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Malliavin Calculus and Its Applications B.Sc. Thesis Malliavin calculus and its applications Ad´amGyenge´ Supervisor: Dr. Tam´asSzabados Associate professor Institute of Mathematics Department of Stochastics Budapest University of Technology and Economics 2010 Contents 1 Introduction 1 1.1 The structure of the thesis . 1 2 Theory of Malliavin calculus 3 2.1 It^ointegral . 3 2.1.1 Brownian motion . 3 2.1.2 Construction of the It^ointegral . 4 2.2 Isonormal Gaussian processes . 5 2.3 Wiener chaos expansion . 6 2.3.1 Iterated It^o-and multiple stochastic integrals . 8 2.3.2 Wiener chaos and iterated It^ointegrals . 9 2.4 The derivative operator . 11 2.4.1 Closability and other properties of the derivative operator . 12 2.4.2 The derivative in the white noise case . 14 2.5 The divergence operator and the Skorohod integral . 16 2.5.1 The Skorohod integral . 17 2.6 The Clark-Ocone formula . 20 3 Applications 22 3.1 Stochastic differential equations . 22 3.1.1 Absolute continuity of distributions . 22 3.1.2 Lie bracket and H¨ormander'scondition . 24 3.1.3 Absolute continuity under H¨ormander'scondition . 25 3.2 Relation to Stein's method . 28 3.2.1 Stein's lemma . 28 3.2.2 Ornstein-Uhlenbeck operators . 29 3.2.3 Bounds on the Kolmogorov distance . 30 3.3 Financial mathematics . 31 3.3.1 Investments . 31 3.3.2 Black-Scholes model . 34 4 Summary and conclusions 36 i Chapter 1 Introduction The Malliavin calculus (or the stochastic calculus of variations) is an infinite dimensional differential calculus on the Wiener space. The foundations of the subject were developed in the late 1970's, mainly in the two seminal works [6] and [7], in order to study the existence and smoothness of density for the probability laws of random vectors. The original motivation and the most important application of this theory has been to provide a probabilistic proof of H¨ormander'shypoellipticity theorem. The theory was further developed by Stroock, Bismut, Watanabe and others, and has gained importance in several domains of applications far beyond the original problem. Because of the many different branches of research and huge number of interesting results in the topic it is not possible to cover everything in this short paper. We focused rather on the most important concepts and the logical relations of the results. During the project the following research questions were stated: 1. What are the key concepts and results in the recently developed mathematical field of Malliavin calculus? 2. In which application areas has the theory been adapted for use? 1.1 The structure of the thesis The rest of this thesis is organized as follows: • Chapter 2 introduces the theory of Malliavin calculus in the framework of isonor- mal Gaussian processes. We tried to collect the most important concepts and theo- rems without going into the technical details. Many theorems are presented without proofs, or with only an outline of the proof. Instead, the focus is rather on showing the importance of the different notions and results, and the relations between them. The chapter follows at some points the excellent monograph of David Nualart [9], that contains all the missing proofs and much more. The notations and the under- lying concepts are usually from functional analysis and stochastic calculus. For a reference on these topics we refer to [11] and [5]. 1 CHAPTER 1. INTRODUCTION 2 • In the recent years, the theory of Malliavin calculus has been applied in several different areas. Out of the many important and interesting applications three are summarized in Chapter 3. This includes the original motivating domain of stochas- tic differential equations, as well as two more recent fields: the connection to Stein's method for limit theorems and applications in financial mathematics. • Finally, in Chapter 4 the thesis is summarized and some directions for future research are proposed. I am thankful to my supervisor Tam´asSzabados, who encouraged me to study stochas- tic calculus and guided and supported me from the beginning. I am also grateful to Pro- fessor D´enes Petz, from whom I have learnt functional analysis and who helped me in several aspects. Lastly, I offer my regards to all of those who supported me in any respect during the completion of the project. Budapest, 10th May 2010 Adam Gyenge Chapter 2 Theory of Malliavin calculus This chapter presents the most important theoretical concepts and results of the stochastic calculus of variations, also known as Malliavin calculus. The introduction is based on the monograph of Nualart [9], and uses the context of isonormal Gaussian processes. After defining the isonormal Gaussian processes, the Wiener chaos decomposition and the different operators (derivative and divergence) are concerned. 2.1 It^ointegral The Brownian motion, or Wiener process is definitely one of the most important stochastic processes and it is the basic model in many areas of stochastic analysis. It is not possible to define the Lebesgue-Stieltjes integral of a process in all cases with respect to the Brownian pathes or in a more general case, with respect to semimartingales. Instead, the It^ointegral extends the methods of calculus to these stochastic processes. 2.1.1 Brownian motion + Definition 2.1. The Brownian motion fW (t); t 2 R g is a stochastic process with the following defining properties [5]: 1. (Independence of increments) W (t) − W (s) is independent of Fs for all t > s, where Fs is the σ-field generated by the past: Fs = σfW (u); u ≤ sg. 2. (Normal increments) W (t) − W (s) has a Normal distribution with mean 0 and vari- ance t − s. This implies that W (t) has a N(0; t) distribution. 3. (Continuity of paths) W (t) is almost surely continuous everywhere in t. Beyond these axioms it is usually also assumed that the process starts at W (0) = 0. The time interval on which the Brownian motion is defined is [0;T ] for some T > 0, which is allowed to be infinite. It can be shown that such a process exist. A slightly different approach is the canonical space of the Brownian motion, where it is assumed that continuous functions are the possible outcomes for the trajectories, and we 3 CHAPTER 2. THEORY OF MALLIAVIN CALCULUS 4 define the process as a probability space. This means that the probability space is defined on Ω = C0([0;T ]), the set of continuous functions from [0;T ] to R starting at 0 (that is called the classical Wiener space). On this set a topology is induced by the uniform norm: kfk = supx2[0;T ] f(x), and F, the σ-field of measurable sets is the Borel σ-field of this topology, that is, the smallest σ-field containing all the open sets. The probability measure P (or the law of the process) is the Wiener measure defined on the cylinder sets of Ω: P (f!; !(t1) 2 F1;:::;!(tk) 2 Fkg) = P (W (t1) 2 F1;:::;W (tk) 2 Fkg) = Z = ρ(t1; 0; x1)ρ(t2 − t1; x1; x2) ··· ρ(tk − tk−1; xk−1; xk)dx1 ··· dxk ; F1×···×Fk where Fi 2 B(R), 0 ≤ t1 < ··· < