1−α) d dt Z t 0 f(s) (t−s) α ds and the latter admits the Weil representation D α 0+ f(t) = 1 Γ(1−α) f(t) t α +α Z t 0 f(t)−f(s) (t−s) α+1 ds ! . case H < 1/2: By the formula for the inverse of K K −1 Z · 0 X r dr (t) =t H−1/2 I 1/2−H 0+ (t 1/2−H X t )(t) (3.3.13) = t H−1/2 Γ(1/2−H) Z t 0 s 1/2−H X s (t−s) H+1/2 ds. (3.3.14) Since t H−1/2 Z t 0 s 1/2−H (t−s) H+1/2 ds = Z 1 0 s 1/2−H (1−s) H+1/2 ds = B(3/2−H, 3/2 +H) 61 (where B(x,y) is the beta-function defined for x,y > 0) and X t is continuous a.e., the following estimates holds K −1 Z · 0 X s ds (t) ≤ B(3/2−H, 3/2 +H) Γ(1/2−H) sup t∈[0,T ] |X t |<∞ (3.3.15) wherethelasttermdependsonlyonT andH. ThereforeK −1 ( R · 0 X r dr)∈L 2 ([0,T ]) for H < 1/2. case H≥ 1/2: K −1 Z · 0 X r dr (t) =t H−1/2 D H−1/2 0+ (t 1/2−H X t )(t) = t H−1/2 Γ(3/2−H) t 1/2−H X t t H−1/2 + (H− 1/2) Z t 0 t 1/2−H X t −s 1/2−H X s (t−s) H+1/2 ds ! . So K −1 Z · 0 X r dr (t) ≤ t H−1/2 Γ(3/2−H) 1 + (H− 1/2) Z t 0 s 1/2−H −t 1/2−H (t−s) H+1/2 ds ! × sup t∈[0,T ] |X t |. (3.3.16) Moreover t H−1/2 Z t 0 s 1/2−H −t 1/2−H (t−s) H+1/2 ds =t 1/2−H Z 1 0 s 1/2−H − 1 (1−s) H+1/2 =t 1/2−H [B(3/2−H, 3/2 +H)−B(1, 3/2 +H)]. 62 It follows that K −1 ( R · 0 X r dr) ∈ L 2 ([0,T ]) for H ≥ 1/2, since both t H−1/2 and t 1/2−H are square integrable on [0,T ]. Finally, we show strong consistency of the estimator. By Theorem 3.2.11 it is enough to show that R T 0 Q 2 t (X)dt→∞ a.s. under the equivalent measure ˜ P. Then X t is a fractional Brownian motion, and we define e B t := X t . We only show strong consistency for the case H≤ 1/2, for H > 1/2 the proof is similar. For H≤ 1/2, we have Q t ( e B) =K −1 Z · 0 X r dr (t) =t H−1/2 I 1/2−H 0+ (t 1/2−H e B t )(t) = t H−1/2 Γ(1/2−H) Z t 0 s 1/2−H e B s (t−s) H+1/2 ds. By substitution it follows that R T 0 Q 2 t dt =T R 1 0 Q 2 Tt dt and Q Tt ( e B) = (Tt) H−1/2 Γ(1/2−H) Z Tt 0 s 1/2−H e B s (Tt−s) H+1/2 ds = (Tt) H−1/2 Γ(1/2−H) Z t 0 (Ts) 1/2−H e B Ts (Tt−Ts) H+1/2 Tds =T 1/2 t H−1/2 Γ(1/2−H) Z t 0 s 1/2−H e B Ts (t−s) H+1/2 T H ds. By self-similarity of e B t , for every C > 0 we have e P Z T 0 Q 2 t ( e B)dt>C ! = e P   T 2 Z 1 0 " t H−1/2 Γ(1/2−H) Z t 0 s 1/2e B s (t−s) H+1/2 ds # 2 dt>C   > 1−ε 63 for everyε∈ (0, 1), ifT is sufficiently large. Since R T 0 Q 2 t ( e B)dt is monotone increas- ing, it converges almost surely, and therefore R T 0 Q 2 t ( e B)dt→∞ as T→∞. Example3.3.6 (Ornstein-UhlenbeckprocessdrivenbyOrnstein-Uhlenbecknoise). Here X t solves the equation X t =θ Z t 0 X s ds + Z t 0 e η(t−s) dW s , see also Example 2.2.4. Here η∈R is known andθ∈R is unknown. To show that MLE for θ∈R is given by (3.3.6), one needs to verify Q t (X) =K −1 ( R · 0 X s ds)(t)∈ L 2 ([0,T ]). The linear operator K satisfies K −1 h(s) =h 0 (s)−ηh(s) (3.3.17) whenever h(0) = 0 and h is a.c. with derivative in L 2 , see [60]. Therefore Q t (X) =K −1 Z · 0 X s ds (t) =X t −η Z t 0 X s ds (3.3.18) which is in L 2 ([0,T ]) by continuity of X t . Thus the MLE is given by (3.3.6) and can be written as (recalling X t = R t 0 e η(t−s)f W t ) b θ T = R T 0 Q t d f W t R T 0 Q 2 t dt = R T 0 (X t −η R t 0 X s ds)d f W t R T 0 (X t −η R t 0 X s ds) 2 dt = R T 0 (X t −η R t 0 X s ds)d(X t −η R t 0 X s ds) R T 0 (X t −η R t 0 X s ds) 2 dt . 64 By Theorem 3.2.11 it is enough to show that R T 0 Q 2 t (X)dt→∞ a.s. under the equivalent measure ˜ P. Then X t = R t 0 e η(t−s) d f W s is an Ornstein-Uhlenbeck process with parameter η and thus has differential dX t =ηX t dt + f W t . Therefore Z T 0 Q 2 t (X)dt = Z T 0 (X t −η Z t 0 X s ds) 2 dt = Z T 0 f W 2 t dt→∞ e P−a.s. Hence b θ T is strongly consistent estimator. 3.4 Maximum likelihood estimation of harmonic oscillator driven by fBm Let B ={B t ,t∈ [0,∞)} be a fractional Brownian motion with index H∈ (0, 1). That means that B is a mean-zero Gaussian process whose covariance kernel is given by R(t,s) =E(B s B t ) := V H 2 s 2H +t 2H −|t−s| 2H , where V H := Γ(2− 2H) cos(πH) πH(1− 2H) . Consider the differential equation ¨ X t =θb(X t ) + ˙ B t , t> 0, X 0 = ˙ X 0 = 0, (3.4.1) 65 understood as system of linear integral equations X t = Z t 0 ˙ X s ds, ˙ X t =θ Z t 0 b(X s )ds +B t . (3.4.2) If b is globally Lipschitz, then (3.4.2) has a unique strong solution. This can be shown as in the deterministic case for each ω by using Picard iterates since the sample paths of B t are continuous a.e. 3.4.1 Girsanov theorem Note that ˙ X t can be written as ˙ X t =θ Z t 0 b(X s )ds +B t =θ Z t 0 b(X s )ds + Z t 0 K(t,s)dW s = Z t 0 K(t,s)d f W s , (3.4.3) assuming the following is an Itô-process f W t =θ Z t 0 K −1 Z · 0 b(X r )dr (s) ds +W t . (3.4.4) Moreover since ˙ X has representation (3.4.3), X t = R t 0 ˙ X s ds is (F e W t ) t≥0 -adapted, so (3.4.4) is a process of the diffusion type in the sense of Definition 3.2.2. To simplify notation define a process Q(X) ={Q t (X),t∈ [0,∞)} as Q t (X) :=K −1 Z · 0 b(X r )dr (t), 0≤t<∞ 66 so that f W t =θ Z t 0 Q s (X)ds +W t . (3.4.5) This observation leads to the following corollary, which is a suitable adaption to Girsanov theorem 3.2.4. Corollary 3.4.1. Assume that P( R T 0 Q 2 s (X)ds <∞) = 1. Then the measure P e W in the space of continuous functions C([0,T ]) generated by f W ={ f W t ,t∈ [0,T ]} is absolutely continuous to the measure P W generated by W ={W t ,t∈ [0,T ]} P e W P W . The Radon-Nikodym derivative is given by dP e W dP W (T, f W ) = exp ( θ Z T 0 Q s (X)d f W s − θ 2 2 Z T 0 Q 2 s (X)ds ) . The stochastic process ˙ X t is a fractional Brownian motion with index H under the measure d e P(ω) = dP e W dP W (T,ω)dP(ω), and thus the stochastic processX t is an integrated fractional Brownian motion under the measure e P. 67 3.4.2 Maximum likelihood estimation By Corollary 3.4.1 the maximum likelihood estimator is given by b θ T = R T 0 Q t d f W t R T 0 Q 2 t dt . (3.4.6) f W t has representation f W t = R t 0 ((K ∗ ) −1 1 [0,t] (·))(s)dX s withK ∗ defined in Chapter 1 in (2.3.11), which is the same as the operator K ∗ in [3]. Note that this implies that the estimator b θ T is observable, that means it can be expressed in terms of the observations (X t , 0≤t≤T ). One needs to verify Q t (X) = K −1 ( R · 0 b(X s )ds)(t)∈ L 2 ([0,T ]). The following identities hold (see [67, page 186]): K −1 h(s) =s H−1/2 I 1/2−H 0+ (s 1/2−H h 0 (s))(s), H≤ 1/2 (3.4.7) K −1 h(s) =s H−1/2 D H−1/2 0+ (s 1/2−H h 0 (s))(s), H≥ 1/2 (3.4.8) where the fractional Riemann-Liouville integral and derivative are given for a func- tion f∈L 1 ([0,t]) and α> 0 by I α 0+ f(t) = 1 Γ(α) Z t 0 f(s) (t−s) 1−α ds D α 0+ f(t) = 1 Γ(1−α) d dt Z t 0 f(s) (t−s) α ds 68 and the latter admits the Weil representation D α 0+ f(t) = 1 Γ(1−α) f(t) t α +α Z t 0 f(t)−f(s) (t−s) α+1 ds ! . case H < 1/2: By the formula for the inverse of K K −1 Z · 0 b(X r )dr (t) =t H−1/2 I 1/2−H 0+ (t 1/2−H b(X t ))(t) (3.4.9) = t H−1/2 Γ(1/2−H) Z t 0 s 1/2−H b(X s ) (t−s) H+1/2 ds. (3.4.10) b(x) is Lipschitz continuous, so it satisfies the linear growth condition|b(x)| ≤ C(1 +|x|) for all x∈R for some constant C > 0. Since t H−1/2 Z t 0 s 1/2−H (t−s) H+1/2 ds = Z 1 0 s 1/2−H (1−s) H+1/2 ds = B(3/2−H, 3/2 +H) (where B(x,y) is the beta-function defined for x,y > 0) and X t is continuous a.s., the following estimates holds K −1 Z · 0 b(X s )ds (t) ≤ B(3/2−H, 3/2 +H) Γ(1/2−H) C 1 + sup t∈[0,T ] |X t | ! <∞ (3.4.11) where the last term depends only on T and H. Therefore K −1 ( R · 0 b(X r )dr) ∈ L 2 ([0,T ]) for H < 1/2. 69 case H≥ 1/2: K −1 Z · 0 b(X r )dr (t) =t H−1/2 D H−1/2 0+ (t 1/2−H b(X t ))(t) = t H−1/2 Γ(3/2−H) t 1/2−H b(X t ) t H−1/2 + (H− 1/2) Z t 0 t 1/2−H b(X t )−s 1/2−H b(X s ) (t−s) H+1/2 ds ! . The estimate is then similar as in Example 3.3.5, the only difference is that in (3.3.16) the last term sup t∈[0,T ] |X t | is replaced by sup t∈[0,T ] |b(X t )|. This term is also finite, since by the linear growth of b(x) there exists a constant C > 0 such that sup t∈[0,T ] |b(X t )|≤C sup t∈[0,T ] |1 +X t |<∞ a.s. 3.4.3 Properties of integrated fractional Brownian motion and strong consistency Under the equivalent measure e P, ˙ X t becomes a fractional Brownian motion and X t becomes an integrated fractional Brownian motion. Its properties are useful in proofs, but are also interesting on its own. In the following set e B t := ˙ X t . Since X t can be written under e P as X t = Z t 0 e B s dt = Z t 0 (t−s)d e B s , 70 ithasaGaussiandistribution. ThemeaniszeroandthecovariancefunctionM(t,s) is given by M(t,s) := Z t 0 Z s 0 e E( e B u e B u 0)dudu 0 = Z t 0 Z s 0 R(u,u 0 )dudu 0 . If s =t, M(t,s) reduces to the simple expression M(t,t) = V H t 2H+2 2(H + 1) . It follows by self-invariance of fractional Brownian motion that for any c> 0 X ct = Z ct 0 e B r dr = Z t 0 c H e B cr c H cdr L =c 1+H Z t 0 e B r dr =c 1+H X t , therefore the self-invariance property of integrated fractional Brownian motion is {c 1+H X t ,t≥ 0} L ={X ct ,t≥ 0}. In the following we assume that b(x) =x, thenX t becomes the harmonic oscil- lator driven by fractional Brownian motion under P. To show strong consistency, by Theorem 3.2.11 it is enough to show that R T 0 Q 2 t (X)dt→∞ a.s. under the equivalent measure ˜ P. In that caseX t is an integrated fractional Brownian motion. 71 We only show strong consistency for the case H≤ 1/2, for H > 1/2 the proof is similar. For H≤ 1/2, we have Q t (X) =K −1 Z · 0 X r dr (t) =t H−1/2 I 1/2−H 0+ (t 1/2−H X t )(t) = t H−1/2 Γ(1/2−H) Z t 0 s 1/2−H X s (t−s) H+1/2 ds. By substitution it follows that R T 0 Q 2 t dt =T R 1 0 Q 2 Tt dt and Q Tt (X) = (Tt) H−1/2 Γ(1/2−H) Z Tt 0 s 1/2−H X s (Tt−s) H+1/2 ds = (Tt) H−1/2 Γ(1/2−H) Z t 0 (Ts) 1/2−H X Ts (Tt−Ts) H+1/2 Tds =T 3/2 t H−1/2 Γ(1/2−H) Z t 0 s 1/2−H X Ts (t−s) H+1/2 T 1+H ds. By self-similarity of X t , for every C > 0 we have e P Z T 0 Q 2 t (X)dt>C ! = e P   T 4 Z 1 0 " t H−1/2 Γ(1/2−H) Z t 0 s 1/2 X s (t−s) H+1/2 ds # 2 dt>C   > 1−ε for everyε∈ (0, 1), ifT is sufficiently large. Since R T 0 Q 2 t (X)dt is monotone increas- ing, it converges almost surely, and therefore R T 0 Q 2 t (X)dt→∞ as T→∞. 72 Chapter 4 Statistical inference of damped harmonic oscillator driven by Gaussian white noise 4.1 Introduction Let (Ω,F, (F t ),P)beastochasticbasissatisfyingusualassumptions. Inthischapter we consider the following equation ¨ X(t) +a ˙ X(t) +θX(t) = ˙ W (t), t≥ 0, X(0) = ˙ X(0) = 0, (4.1.1) with standard Brownian motion W (t) with respect to (F t ). Equation (4.1.1) is understood as system of stochastic integral equations X(t) = Z t 0 ˙ X(s)ds, ˙ X(t) =−a Z t 0 ˙ X(s)ds−θ Z t 0 X(s)ds +W (t), (4.1.2) 73 We call X(t) solving equation (4.1.1) damped oscillator; If a = 0 harmonic oscillator. Recently, there has been an increasing interest in statistical inference of os- cillators. Asymptotic properties for estimators of coefficients of multidimensional diffusions are studied in [6]. In the non-ergodic case, the asymptotic distribution and strong consistency of the estimators are discussed in [41]. Oscillator driven by Gaussian white noise have applications in physics, engineer- ing (where they are called random vibrations) and economics. According to [39,47] most vibratory environments of vehicles consist of damped harmonic oscillations driven by a random term, and the majority of signals measured in those environ- ments are Gaussian. Some examples are vehicle driving on a rough road, noise of the engines of an aircraft, aerodynamic turbulent flow around the wings of an aircraft and vibrations of internal components of guided missiles (see [39, p. 2] and references therein). There is a variety of different methods to analyze damped or harmonic oscillator driven by a random noise term. Two parameters most frequently used in practice are the root mean square of the oscillator to study the time domain and the power spectral density to study the frequency domain. A result about the first method is discussed in this chapter, more specifically we compute the Laplace transform of the density of root mean square and related quadratic functionals of the damped and harmonic oscillator. 74 For first order equations, the Laplace transform of the integrated square of the solution has been computed in [43]. [16] inverted the Laplace transform to obtain asymptotics of the density. Results on asymptotics of Laplace transforms are given in [70]. The novelty of this chapter is to compute the Laplace transform for the second order equation. An asymptotic inversion of the Laplace transform using complex integration techniques as in [70] however could not be accomplished so far. A com- mon way to obtain a relationship between the asymptotics of the Laplace transform and the distribution is to use Tauberian theorems. To our knowledge Tauberian the- orems however require a Laplace transform that is slowly varying at infinity, i.e. lim t→∞ L(at) L(t) = 1 for all a > 0 or at least of regular variation at infinity, i.e. there is p∈R so that lim t→∞ L(at) L(t) =a p for all a> 0. Neither of these is true here, since the expressions involve terms such as sinh or cosh which are not slowly or at least regular varying at infinity. Since Tauberian theorems can not be applied directly to the Laplace transform, we can instead look at log-asymptotics. It turns out that the logarithm of the Laplace transform is regular varying at infinity. This can be exploited by apply- ing an exponential Tauberian Theorem which we use to obtain asymptotics. A by-product of these asymptotics are the small ball probabilities of the quadratic functionals. A comprehensive overview of applications of small ball probabilities is [40]. 75 Inthethirdsection, weconsiderthemaximumlikelihoodestimatorofparameter θ of the harmonic oscillator. We study the convergence of moments of the estimator and find the asymptotics for the first and the second moment. The main tool in the proof here is the Laplace transform developed in section 1. The idea originates from [43, Chapter 17], who obtained these asymptotics in the first order case, i.e. an Ornstein-Uhlenbeck process. 4.2 Laplace transform of quadratic functionals In the first section the Laplace transform of the density of quadratic functionals of the damped/harmonic oscillator X t in (4.1.1) are studied. 4.2.1 Properties of the solution Lemma 4.2.1. The solution of (4.1.1) is given by X(t) = Z t 0 φ(t−s)dW (s), (4.2.1) where φ solves the second order differential equation ¨ φ +a ˙ φ +θφ = 0, φ(0) = 0, ˙ φ(0) = 1. (4.2.2) 76 Proof. Integration by parts yields X(t) = Z t 0 φ(t−s)dW (s) =φ(0)W (t)−φ(t)W (0)− Z t 0 W (s)dφ(t−s) = Z t 0 ˙ φ(t−s)W (s)ds. (4.2.3) So ˙ X(t) = Z t 0 ¨ φ(t−s)W (s)ds + ˙ φ(0)W (t)− ˙ φ(t)W (0) = Z t 0 −a ˙ φ(t−s)W (s)−θφ(t−s)W (s)ds +W (t) =−a Z t 0 ˙ X(s)ds−θ Z t 0 X(s)ds +W (t), where the last equality follows by Z t 0 X(s)ds = Z t 0 Z s 0 ˙ φ(s−r)W (r)drds = Z t 0 Z t r ˙ φ(s−r)W (r)dsdr = Z t 0 W (r) Z t−s 0 ˙ φ(u)dudr = Z t 0 φ(t−s)W (s)ds Z t 0 Y (s)ds =X(t) = Z t 0 ˙ φ(t−s)W (s)ds. 77 Lemma 4.2.2. We have the following relations Z t 0 X(s)d ˙ X(s) =X(t) ˙ X(t)− Z t 0 ˙ X 2 (s)ds, (4.2.4a) Z t 0 ˙ X(s)d ˙ X(s) = ˙ X 2 (t)−t 2 , (4.2.4b) Z t 0 X(s) ˙ X(s)ds = X 2 (t) 2 . (4.2.4c) Proof. Equality (4.2.4a) and (4.2.4c) follow after integration by parts, (4.2.4b) fol- lows by the Itô formula applied to ˙ X 2 (t). Lemma 4.2.3. (X(t), ˙ X(t)) T is a Gaussian vector with covariance matrix Cov(X(t),Y (t)) =      R t 0 φ 2 (s)ds R t 0 φ(s) ˙ φ(s)ds R t 0 φ(s) ˙ φ(s)ds R t 0 ˙ φ 2 (s)ds      . (4.2.5) Proof. Using integration by parts and the fact that φ is infinitely differentiable, X(t) = Z t 0 φ(t−s)dW (s) =φ(0)W (t)−φ(t)W (0) + Z t 0 ˙ φ(t−s)W (s)ds = Z t 0 ˙ φ(t−s)W (s)ds. (4.2.6) So ˙ X(t) = Z t 0 ¨ φ(t−s)W (s)ds + ˙ φ(0)W (t)− ˙ φ(t)W (0) = Z t 0 ˙ φ(t−s)dW (s). 78 Now for every t = (t 1 ,t 2 ) T ∈R 2 t T (X(t), ˙ X(t)) =t 1 X(t) +t 2 ˙ X(t) = Z t 0 t 1 φ(t−s) +t 2 ˙ φ(t−s) dW (s) is normally distributed, so (X(t), ˙ X(t)) T is jointly normally distributed. Itô isom- etry implies (4.2.5). Lemma 4.2.4. The solution X(t) of (4.1.1) has autocovariance function Cov(X(t)X(s)) = Z min(s,t) 0 φ(t−r)φ(s−r)dr. or equivalently for an h≥ 0 Cov(X(t +h)X(t)) = Z t 0 φ(t +h−r)φ(t−r)dr = Z t 0 φ(h +r)φ(r)dr. (4.2.7) Proof. Using Itô isometry, the covariance function can be computed as Cov(X(t),X(s)) =E Z t 0 φ(t−r)dW (r) Z s 0 φ(s−r)dW (r) = Z min(s,t) 0 φ(t−r)φ(s−r)dr. Remark 4.2.5. As can be seen in (4.2.7) the autocovariance depends on t, so the process (X(t), ˙ X(t)) T is nonstationary. It seems that in (4.2.7) for large values oft 79 (ifφ is sufficiently decreasing) the autocovariance is almost constant . This happens in the case when damping is present, i.e. a> 0, θ6= 0. Lemma 4.2.6. Assumea> 0 andθ6= 0, then the variance of the Gaussian process (X(t), ˙ X(t)) T is asymptotically given by      1 2aθ 0 0 1 2a      as t→∞. Proof. Assume that a> 0, 4θ−a 2 > 0. Then the solution of (4.2.2) is given by φ(t) =e −at/2 sin(t q θ−a 2 /4) q θ−a 2 /4 . The variance of X(t) can be computed by (4.2.5) Var(X(t)) = Z t 0 φ 2 (s)ds = 1 2aθ − e −at a √ 4θ−a 2 sin t √ 4θ−a 2 +a 2 − cos t √ 4θ−a 2 + 4θ 2aθ (4θ−a 2 ) = 1 2aθ −e −at    2a 4θ−a 2 + sin t √ 4θ−a 2 2θ √ 4θ−a 2 − a cos t √ 4θ−a 2 2θ (4θ−a 2 )    So for a large enough t, the variance is approximately constant equal to 1 2aθ . The covariance of X(t) and ˙ X(t) can be computed by (4.2.5) Cov(X(t), ˙ X(t)) = Z t 0 φ(s) ˙ φ(s)ds = φ 2 (t) 2 =e −at sin 2 (t q θ−a 2 /4) 2θ−a 2 /2 . So for t large enough, the covariance of X(t) and ˙ X(t) is close to 0. 80 Finally the variance of ˙ X(t) can be computed by (4.2.5) Var( ˙ X(t)) = Z t 0 ˙ φ 2 (s)ds =− e −at a √ 4θ−a 2 sin t √ 4θ−a 2 +a 2 cos t √ 4θ−a 2 − 4θ 2 (a 3 − 4aθ) − 4θ−a 2 2a (a 2 − 4θ) = 1 2a − e −at a √ 4θ−a 2 sin t √ 4θ−a 2 +a 2 cos t √ 4θ−a 2 − 4θ 2 (a 3 − 4aθ) = 1 2a −e −at    2aθ 4θ−a 2 + sin t √ 4θ−a 2 2 √ 4θ−a 2 − a cos t √ 4θ−a 2 2 (4θ−a 2 )    So for a large enought, the variance of ˙ X(t) is approximately constant equal to 1 2a . Note that in the case of no damping (a = 0) the variance becomes by L’Hôpital rule lim a→0 Var(X(t)) = 2t θ − sin 2t √ θ 2θ 3/2 . Moreover lim a→0 Var( ˙ X(t)) = t 2 + sin 2t √ θ 4 √ θ lim a→0 Cov(X(t), ˙ X(t)) = sin 2 (t √ θ) 2θ So the process does not converge to a stationary process as t increases. 81 Thirdly assume thata> 0,θ6= 0 anda 2 − 4θ> 0. Then the solution of (4.2.2) is given by φ(t) =e −at/2 sinh(t q a 2 /4−θ) q a 2 /4−θ . The variance of X(t) can be computed by (4.2.5) Var(X(t)) = Z t 0 φ 2 (s)ds = 1 2aθ + e −at −a √ a 2 − 4θ sinh t √ a 2 − 4θ +a 2 − cosh t √ a 2 − 4θ + 4θ 2aθ (a 2 − 4θ) = 1 2aθ −e −at    2a 4θ−a 2 + sinh t √ a 2 − 4θ 2θ √ a 2 − 4θ − a cosh t √ a 2 − 4θ 2θ (a 2 − 4θ)    So for a large enough t, the variance is approximately constant equal to 1 2aθ . The covariance of X(t) and ˙ X(t) can be computed by (4.2.5) Cov(X(t), ˙ X(t)) = Z t 0 φ(s) ˙ φ(s)ds = φ 2 (t) 2 =e −at sinh 2 (t √ a 2 − 4θ) a 2 /2− 2θ . So for t large enough, the covariance of X(t) and ˙ X(t) is close to 0. Finally the variance of ˙ X(t) can be computed by (4.2.5) Var( ˙ X(t)) = Z t 0 ˙ φ 2 (s)ds = 1 2a + 1 2 e −at    4θ a 3 − 4aθ + sinh t √ a 2 − 4θ √ a 2 − 4θ − a cosh t √ a 2 − 4θ a 2 − 4θ    82 So for a large enought, the variance of ˙ X(t) is approximately constant equal to 1 2a . Furthermore in the case a 2 /4−θ = 0, by letting θ→ a 2 /4 (or computing the solution φ(t) =e −at t): lim θ→a 2 /4 Var(X(t)) = 2 a 3 + e −at (−a 2 t 2 − 2at− 2) a 3 = 1 2aθ −e −at ( a 2 t 2 + 2at + 2 a 3 ) lim θ→a 2 /4 Var( ˙ X(t)) = 1 2a + e −at (−a 2 t 2 + 2at− 2) 4a = 1 2a −e −at ( a 2 t 2 − 2at + 2 4a ) lim θ→a 2 /4 Cov(X(t), ˙ X(t)) = 1 2 t 2 e −at Summingup, thesolutionisaGaussianprocessandifa> 0,θ6= 0itscovariance is given asymptotically by      1 2aθ 0 0 1 2a      for large t. 4.2.2 Laplace transform In the following we compute the Laplace transform Ψ t (a,θ;x,y,z) =E exp       − Z t 0      X(s) ˙ X(s)      T     y x x z           X(s) ˙ X(s)      ds       (4.2.8) 83 where the matrix      y x x z      ∈R 2×2 and X(t) is the solution of (4.1.1). The (only) known results correspond to (a) Brownian motion: a = θ = 0, y = x = 0; (b) Ornstein-Uhlenbeck process: θ = 0, y = x = 0; (c) Integrated Brownian motion: a =θ = 0, z =x = 0. The proof is a suitable modification of the argument from [43, proof of Lemma 17.3]. Theorem 4.2.7. The Laplace transform (4.2.8) is given by Ψ t (a,θ;x,y,z) =e −(b−a)t/2 [det(I +R(t)J)] −1/2 (4.2.9) where b = q a 2 + 2ρ + 2z, ρ =λ−θ, λ = q θ 2 + 2y, (4.2.10) I =      1 0 0 1      , J =      2x +aθ−bλ −ρ ρ a−b.      , (4.2.11) and R(t) is the covariance matrix of the Gaussian vector (X ∗ (t), ˙ X ∗ (t)) T solving dX ∗ (t) = ˙ X ∗ (t)dt, d ˙ X ∗ (t) =−b ˙ X ∗ (t)−λX ∗ (t)dt +dW (t), X ∗ (0) = ˙ X ∗ (0) = 0. (4.2.12) 84 Proof. Let X ∗ be the solution of ¨ X ∗ +b ˙ X ∗ +λX ∗ = ˙ W (s), X ∗ (0) = ˙ X ∗ (0) = 0, b,λ∈R. The goal is to eliminate the expressionsy R t 0 X 2 (s)ds andz R t 0 ˙ X 2 (s)ds using Lemma 4.2.2 and a suitable change of measure using the process X ∗ with suitableb andλ. By [44, Theorem 7.19], the measureP generated by the process ˙ X in the space of continuous functions on (0,t) is equivalent to the measure P ∗ generated by the process ˙ X ∗ and dP dP ∗ ( ˙ X ∗ ) = exp{− Z t 0 [(θX ∗ (s) +a ˙ X ∗ (s))− (λX ∗ (s) +b ˙ X ∗ (s))]d ˙ X ∗ (s) − 1 2 Z t 0 [(θX ∗ +a ˙ X ∗ (s)) 2 − (λX ∗ (s) +b ˙ X ∗ (s)) 2 ]ds} = exp{(λ−θ) Z t 0 X ∗ (s)d ˙ X ∗ (s) + (b−a) Z t 0 ˙ X ∗ (s)d ˙ X ∗ (s) + λ 2 −θ 2 2 Z t 0 X 2 ∗ (s)ds + (bλ−aθ) Z t 0 X ∗ (s) ˙ X ∗ (s)ds + b 2 −a 2 2 Z t 0 ˙ X 2 ∗ (s)ds} By Lemma 4.2.2, dP dP ∗ ( ˙ X ∗ ) = exp   (λ−θ) X ∗ (t) ˙ X ∗ (t)− Z t 0 ˙ X 2 ∗ (s)ds + b−a 2 ˙ X 2 ∗ (t)−t + λ 2 −θ 2 2 Z t 0 X 2 ∗ (s)ds + bλ−aθ 2 X 2 ∗ (t) + b 2 −a 2 2 Z t 0 ˙ X 2 ∗ (s)ds   . (4.2.13) 85 Note that X(t) = R t 0 ˙ X(s)ds, and thus the Laplace transform Ψ t is a functional of ˙ X(t). Then Ψ t (a,θ;x,y,z) =E exp −y Z t 0 X(s) 2 ds−z Z t 0 ˙ X 2 (s)ds− 2x Z t 0 X(s) ˙ X(s)ds =E exp −y Z t 0 X ∗ (s) 2 ds−z Z t 0 ˙ X 2 ∗ (s)ds− 2x Z t 0 X ∗ (s) ˙ X ∗ (s)ds dP dP ∗ ( ˙ X ∗ ) ! Using (4.2.13) and choosing b,λ according to (4.2.10), I find: Ψ t (a,θ;x,y,z) =e −(b−a)t/2 E exp   (λ−θ)X ∗ (t) ˙ X ∗ (t) + b−a 2 ˙ X 2 ∗ (t) + bλ−aθ 2 −x ! X 2 ∗ (t)   . (4.2.14) Now recall that (X ∗ (t), ˙ X ∗ (t)) T is Gaussian vector satisfying (4.2.12). Denote by R(t) the covariance matrix of (X ∗ (t), ˙ X ∗ (t)) T . 86 Then E exp   (λ−θ)X ∗ (t) ˙ X ∗ (t) + (b−a) 2 ˙ X 2 ∗ (t) + (bλ−aθ− 2x) 2 X 2 ∗ (t)   = Z ∞ −∞ Z ∞ −∞ exp   ρuv + (b−a)v 2 2 + (bλ−aθ− 2x)u 2 2   × 1 (2π)(detR(t)) 1/2 exp   − 1 2 (u,v)R(t) −1 (u,v) T   dudv = Z ∞ −∞ Z ∞ −∞ exp − 1 2 (u,v)(R(t) −1 +J)(u,v) T (2π)(detR(t)) 1/2 dudv = (det(R(t) −1 +J) −1 ) 1/2 (detR(t)) 1/2 = (det(I +R(t)J)) −1/2 . Let us now look closer at the determinant det(I +R(t)J). Lemma 4.2.8. lim t→∞ det(I +R(t)J) = 1 4 + a 4 r a 2 + 2z + 2 −θ + √ 2y +θ 2 + 2ax +a 2 θ− −θ + √ 2y +θ 2 2 + 2x +aθ 4 √ 2y +θ 2 a 2 + 2z + 2 −θ + √ 2y +θ 2 . (4.2.15) Proof. Using Lemma 4.2.6 the covariance matrix will be approximately constant for large t, since b> 0 and λ6= 0 R(t)∼      1/(2bλ) 0 0 1/(2b)      . (4.2.16) 87 plugging in the values leads to above expression. lim t→∞ det(I +R(t)J) = a 2 θ +abθ +abλ + 2ax +b 2 λ + 2bx−ρ 2 4b 2 λ (4.2.17) = 1 4 + a 4b + ax 2b 2 λ + x 2bλ + a 2 θ 4b 2 λ + aθ 4bλ − ρ 2 4b 2 λ (4.2.18) = 1 4 + a 4 r a 2 + 2z + 2 −θ + √ 2y +θ 2 (4.2.19) + 2ax +a 2 θ− −θ + √ 2y +θ 2 2 + 2x +aθ 4 √ 2y +θ 2 a 2 + 2z + 2 −θ + √ 2y +θ 2 . (4.2.20) Finally we compute some explicit cases choosing specific a,θ,x,y,z. To simpli- fy notation we drop the variables in expression Ψ t (a,θ;x,y,z) whenever one of a,θ,x,y,z is zero. Brownian motion Ψ t (z) = 1 q cosh(t √ 2z) . Ornstein Uhlenbeck process Ψ t (a;z) = 1 s e −at asinh ( t √ a 2 +2z ) √ a 2 +2z +cosh(t √ a 2 +2z) =e at/2 √ 2 4 √ a 2 +2z q ( √ a 2 +2z+a) exp(t √ a 2 +2z)+( √ a 2 +2z−a) exp(−t √ a 2 +2z) Integrated Brownian motion Ψ t (y) = 2 q cos(2 3/4 t 4 √ y)+cosh(2 3/4 t 4 √ y)+2 Harmonic oscillator Ψ t (θ;y) = 2 4 √ θ 2 +2y r ( √ θ 2 +2y−θ) cos( √ 2t q √ θ 2 +2y+θ)+( √ θ 2 +2y+θ) cosh( √ 2t q √ θ 2 +2y−θ)+2 √ θ 2 +2y 88 Joint integrated Brownian motion and Brownian motion Ψ t (y,z) =                                                      2 3/4 √ 2y−z 2 q (− √ yz+ √ 2y− √ 2z 2 ) cos(t √ 2 √ 2 √ y−2z)+( √ yz+ √ 2y− √ 2z 2 ) cosh(t √ 2 √ 2 √ y+2z)+2 √ 2y , b 2 − 4λ = 2z− 2 √ 2y< 0 2 3/4 √ 2y−z 2 q (− √ yz+ √ 2y− √ 2z 2 ) cosh(t √ 2z−2 √ 2 √ y)+( √ yz+ √ 2y− √ 2z 2 ) cosh(t √ 2 √ 2 √ y+2z)+2 √ 2y , b 2 − 4λ = 2z− 2 √ 2y> 0 2 q √ 2t 2 √ y+ 3 2 cosh(2 4 √ 2t 4 √ y)+ 5 2 = 2 q t 2 z+ 3 2 cosh(2t √ z)+ 5 2 ,b 2 − 4λ = 2z− 2 √ 2y = 0. Next we compute asymptotic behavior of the Laplace transform Ψ t (a,θ;y,z) =E exp −y Z t 0 X 2 (s)ds−z Z t 0 ˙ X 2 (s)ds (4.2.21) =e −(b−a)t/2 [det(I +R(t)J)] −1/2 , y,z > 0. (4.2.22) Four cases (and more) can be considered: Letting y→∞ while z > 0 is fixed, z→∞ while y> 0 is fixed, y =z→∞ and √ 2y =z→∞. Note that e −(b−a)t/2 =e − 1 2 t( q a 2 +2( √ 2y+θ 2 −θ)+2z−a) ∼                                      e −t( 4 √ y/2−a/2) as y→∞, z fixed e −t( √ z/2−a/2) as z→∞, y fixed e −t( √ z/2−a/2+1/2) as y =z→∞ e −t( √ z−a/2) as √ 2y =z→∞. (4.2.23) 89 Similar as in Lemma 4.2.8 one can show that the asymptotics of the determinant are of maximal polynomial growth. Therefore the log-asymptotics of the determinant are given by log Ψ t (a,θ;y,z)∼                                      −t 4 q y/2 as y→∞, z fixed −t q z/2 as z→∞, y fixed −t q z/2 as y =z→∞ −t √ z as √ 2y =z→∞. (4.2.24) Remark4.2.9. ForcomparisonOrnstein-Uhlenbeckprocesshasasymptotics−t q z/2. 4.2.3 Application to small deviations Let V be a positive random variable on (Ω,F,P). The small deviation studies the behaviour of: logP(V≤ε) (4.2.25) as ε→∞, see [40]. To analyze the small deviation (4.2.25) we can make use of its following relation to the Laplace transform [40, Theorem 3.5]: 90 Theorem 4.2.10. For α> 0 and β∈R logP(V≤ε)∼−C V ε −α | logε| β ε→ 0 + (4.2.26) if and only if logE exp (−yV )∼−(1 +α)α −α/(1+α) C 1/(1+α) V y α/(1+α) (logy) β/(1+α) y→∞. (4.2.27) Theorem 4.2.10 is also called exponential Tauberian theorem. Example 4.2.11. The small deviations of X defined in (4.1.1) in L 2 ([0,t])-norm (denoted byk·k) have asymptotics logP(kXk 2 ≤ε)∼− 3 8 t 4/3 ε −1/3 as ε→ 0 + . (4.2.28) Proof. Define α = 1/3 and let β = 0 then as y→∞ logE exp (−ykXk 2 ) = log Ψ t (a,θ; 0,y, 0)∼ 1 2 log 8 +at/2− (y/2) 1/4 t∼−(y/2) 1/4 t (4.2.29) ! =−(4/3)3 1/4 C 3/4 y 1/4 (4.2.30) So C 3/4 = ( 1 2 ) 1 4 t 3 4 ( 1 3 ) 1 4 or: C = t 4 3 ( 1 6 ) 1 3 ( 3 4 ) 4 3 = 3 8 t 4 3 . Hence by Theorem (4.2.10) the result follows. 91 Example 4.2.12. Compare this to the L 2 ([0,t])-norm of an Ornstein Uhlenbeck process ˙ X(t) (θ = 0), where with α = 1, β = 0: logE exp (−zk ˙ Xk 2 ) = Ψ t (a, 0; 0, 0,z)∼ 1 2 log 2 +at/2− (z/2) 1/2 t (4.2.31) ∼−(z/2) 1/2 t ! =−2C 1/2 z 1/2 (4.2.32) and so by Theorem (4.2.10) (C = 1 8 t 2 ) logP(k ˙ Xk 2 ≤ε)∼− 1 8 t 2 ε −1 as ε→ 0 + . (4.2.33) Example 4.2.13. Settingy =z one can compute the asymptotics of the small devi- ation of R t 0 X 2 (s)ds+ R t 0 ˙ X 2 (s)ds. The asymptotics of the exponential Laplace trans- form are logE exp −z kXk 2 +k ˙ Xk 2 ∼−(z/2) 1/2 t , hence the Sobolev norm has the same asymptotics ask ˙ Xk 2 : logP(kXk 2 +k ˙ Xk 2 ≤ε)∼ logP(k ˙ Xk 2 ≤ε)∼− 1 8 t 2 ε −1 as ε→ 0 + . (4.2.34) Example 4.2.14. Setting y = z 2 /2 the asymptotics of the exponential Laplace transform are logE exp −z 2 kXk 2 /2−zk ˙ Xk 2 ∼−z 1/2 t as z→∞, so logP(kXk 2 /2 +k ˙ Xk 2 ≤ε)∼− 1 4 t 2 ε −1 as ε→ 0 + . 92 4.3 Parameter estimation of harmonic oscillator driven by Gaussian white noise Let θ > 0 and define X T = R T 0 φ(t−s)dW s with φ(t) = sin(t √ θ) √ θ . Then X t is the solution to ¨ X t +θX t = ˙ W t , X 0 = ˙ X 0 = 0. In the following we analyze the behavior of a maximum likelihood estimator obtained in [41]. It is given by b θ T =              − R T 0 Xtd ˙ Xt R T 0 X 2 t dt R T 0 X t d ˙ X t ≤ 0, 0 R T 0 X t d ˙ X t > 0. Moreover the authors show in [41, Thm 2.1] that T ( b θ T −θ) L → 2 √ θ R 1 0 W 2 1 (t)dW 2 (t)− R 1 0 W 2 2 (t)dW 1 (t) R 1 0 W 2 1 (t)dt + R 1 0 W 2 2 (t)dt . 4.3.1 Convergence of moments Theorem 4.3.1. All moments of T ( b θ T −θ) converge, that means if Υ denotes the limiting distribution of T ( b θ T −θ), then E T ( b θ T −θ) p →E (Υ) p for any p≥ 1. 93 In the following define f(T ) = O(g(T )) as T →∞ if and only if there exists constant M > 0 and T 0 > 0 such that |f(T )|≤M|g(T )| for all T >T 0 . Lemma 4.3.2. Let X t be defined as above. Then for any q≥ 1 E    1 R T 0 X 2 t dt q/2    =O 1 T q as T→∞. Theorem 4.3.3. If M t is a continuous L 2 -martingale and 1≤ p < q, then there exists a constant C(p,q) such that M t hMi t p ≤C(p,q) 1 q hMi t q . HerekXk p = (E(X p )) 1/p . Proof. This is a consequence of [?, Thm 12.1]. A standard result says, that a family of random variables{Y t } is uniformly integrable if and only if there exists a nonnegative increasing convex function G(y) such that lim y→∞ G(y) y =∞ and sup t E(G(|Y t |))<∞. We conclude the following: 94 Lemma 4.3.4. If{X t } converges to X in distribution and sup t E(|X t | k+δ ) <∞ for some δ> 0, then E(X r t )→E(X) for all 1≤r≤k. Proof of Theorem 4.3.1. By Lemma 4.3.4 it is enough to show that sup T E(|T ( b θ T −θ)| p )<∞ for any p≥ 1. Letp≥ 1 arbitrary and letq>p. By Theorem 4.3.3 applied to b θ T −θ =− R T 0 XtdWt R T 0 X 2 t dt there exists a constant C(p,q) such that E(|T ( b θ T −θ)| p ) 1/p ≤C(p,q)T E 1 ( R T 0 X 2 s dt) q/2 !! 1/q . Finally by Lemma 4.3.2 we have T E 1 ( R T 0 X 2 s dt) q/2 !! 1/q =O(1) as T→∞. Proof of Lemma 4.3.2. Define Ψ t (θ;z) = E(−z R t 0 X 2 s ds). We have by Theorem 4.2.7 Ψ t (θ;z) = 2 4 √ θ 2 +2z q ( √ θ 2 +2z−θ) cos( √ 2t √ √ θ 2 +2z+θ)+( √ θ 2 +2z+θ) cosh( √ 2t √ √ θ 2 +2z−θ)+2 √ θ 2 +2z . 95 We can rewrite Ψ t (θ;z) as Ψ t (θ;z) = exp ( − t √ 2 q √ θ 2 + 2z−θ ) g t (z), (4.3.1) with g t (z) chosen appropriately. Here g t (z) also depends on θ, but we consider it as a fixed variable. We can show that for t and z≥ 0, g t (z)≤ 2 √ 2 =:C 1 . To show this, let g t (z) = 2{h(t,z)} −1/2 with h(t,z) = exp − √ 2t q √ θ 2 + 2z−θ " 1− θ √ θ 2 + 2z ! cos( √ 2t q √ θ 2 + 2z +θ) + 1 + θ √ θ 2 + 2z ! cosh( √ 2t q √ θ 2 + 2z−θ) + 2 # = 1 2 1 + θ √ θ 2 + 2z ! + exp − √ 2t q √ θ 2 + 2z−θ · " 1− θ √ θ 2 + 2z ! cos( √ 2t q √ θ 2 + 2z +θ) + 1 2 exp − √ 2t q √ θ 2 + 2z−θ + 2 # (∗) ≥ 1 2 1 + θ √ θ 2 + 2z ! ≥ 1 2 . To show inequality (∗), it is enough to see that the term in brackets is positive, since 1 2 exp − √ 2t q √ θ 2 + 2z−θ + 2> 1− θ √ θ 2 + 2z ∈ (0, 1). 96 We can conclude g t (z)≤ 2 √ 2. Now E 1 ( R t 0 X 2 s ds) q/2 ! = 1 Γ(q/2) Z ∞ 0 z q/2−1 Ψ t (θ;z)dz = 1 Γ(q/2) Z ∞ 0 z q/2−1 exp − √ 2t q √ θ 2 + 2z−θ g(t,z)dz ≤ C 1 Γ(q/2) Z ∞ 0 z q/2−1 exp − √ 2t q √ θ 2 + 2z−θ dz. Substitute u = q √ θ 2 + 2z−θ, then z = (u 2 +θ) 2 −θ 2 /2 = (u 4 + 2u 2 θ)/2 =u 2 (u 2 + 2θ)/2 and dz = 2u (u 2 +θ)du. Using the estimate 2 −q/2+2 u 4 + 2u 2 θ q/2−1 u u 2 +θ = 2 −q/2+2 u q−1 u 2 + 2θ q/2 ≤C 2 (q) u q−1 +u 2q−1 , with some constant C 2 (q)> 0, we get E 1 ( R t 0 X 2 s ds) q/2 ! ≤ C 1 C 2 (q) Γ(q/2) Z ∞ 0 u q−1 +u 2q−1 exp n − √ 2tu o du. 97 Substituting z = √ 2tu yields E 1 ( R t 0 X 2 s ds) q/2 ! ≤ C 1 C 2 (q) Γ(q/2) ( √ 2t) −q Z ∞ 0 z q−1 exp{−z}dz + ( √ 2t) −2q Z ∞ 0 z 2q−1 exp{−z}dz = C 1 C 2 (q) Γ(q/2) ( √ 2t) −q Γ(q) + ( √ 2t) −2q Γ(2q) =O(t −q ) 4.3.2 Asymptotics of the first and second moment Denote by f(t) ∼ g(t) as t → ∞ if and only if lim t→∞ f(t) g(t) = 1. Denote by f(t) =o(g(t)) as t→∞, if and only if lim t→∞ f(t) g(t) = 0. Theorem 4.3.5. As T→∞ E b θ T −θ ∼− 3π 3 √ 2 1 T 6 , (4.3.2) E b θ T −θ 2 ∼ 8Cθ 1 T 2 , (4.3.3) where C≈ 0.9159 is the Catalan constant. Proof. From [43, Thm 17.2] we can deduce that −E b θ t −θ = Z ∞ 0 ∂ ∂θ Ψ t (θ;y)dy. 98 Define d ± = √ 2t q √ θ 2 + 2y±θ, then ∂ ∂θ Ψ t (θ;y) = t r y(θ 2 +2y) √ θ 2 +2y−θ sin(d + )+2y cos(d + )+t r y(θ 2 +2y) θ+ √ θ 2 +2y sinh(d − )−2y cosh(d − ) (θ 2 +2y) 3/4 √ θ 2 +2y−θ cos(d + )+ θ+ √ θ 2 +2y cosh(d − )+2 √ θ 2 +2y 3/2 . Substitutingu =t q √ θ 2 + 2y−θ yieldsy = u 2 t 2 θ + u 2 2t 2 anddy = 2u 3 t 4 + 2θu t 2 du. Under the integral sign we get the function ∂ ∂θ Ψ t θ; u 2 t 2 θ + u 2 2t 2 !! × 2u 3 t 4 + 2θu t 2 ! = u 2 (t 2 +u 2 ) 2u 3 t 4 + 2θu t 2 √ 2 sinh √ 2u (2θ 2 t 4 + 3θt 2 u 2 +u 4 ) t 4 (θt 2 +u 2 ) 2 (θt 2 +u 2 ) + cosh √ 2u (2θt 2 +u 2 ) 3/2 +o 1 t 6 This function, multiplied by t 6 , converges to u 3 sinh √ 2u √ 2 cosh 2 u √ 2 3/2 . An application of dominated convergence theorem gives lim t→∞ t 6 Z ∞ 0 ∂ ∂θ Ψ t θ; u 2 t 2 θ + u 2 2t 2 !! × 2u 3 t 4 + 2θu t 2 ! du = Z ∞ 0 u 3 sinh √ 2u √ 2 cosh 2 u √ 2 3/2 du = 3π 3 √ 2 . Furthermore from [43, Thm 17.2] we can deduce that E b θ t −θ 2 = Z ∞ 0 Ψ t (θ;y)dy + Z ∞ 0 y ∂ 2 ∂θ 2 Ψ t (θ;y)dy. 99 First we analyze R ∞ 0 Ψ t (θ;y)dy. Substituting u =t q √ θ 2 + 2y−θ yields the terms y = u 2 t 2 θ + u 2 2t 2 and dy = 2u 3 t 4 + 2θu t 2 du. Therefore we get under the integral sign Ψ t θ; u 2 t 2 θ + u 2 2t 2 !! × 2u 3 t 4 + 2θu t 2 ! = 4u (θt 2 +u 2 ) 3/2 t 4 s 2 (θt 2 +u 2 ) +u 2 cos t q 4θ + 2u 2 t 2 + cosh √ 2u (2θt 2 +u 2 ) . Observe that this function multiplied by t 2 converges to 4uθ r 2 + cosh √ 2u 2 . An application of dominated convergence theorem gives lim t→∞ t 2 Z ∞ 0 Ψ t θ; u 2 t 2 θ + u 2 2t 2 !! × 2u 3 t 4 + 2θu t 2 ! du = Z ∞ 0 4uθ r 2 + cosh √ 2u 2 du = 8Cθ, where C≈ 0.9159 is the Catalan constant. Next we analyze R ∞ 0 y ∂ 2 ∂θ 2 Ψ t (θ;y)dy. The calculations for the asymptotics are almost identical to the previous integrals. It turns out that R ∞ 0 y ∂ 2 ∂θ 2 Ψ t (θ;y)dy = o 1 t 3 . Combining the asymptotics leads to the result. 100 Chapter 5 Hypothesis testing in a fractional Ornstein-Uhlenbeck model Consider an Ornstein Uhlenbeck process driven by fractional Brownian motion. It is an interesting problem to find criterias wether the process is stable or has a unit root, given a finite sample of observations. Recently various asymptotic dis- tributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein’s bound that these asymptotic distributions are inadequate approximations of the finite sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power. 101 5.1 Introduction Stability properties of the ordinary differential equationx 0 (t) =θx(t) depend on the sign of the parameter θ: the equation is asymptotically stable if θ < 0, neutrally stable if θ = 0, and instable if θ > 0. These stability results carry over to the stochastic process X(t) =X(0) +θ Z t 0 X(s)ds +Z(t) (5.1.1) drivenbynoiseZ =Z(t). Whenthevalueofθ isnotknownandatrajectoryofX = X(t) is observed over a finite time intervalt∈ [0,T ], a natural problem is to develop the zero-root test, that is, a statistical procedure for testing the hypothesis θ = 0 versus one of the possible alternatives θ6= 0, θ > 0, or θ < 0. While the classical solution to this problem is well-known (to use the maximum likelihood estimator (MLE)oftheparameterθ astheteststatistic), furtheranalysisisnecessarybecause the exact distribution of the MLE is usually too complicated to allow an explicit computation of either the critical region or the power of the test. More specifically, an approximate distribution of the MLE must be introduced an investigated, both in the finite sample asymptotic and in the limitT→∞. There are other potential complications, such as when MLE is not available (for example, if Z is a stable Levy process, see [28]) or when the MLE is difficult to implement numerically (for example, if Z is fractional Brownian motion, see [13]). 102 Theobjectiveofthisworkisanalysisandimplementationofthezeroroottestfor (5.1.1) whenZ =W (H) , the fractional Brownian motion with the Hurst parameter H, and 1/2≤ H < 1. When 0 < H < 1/2, the integral transformation of C. Jost [32, Cor. 5.2] reduces the corresponding model back to H≥ 1/2 (see [13]). Recall that the fractional Brownian motion W (H) =W (H) (t), t≥ 0, is a Gaus- sian process with W (H) (0) = 0, mean zero, and covariance E W (H) (t)W (H) (s) = 1 2 |s| 2H +|t| 2H −|t−s| 2H :=R H (t,s). Direct computations show that, for every continuous process Z, equation (5.1.1) has a closed-form solution that does not involve stochastic integration X(t) =X(0)e θt +Z(t)−Z(0) +θ Z t 0 e θ(t−s) Z(s)ds. (5.1.2) WhenZ =W (H) , letX (H) denote the corresponding fractional Ornstein-Uhlenbeck process: X (H) (t) =X (H) (0)e θt +W (H) (t) +θ Z t 0 e θ(t−s) W (H) (s)ds, (5.1.3) 103 and let U (H) θ =U (H) θ (t) (now with explicit dependence on the parameter θ) denote the particular case of (5.1.3) with zero initial condition: U (H) θ (t) =W (H) (t) +θ Z t 0 e θ(t−s) W (H) (s)ds. (5.1.4) Define random variables e θ T,H = (X (H) (T )) 2 − (X (H) (0)) 2 2 R T 0 (X (H) (t)) 2 dt − 1 HΓ(2H)T Z T 0 (X (H) (t)) 2 dt ! − 1 2H . (5.1.5) and Ψ H,c = (U (H) c (1)) 2 2 R 1 0 (U (H) c (t)) 2 dt − 1 HΓ(2H) Z 1 0 (U (H) c (t)) 2 dt ! − 1 2H , (5.1.6) where c∈ R. To motivate the main result of this paper, note that if H = 1/2, then e θ T,1/2 is the maximum likelihood estimator of θ based on the observations (X (1/2) (t), 0≤t≤T ) (see [43, Section 17.3]): e θ T,1/2 = (X (1/2) (T )) 2 − (X (1/2) (0)) 2 −T 2 R T 0 (X (1/2) (t)) 2 dt = R T 0 X (1/2) (t)dX (1/2) (t) R T 0 (X (1/2) (t)) 2 dt . (5.1.7) 104 While the exact distribution of e θ T,1/2 is not known, the following asymptotic rela- tions hold as T→∞: q |θ|T ( e θ T,1/2 −θ) L −→N 0, 2θ 2 , θ< 0; (5.1.8) T e θ T,1/2 L −→ Ψ 1/2,0 , θ = 0; (5.1.9) e θT ( e θ T,1/2 −θ) L −→ 2θC(1), θ> 0, X (1/2) (0) = 0; (5.1.10) whereN is the normal distribution, Ψ 1/2,0 is from (5.1.6) with H = 1/2 and b = 0,C(1) is the standard Cauchy distribution with probability density function 1/(π(1 +x 2 )), x∈R. While (5.1.9) suggests that the distribution Ψ 1/2,0 can be used to construct an asymptotic zero-root test, it turns out that neither (5.1.8) nor (5.1.10) is a good choice for analyzing the power of the resulting test for small values of the product θT. There are two reasons: (a) both (5.1.8) and (5.1.10) suggest that the prod- uct θT should be sufficiently large for the corresponding approximation to work; (b) it follows from (5.1.8)–(5.1.10) that the limit distribution of the appropriately normalized residual e θ T,1/2 −θ has a discontinuity near θ = 0, while, by (5.1.5), the distribution of e θ T,1/2 −θ is a continuous function ofθ for each fixedT > 0. Further discussion of the finite sample statistical inference is in Sections 5.2 and 5.3. In particular, Table 5.1 and Figure 5.1 in Section 5.3 provide some numerical results when θ< 0. 105 It is therefore natural to derive a different family of asymptotic distributions, one that depends continuously on the parameter θ near θ = 0. To this end, let θ =θ(T ) depend on the observation time T and lim T→∞ θ(T ) = 0. Then, for each fixedT, equality (5.1.3) still defines the fractional Ornstein-Uhlenbeck process, but the asymptotic behavior of the estimator (5.1.7) changes. The following is the main result of the paper. Theorem 5.1.1. Assume that 1/2 ≤ H < 1 and let θ = θ(T ) be a family of parameters such that lim T→∞ Tθ(T ) = c for some c ∈ R. Then, as T → ∞, e θ T,H → 0 with probability one and T e θ T,H L −→ Ψ H,c . (5.1.11) An alternative least-squares type estimator has been considered in [50] and is given by b θ T,H = R T 0 X (H) (t)dX (H) (t) R T 0 (X (H) (t)) 2 dt = R T 0 X (H) (t)dW (H) (t) R T 0 (X (H) (t)) 2 dt +θ. (5.1.12) The stochastic integral in (5.1.12) is understood as divergence integral forH > 1/2 (see Section 5.3) and as Itô-integral for H = 1/2. A serious drawback of this estimator is that, unless H = 1/2, there is no computable representation of b θ T,H given the observations (X (H) (t), 0≤ t≤ T ) because there is no known way to computethedivergenceintegral. IfH = 1/2, then b θ T,1/2 = e θ T,1/2 andiscomputable. 106 Nonetheless, there is an analogue of Theorem 5.1.1 for all H≥ 1/2. To state the result, define the random variable Π H,c = ((U (H) (1)) 2 − 1 F 1 (2H− 1, 2H + 1,c) 2 R T 0 (U (H) (t)) 2 dt , (5.1.13) where c∈R and 1 F 1 (a,b,z) is Kummer’s hypergeometric function, see Section 5.3 for details. Theorem 5.1.2. Assume that 1/2 ≤ H < 1 and let θ = θ(T ) be a family of parameters such that lim T→∞ Tθ(T ) = c for some c ∈ R. Then, as T → ∞, b θ T,H → 0 with probability one and T b θ T,H L −→ Π H,c . (5.1.14) 5.2 Strongconsistencyandlargesampleasymptotics Theorem 5.2.1. Let 1/2≤H < 1 andX (H) =X (H) (t) be the fractional Ornstein- Uhlenbeck process defined in (5.1.3). Then e θ T,H defined in (5.1.7) is strongly con- sistent estimator for all θ∈R, i.e. for all θ∈R lim T→∞ e θ T,H =θ (5.2.1) with probability one. 107 Proof. If θ< 0 and X (H) (0) = 0, then, by [29, Lemma 3.3], lim T→∞ 1 T Z T 0 (X (H) (t)) 2 dt =θ −2H HΓ(2H) (5.2.2) with probability one; analysis of the proof shows that (5.2.2) also holds when X (H) (0)6= 0. Also, lim T→∞ (X (H) (T )) 2 − (X (H) (0)) 2 2 R T 0 (X (H) (t)) 2 dt = 0 (5.2.3) with probability one; see [29, the remark after the proof of Theorem 3.4]. Then (5.2.1) follows from (5.2.2) and (5.2.3). If θ> 0 and X (H) (0) = 0, then, by [7, Theorem 1] lim T→∞ (X (H) (T )) 2 2 R T 0 (X (H) (t)) 2 dt =θ (5.2.4) with probability one; analysis of the proof shows that (5.2.4) also holds when X (H) (0)6= 0. Also, by [7, Lemmas 2 and 3], the finite limit ξ = lim t→∞ Z t 0 e −θs dW (H) (s) (5.2.5) 108 exists with probability one. Therefore lim t→∞ e −2θt Z t 0 (X (H) (s)) 2 ds = lim t→∞ e −2θt Z t 0 e 2θs X (H) (0) + Z s 0 e −θr dW (H) (r) 2 ds = X (H) (0) +ξ 2 2θ , lim T→∞ 1 HΓ(2H)T Z T 0 (X (H) (t)) 2 dt ! − 1 2H = 0, lim T→∞ (X (H) (0)) 2 2 R T 0 (X (H) (t)) 2 dt → 0, (5.2.6) all with probability one, and (5.2.2) follows. If θ = 0, then X (H) (t) =X (H) (0) +W (H) (t), and the law of iterated logarithm for self-similar Gaussian processes [5, Theorem 3.1] implies lim T→∞ (W (H) (T )) 2 T 2H+ε = 0 with probability one for every ε> 0 and lim sup T→∞ 1 T 2H ln lnT Z 1 0 (W (H) (Ts)) 2 ds<∞ Since R T 0 (W (H) (t)) 2 dt =T R 1 0 (W (H) (Ts)) 2 ds, equality (5.2.2) follows. We have the following asymptotic distributions and convergence rates of e θ T,H . 109 Theorem 5.2.2. As T→∞: q |θ|T ( e θ T,H −θ) L −→N 0, θ 2 (2H) 2 σ 2 H ! , θ< 0, H∈ [1/2, 3/4), (5.2.7) T e θ T,H L −→ Ψ H,0 , θ = 0; (5.2.8) e θT ( e θ T,H −θ) L −→ 2θ η 1 η 2 +X (H) (0)b H , θ> 0, (5.2.9) with σ 2 H := (4H− 1) 1 + Γ(3− 4H)Γ(4H− 1) Γ(2− 2H)Γ(2H) ! , b H := θ H q HΓ(2H) , (5.2.10) and η 1 and η 2 are independent standard normally distributed. Proof. Set A(T ) = (X (H) (T )) 2 − (X (H) (0)) 2 2 R T 0 (X (H) (t)) 2 dt and B(T ) = 1 HΓ(2H)T Z T 0 (X (H) (t)) 2 dt ! − 1 2H . Then e θ T,H =A(T )−B(T ). Caseθ< 0. By [50, Theorem 4.1] we have √ T (B(T )+θ) L −→N 0, |θ| (2H) 2 σ 2 H , if H∈ (1/2, 3/4). Hence for (5.2.7) it is enough to show that √ TA(T )→ 0. We have 1 T R T 0 (X (H) (t)) 2 dt→ θ −2H HΓ(2H) as T →∞. By [50, (3.7)] combined with [50, Corollary 5.2] in the web-only appendix it follows that (X (H) (T )) 2 / √ T→ 0 almost surely, the result follows. 110 Case θ > 0. From [7, Theorem 5], with the obvious modification to non-zero initial condition, e θT A(T )→ 2θη 1 /(η 2 +X (H) (0)b H ), with b H defined in (5.2.10). Moreover using the first convergence in (5.2.6) and 1/H > 1, e θT B(T ) =e θT 1 HΓ(2H)T Z T 0 (X (H) (t)) 2 dt ! − 1 2H =e θT e −θT/H (HΓ(2H)T ) 1 2H e −2θT Z T 0 (X (H) (t)) 2 dt ! − 1 2H =e −(1/H−1)θT (HΓ(2H)T ) 1 2H e −2θT Z T 0 (X (H) (t)) 2 dt ! − 1 2H → 0, T→∞, (5.2.11) almost surely. Caseθ = 0. Equality (5.2.8) follows from Theorem 5.1.1, which is proved later. Whileboth(5.2.9)and(5.2.9)suggestthattherateofconvergenceisdetermined by the product|θ|T, more precise estimates are possible when θ< 0 andH = 1/2. If H = 1/2, then (5.1.7) implies √ T ( e θ T −θ) = T −1/2 R T 0 X(t)dW (t) T −1 R T 0 X 2 (t)dt , (5.2.12) whereW =W (1/2) is the standard Brownian motion,X =X (1/2) is the correspond- ing Ornstein-Uhlenbeck process, and e θ T = e θ T,1/2 . In the following we show that, when θ< 0 and H = 1/2, the rate of convergence of (a) the numerator of (5.2.12) 111 to the normal distribution and (b) the denominator of to a constant indeed depends on how large the term|θ|T is. For (a) we use use elements of Stein’s method on Wiener chaos (see [49]). To simplify the notations, we switch from θ to−θ and assume zero initial condition, that is, consider X(t) =−θ Z t 0 X(s)ds +W (t), X(0) = 0, where θ> 0. For random variables (X,Y ) on (Ω,F,P) we define the total variation distance kL(X)−L(Y )k TV := sup A∈F |P (X∈A)−P (Y∈A)|. Note that T 2θ 1/2 ( e θ T −θ) =− 2θ T 1/2R T 0 X(t)dW (t) 2θ T R T 0 X(t) 2 dt =− 2θ T 1/2R T 0 R t 0 e −θ(t−s) dW (s)dW (t) 2θ T R T 0 X(t) 2 dt − = θ 2T 1/2 I 2 (e −θ|t−s| ) 2θ T R T 0 X(t) 2 dt , (5.2.13) whereI 2 (f(s,t)) = 2 R T 0 R t 0 f(s,t)dW (s)dW (t) denotes the iterated Wiener inte- gral for symmetric square integrable functions f(s,t). Then one can use that the numerator F T := θ 2T ! 1/2 I 2 (e −θ|t−s| ) 112 converges to a normal distribution, and the denominator I T := 2θ T ! Z T 0 X 2 t dt converges to a constant, both almost surely and in mean square. By an estimate from [12, Section 2] we get using an application of Chebyshev’s inequality P{|I T − 1|≥δ}≤ 2 δ 2 ( 4(2θT− 1 +e −2θT ) 4θ 2 T 2 + 3(1−e −2θT ) 2 4θ 2 T 2 ) ≤ C δ 2 min 1, 1 θT . (5.2.14) Thus the convergence of the denominator depends on the size of θT. For the numerator, we need some additional computations to get the rate of convergence to the normal. Note that F T is in the second Wiener chaos of W, see [50, Section 1.1.2] for Wiener chaos for the white noise case. Hence by [49, Theorem 1.5] we have kL(F T )−L(Z)k TV ≤ 2 s 1 6 |E(F 4 T )− 3| + 3 +E(F 2 T ) 2 |E(F 2 T )− 1|, (5.2.15) where Z is a generic standard normally distributed random variable. 113 We have E I 2 (e −θ|t−s| ) 2 =E 2 Z T 0 X t dW t ! 2 = 4E Z T 0 X 2 t dt = 4 Z T 0 1−e −2θt 2θ dt = 2 θ T− 1−e −2θT 2θ ! . Therefore E(F 2 T ) = θ 2T ! E I 2 (e −θ|t−s| ) 2 = θ 2T ! 2 θ T− 1−e −2θT 2θ ! = 1− 1−e −2θT 2Tθ , (5.2.16) that is, |E(F 2 T )− 1|≤C min 1, 1 θT (5.2.17) We also obtain E(F 4 T ) = θ 2T ! 2 E I 2 (e −θ|t−s| ) 4 = θ 2T ! 2 h E I 2 (e −θ|t−s| ) 2 i 2 3 +3 4e −2θT (4θ 2 T 2 + 10θT + 7) + (20θT− 29) +e −4θT θ 4 ) = 3 h E(F 2 T ) i 2 + 3 θ 2T ! 2 4e −2θT (4θ 2 T 2 + 10θT + 7) + (20θT− 29) +e −4θT θ 4 = 3 " 1− 1−e −2θT 2θT # 2 + 3 4e −2θT (4θ 2 T 2 + 10θT + 7) + (20θT− 29) +e −4θT 4θ 2 T 2 , (5.2.18) that is, |E(F 4 T )− 3|≤C min 1, 1 θT (5.2.19) 114 Combining (5.2.17), (5.2.19), and (5.2.15), we conclude that kL(F T )−L(Z)k TV ≤C min 1, 1 √ θT ! ; (5.2.20) an explicit value of C can be recovered from the above computations. Both (5.2.14) and (5.2.20) suggest that the distribution of e θ T,H will be rather different from normal for moderate values of 1/ √ θT. This conclusion is consistent with Monte Carlo simulations: ifθ = 1 andT = 200 (so that 1/ √ θT≈ 1/14, mod- erate indeed), then the normality assumption for e θ T,H is rejected at the significance level 5%, see Table 5.1. In the next section we study the asymptotic distribution of the statistic e θ T,H and obtain a better finite sample distribution approximation of e θ T,H . Table 5.1: p-values (in %) for test of normality (Jarque-Bera test) with H = 0.6. Sample size is 1000 for each p-value. Only when θ =−3 and T = 200 or θ =−5 and T = 100 normality gets not rejected at the 5%-significance level. θ\T 25 50 100 200 -0.01 < 0.1 < 0.1 < 0.1 0.40 -1 < 0.1 < 0.1 < 0.1 1.26 -3 < 0.1 < 0.1 2.25 15.58 -5 < 0.1 2.45 5.56 50 115 Figure 5.1: Quantile-quantile plot of 1000 samples of √ T ( e θ T,H −θ) versus standard normal distribution withθ =−0.5 on the left andθ =−5 on the right,T = 25 and H = 0.7. Ifthedistributionisnormal, thesimulatedpointsshouldlayonthedotted red line. The left plot suggests that the sample data is not normally distributed. The right plot suggests that the data is approximately normally distributed. −4 −2 0 2 4 −4 −2 0 2 4 Standard Normal Quantiles Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal −4 −2 0 2 4 −4 −2 0 2 4 Standard Normal Quantiles Quantiles of Input Sample QQ Plot of Sample Data versus Standard Normal 5.3 Finite sample approximation and hypothesis testing In this section we develop approximations of the finite sample distribution of the estimatorwhicharedifferentfrom(5.2.7)and(5.2.9). Theapproximatedistribution is continuous as a function of the suitable parameter and, according to Monte-Carlo simulations, works well when (5.2.7) and (5.2.9) do not. As a motivation, recall an analogous result for the first-order stochastic differ- ence equation y n =αy n−1 +e n , n = 1,...,N, 116 where e n are i.i.d. normally distributed random variables with mean zero and variance 1, andα is an unknown parameter. The maximum likelihood estimator is the least-squares estimator (see [46]): b α N = N X n=1 y n y n−1 N X n=1 y 2 n−1 ! −1 . It is known that b α T is consistent estimator ofα, i.e. lim N→∞ b α N =α in probability. Moreover the asymptotic distribution as N→∞ of b α n is given by √ N(b α N −α) L −→N 0, (1−α 2 ) |α|< 1, (5.3.1) Nb α N L −→ 1− (W 1 ) 2 2 R 1 0 (W u ) 2 du , α = 1, (5.3.2) |α| N α 2 − 1 (b α N −α) L −→C(1). |α|> 1. (5.3.3) (5.3.1) has been proven in [46], and (5.3.2) and (5.3.3) in [71]. Several authors deal with asymptotic distributions in the case that α is near 1, which has been extensively studied in [55–57]. The idea is to choose the parameter c according to α = exp(c/N), (5.3.4) whereN is the sample size. Note that this family of parameters satisfies the prop- erty lim N→∞ N(α(N)− 1) =c, especially the autoregressive parameter α(N) con- verges to 1 as the sample size increases. c < 0 corresponds to stationary case, 117 c = 0 corresponds to the unit root, c > 0 corresponds to explosive case. The distribution of Nb α N converges to a functional of the Ornstein Uhlenbeck process, see [55, Theorem 1(b)]: Nb α N L −→ Ψ 1/2,c , α∈R, (5.3.5) where Ψ 1/2,c is given in (5.1.6) with H = 1/2. It is shown, that this distribution is a better asymptotic distribution than (5.3.1) and (5.3.3) for moderate values of α and N. In the continuous time case we have a similar issue as in the discrete case, whenever the parameter θ or the sample size T is small. This is illustrated in Section 5.2. To proceed, we need a better understanding of stochastic integration with re- spect to the fractional Brownian motion. This understanding is necessary both to further analyze (5.1.5) and to establish the connection between (5.1.5) and (5.1.12). Similar to [11,23,50], we follow the Malliavin calculus approach. 118 5.3.1 Stochastic integration with respect to fBm As before denote W (H) = W (H) (t) a fractional Brownian motion with index H∈ [1/2, 1). It can be shown thatW (H) has stationary increments and it is self-similar, in the sense that for every c> 0 {W (H) (ct),t≥ 0} L ={c H W (H) (t),t≥ 0}. (5.3.6) Assume furthermore that the sigma-fieldF is generated by W (H) (t). LetE be the set of real valued step function on [0,T ] and letH be the real separable Hilbert space defined as the closure ofE with respect to the scalar product h1 [0,t] , 1 [0,s] i H =R H (t,s), and denote byW (H) (h) the image of an elementh∈H under the mapW (H) :H→ L 2 (Ω). The spaceH is not a space of functions only, it also contains distributions, see [59]. LetS denote the space of smooth and cylindrical random variables of the form F =f(W (H) (h 1 ),...,W (H) (h n )), h i ∈H, where f ∈ C ∞ p (R n ), the space of infinitely differentiable functions f : R n → R, where f and all its derivatives have at most polynomial growth. 119 Define the derivative operator of such F as theH-valued random variable D (H) F = n X j=1 ∂f ∂x j W (H) (h 1 ),...,W (H) (h n ) h j . The derivative operatorD (H) is a closable unbounded operator fromL p (Ω,F,P) to L p (Ω,F,P;H) for any p≥ 1. Define as D H,1,p (H), p≥ 1, the closure ofS with respect to the norm kFk p 1,p =kFk p L p (Ω) +kDFk p L p (Ω;H) . Denote byδ (H) the adjoint of the operatorD (H) . The domain dom(δ (H) ) are all u∈L 2 (Ω,F,P;H) such that there exists a constant C > 0 such that hD (H) F,ui L 2 (Ω;H) ≤CkFk L 2 (Ω) , for all F∈S. (5.3.7) For an element u∈ dom(δ (H) ) we can define δ (H) (u) through the relationship hF,δ (H) (u)i L 2 (Ω) =hD (H) F,ui L 2 (Ω;H) , for all F∈D H,1,2 (H). In [50, Proposition 1.3.1] it is shown that for any u∈D H,1,2 (H) kδ (H) (u)k L 2 (Ω;H) ≤kuk 1,2 , 120 and hence the space D H,1,2 (H) is contained in dom(δ). In fact, for any simple function F∈D H,1,2 (H) of the form F = n X i=1 F i h i , F i ∈S, h i ∈H, i = 1,...,n we have δ (H) (F ) = n X i=1 F i W (H) (h i )− n X i=1 hD (H) F j ,h j i L 2 (Ω;H) . For any u∈ dom(δ (H) ) we call δ (H) (u) the divergence integral and write δ (H) (u) = Z T 0 u(t)dW (H) (t). Let f,g : [0,T ]→ R be Hölder continuous functions of order α∈ (0, 1) and β ∈ (0, 1) respectively, with α +β > 1. [72] proved that the Riemann-Stieltjes integral(alsocalledYoungintegral) R T 0 f(s)dg(s)exists. ForH > 1/2, wecandefine the pathwise Riemann-Stieltjes integral for any process u ={u(t), 0≤t≤T} that has Hölder-continuous paths of order α> 1−H by Z T 0 u(t)◦dW (H) (t) = lim n→∞ n−1 X i=0 u(t i ) W (H) (t i+1 )−W (H) (t i ) in L 2 (Ω), (5.3.8) where 0 = t 0 ≤ ...≤ t n = T is any partition such that max i |t i+1 −t i |→ 0 as n→∞. In [23, Theorem 12] it is shown, that using the right endpoints such as u(t i+1 ) in (5.3.8) does not change the limit. 121 Suppose additionally that u ={u(t), 0≤ t≤ T} is a stochastic process in the spaceD H,1,2 (H) and suppose that Z T 0 Z T 0 D (H) s u(t) |t−s| 2H−2 dsdt<∞ P−a.s. Then as in [7,29] we have the following relation between the two integrals and the Malliavin derivative: Z T 0 u(t)◦dW (H) (t) = Z T 0 u(t)dW (H) (t)+H(2H−1) Z T 0 Z T 0 D (H) s u(t)|t−s| 2H−2 dsdt. 5.3.2 Asymptotic distribution of the statistics Let θ =θ(T ) depend on the observation time T and lim T→∞ θ(T ) = 0. In analogy to the discrete time case (5.3.4), we make the assumption that θ(T ) belongs to a family of parameters that satisfy lim T→∞ Tθ(T ) =c, for some real number c. The parameterθ(T ) will affect the finite sample distribution of the least-squares estimator but will dissipate asymptotically. The additional parameter c allows a greater flexibility of the distribution. The processX(t) depends on the observation 122 time T, so it would be better to write X(t,T ). However, for readability of our presentation we will not write this out explicitely. Lemma 5.3.1. As T→∞ we have the following asymptotic distributions X (H) (T ) T H L →U (H) c (1), (5.3.9) R T 0 (X (H) (t)) 2 dt T 2H+1 L → Z 1 0 U (H) c (t)dt, (5.3.10) where U (H) c =U (H) c (t) is defined in (5.1.3). Proof. X (H) (T ) =X (H) (0)e θT +W (H) (T ) +θ Z T 0 e θ(T−s) W (H) (s)ds =X (H) (0)e θT +W (H) (T ) +θT Z 1 0 e θT (1−s) W (H) (Ts)ds. Using self-similarity of W (H) and the continuous mapping theorem (5.3.9) follows. Moreover R T 0 (X (H) (t)) 2 dt =T R 1 0 (X (H) (Tt)) 2 dt and X (H) (Tt) =X (H) (0)e θTt +W (H) (Tt) +θ Z Tt 0 e θ(Tt−s) W (H) (s)ds =X (H) (0)e θTt +W (H) (Tt) +θT Z t 0 e θT (t−s) W (H) (Ts)ds. Using again self-similarity it follows that X (H) (Tt) T H L →U (H) c (t). 123 Therefore R T 0 (X (H) (t)) 2 dt T 2H+1 = Z 1 0 (X (H) (Tt)) 2 T 2H dt L → Z 1 0 U (H) c (t)dt. Proof of Theorem 5.1.1. The almost sure convergence follows from the law of iter- ated logarithm similar to the proof of Theorem 5.2.1. Convergence in distribution (5.1.11) is a consequence of Lemma 5.3.1 and the continuous mapping theorem. Figure 5.2: Quantile-quantile plot of 1000 samples of T e θ T,H with θ =−0.5 versus the asymptotic distribution Ψ H,c ; c =−12.5, T = 25 and H = 0.7. Note that Tθ =c. The plot suggests a superior fit compared to normal distribution in Figure 5.1, since the simulated quantiles lay on the dotted red line y =x. −50 −40 −30 −20 −10 0 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 5 Quantiles of Asymptotic Distribution (c=−12.5) Quantiles of Input Sample (theta=−0.5, T=25) QQ Plot of Sample Data versus Asymptotic Distribution (H=0.7) 124 Comparing Figure 5.1 and Figure 5.2 suggests, that distribution in (5.2) is a better candidate for the finite sample distribution for θ =−0.5 and T = 25 than the Gaussian distribution in 5.1. 5.3.3 Alternative estimators In the case H > 1/2, estimator (5.1.7) is different from a maximum likelihood estimator. In [36], based on the work of [48], they obtained the maximum likelihood estimator ¯ θ T,H = R T 0 Q(s)dZ s R T 0 Q 2 (s)dw H s , where k H (t,s) =κ −1 H s 1/2−H (t−s) 1/2−H , κ H = 2HΓ(3/2−H)Γ(H + 1/2), w H t =λ −1 H t 2−2H , λ H = 2HΓ(3− 2H)Γ(H + 1/2) Γ(2/3−H) , Q(t) = d dw H t Z t 0 k H (t,s)X s ds, 0≤t≤T, Z t = Z t 0 k H (t,s)dX s . It is moreover shown that this estimator is strongly consistent and the authors find asymptotics of the bias and the mean square. Even though maximum-likelihood estimator have usually very desirable prop- erties, in this case we see a problem in the computation of ¯ θ T,H , which is very complicated given a set of observations (X (H) (t), 0≤t≤T ). 125 An alternative estimator b θ T,H is given in (5.1.12). It has been proposed in [50] and is theoretically very useful and a prerequisite to show asymptotical results of estimator e θ T,H . Theestimatorismotivatedby(formally)minimizingtheleast-squaresfunctional Z T 0 | ˙ X (H) (t)−θX (H) (t)| 2 dt. It is shown in [29], that b θ T is a strongly consistent estimator of θ < 0, that is lim T→∞ b θ T =θ with probability 1. [7] implied strong consistency for the case θ> 0 after a slight modification of the estimator. A feature of much interest is its asymptotic distribution as the observation size becomes large. So far, we have the following asymptotic distributions and convergence rates of b θ T : √ T ( b θ T −θ) L −→N 0,|θ|σ 2 H , as T→∞, θ< 0, H∈ [1/2, 3/4); (5.3.11) e θT ( b θ T −θ) L −→ 2θC(1), as T→∞, θ> 0,X (H) (0) = 0, H∈ [1/2, 1); (5.3.12) where σ 2 H := (4H− 1) 1 + Γ(3− 4H)Γ(4H− 1) Γ(2− 2H)Γ(2H) ! 126 andC(1) is the standard Cauchy distribution with probability density function 1/(π(1 +x 2 )), x∈R. Results (5.2.7) and (5.2.9) have been shown in [29] and [7] respectively. Moreover for H = 1/2 and θ > 0 [12] obtained the error bound for the convergence to normal distribution in the supremums norm as O(T −1/2 ). In the case H = 1/2, estimator (5.1.5) is also the Maximum-Likelihood esti- mator of the parameter θ: Denote by P θ T the measure generated by the Ornstein Uhlenbeck process Z ={Z t , 0≤ t≤ T}, Z t = R t 0 e θ(t−s) dW 1/2 s in the space of con- tinuous functions C([0,T ]). Then the measures P 0 T andP θ T are equivalent and the likelihood function is given by P θ T P 0 T (Z) = exp θ Z T 0 Z t dZ t − θ 2 2 Z T 0 Z 2 t dt ! . (5.3.13) Maximising the density with respect to θ leads to (5.1.5). An extension of this result to second order differential equations is available in [41]. However, wecannotuse b θ T,H inpracticefortworeasons: First, thereisnowayto computethedivergenceintegral R T 0 X (H) (t)dX (H) (t)givenobservations (X (H) (t), 0≤ t≤T ). Secondly, the representation we obtain in this section depends on the un- known parameter θT, and therefore can also not be used to compute the value of b θ T,H . Nonetheless, we find that the finite sample asymptotics for this estimator are an interesting subject to investigate and we can also see similarities to e θ T,H . 127 In the following let 1 F 1 (a,b,z) be Kummer’s confluent hypergeometric function (see [1, Ch. 13], which is given by 1 F 1 (a,b,z) = ∞ X n=0 a (n) z n b (n) n! where a (n) =a(a + 1)(a + 2)··· (a +n− 1). It is an entire function for a,b,z, except for poles at b = 1, 2,.... From the series representation we get that 1 F 1 (0,b,z) = 1. If b > a > 0, 1 F 1 (a,b,z) can be represented as an integral 1 F 1 (a,b,z) = Γ(b) Γ(a)Γ(b−a) Z 1 0 e zu u a−1 (1−u) b−a−1 du. Lemma 5.3.2. The estimator b θ T,H defined in (5.1.12) has representation b θ T = (X (H) (T )) 2 − (X (H) (0)) 2 − 1 F 1 (2H− 1, 2H + 1,θT )T 2H 2 R T 0 (X (H) (t)) 2 dt . 128 Proof. Assume H > 1/2. From the relation between divergence integral and the pathwise Riemann-Stieltjes integral Z T 0 X (H) (t)◦dW (H) (t) = Z T 0 X (H) (t)dW (H) (t) +H(2H− 1) Z T 0 Z T 0 D (H) s X (H) (t)|t−s| 2H−2 dsdt (5.3.14) where D (H) denotes the Malliavin derivative. The latter integral simplifies to Z T 0 Z t 0 e θ(t−s) |t−s| 2H−2 dsdt = Z T 0 Z t 0 e θξ ξ 2H−2 dξdt = Z 1 0 Z Tt 0 e θξ ξ 2H−2 Tdξdt =T 2H Z 1 0 Z t 0 e θTξ ξ 2H−2 dξdt =T 2H Z 1 0 (1−ξ)e θTξ ξ 2H−2 dξ. (5.3.15) Moreover 1 F 1 (2H− 1, 2H + 1,θT ) = Γ(2H + 1) Γ(2H− 1)Γ(2) Z 1 0 (1−ξ)e θTξ ξ 2H−2 dξ = 2H(2H− 1) Z 1 0 (1−ξ)e θTξ ξ 2H−2 dξ. (5.3.16) 129 Moreover since W (H) (t) =X (H) (t)−X (H) (0)−θ R t 0 X (H) (s)ds Z T 0 X (H) (t)◦dW (H) (t) = Z T 0 X (H) (t)◦dX (H) (t)−θ Z T 0 (X (H) (t)) 2 dt = ((X (H) (T )) 2 − (X (H) (0)) 2 )/2−θ Z T 0 (X (H) (t)) 2 dt. (5.3.17) If H = 1/2 then the result follows by Itô’s formula. Proof of Theorem 5.1.2 . This is a consequence of Lemma 5.3.1, 5.3.2 and the con- tinuous mapping theorem. 5.3.4 Hypothesis testing We consider the following problem of testing for the zero root: H 0 :θ = 0, H 1 :θ6= 0. (5.3.18) Another interesting test is testing for stability: H 0 :θ≥ 0, H 1 :θ< 0. (5.3.19) 130 Using the main result Theorem 5.1.1 we can easily construct a statistical decision function for the tests. For an introduction to the statistics of random processes see [38,43]. Define by φ T (X) the statistical decision function, which is 1 ifH 1 is accepted and 0 if not. Denote byP θ T the measure generated by the Ornstein Uhlenbeck processX (H) = X (H) (t), X (H) (t) = X (H) (0)e θt +θ R t 0 e θ(t−s) W (H) (s)ds in the space of continuous functions C([0,T ]). Fix a numberα∈ (0, 1), the so called level of significance, and define byK α the class of tests of asymptotic significance level smaller than α, i.e. K α ={φ T : lim T→∞ E 0 φ T (X)≤α}, where the expectation E θ is the integral taken with respect to the measure P θ T . Denote by ψ α the quantiles of the distribution Ψ H,0 = (W (H) (1)) 2 2 R 1 0 (W (H) (t)) 2 dt − 1 HΓ(2H) Z 1 0 (W (H) (t)) 2 dt ! − 1 2H which can be obtained by Monte-Carlo simulation. 131 Given observations (X (H) (t), 0≤t≤T ) the statistical decision functionφ T (X) for test (5.3.19) is given by φ T (X) =1(T e θ T,H <ψ α ), where 1(A(ω)) = 1 if ω∈A and 0 otherwise. Hence φ T (X)∈K α . Likewise, the statistical decision function φ T (X) for test (5.3.18) is given by φ T (X) =1({T e θ T,H <ψ α/2 }∪{T e θ T,H >ψ 1−α/2 }), Next we analyze the power of the test of a unit root vs. a simple alternative H 0 :θ = 0, H 1 :θ =θ 1 . (5.3.20) Define by β T (φ T ) the probability of the true decision underH 1 , i.e., β T (φ T ) =E θ 1 φ T (X). The value β T (φ T ) is called the power of the test φ T (X). 132 The asymptotic power of φ T (X) can be computed using the asymptotic distri- butions in (5.2.7) and (5.2.9). Consider the hypothesis test (5.3.20). If θ 1 < 0 and H∈ [1/2, 3/4), the asymptotic power of the test is 1: β T (φ T ) =E θ T φ T (X) =P θ 1 T (T e θ T,H <ψ α ) =P θ 1 T ( √ T ( e θ T,H −θ 1 )<ψ α / √ T− √ Tθ 1 )→ 1, (5.3.21) recalling thatθ 1 < 0. A similar calculations shows, that forθ 1 > 0 andH∈ [1/2, 1) the asymptotic power of the test is 1 as well. This is not a very informative result: for every reasonable test one would expect full power in the large-sample asymptotic. More interesting is the power of the test β T (φ) when the sample is finite. Here, the result of Theorem 5.1.1 helps. We get for any θ 1 ∈R β T (φ T ) =E θ 1 φ T (X) =P θ 1 T (T e θ T,H >ψ α )→P(Ψ H,c >ψ α ), c =Tθ 1 , (5.3.22) where Ψ H,c is the asymptotic distribution of T e θ T,H as stated in Theorem 5.1.1. Thus we get a better approximation of the actual power under finite samples. 133 Chapter 6 Open questions Open questions concerning maximum likelihood estimation include: 1. Strong consistency of estimator when the stochastic differential equation of second order has non-linear coefficients and is driven by fractional Brownian motion. 2. Asymptotic distributions and convergence of moments of estimator when the driving process has dependent increments. In [13] these properties are studied for the drift parameter in a partially observed fractional diffusion system. 3. If the Volterra process can be transformed into a semimartingale, this opens up an alternative maximum likelihood estimator (see for example [36] in the case of fractional Brownian motion with H≥ 1/2). 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Abstract (if available)

Abstract The objective of this thesis is to study statistical inference of first and second order ordinary differential equations driven by continuous Gaussian noise under continous time observations. ❧ The Gaussian process can be defined by a stochastic integral of a time-dependent triangular deterministic kernel with respect to standard Brownian motion. Especially, we do not assume the process to be Markovian nor a semimartingale. An important example for such a process is fractional Brownian motion. ❧ The thesis is focussed on (a) properties of these Gaussian processes (b) maximum likelihood estimation (c) asymptotic distribution of finite sample distribution of least squares type estimator.

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Creator Moers, Michael (author)

Core Title Statistical inference of stochastic differential equations driven by Gaussian noise

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 07/25/2012

Defense Date 06/14/2012

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

Tag fractional Brownian motion,Maximum-likelihood Estimation,OAI-PMH Harvest,statistical inference,Volterra processes

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lototsky, Sergey V. (committee chair), Haas, Stephan W. (committee member), Mikulevičius, Remigijus (committee member)

Creator Email mmoers@gmail.com,moers@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-63399

Unique identifier UC11290071

Identifier usctheses-c3-63399 (legacy record id)

Legacy Identifier etd-MoersMicha-980.pdf

Dmrecord 63399

Document Type Dissertation

Rights Moers, Michael

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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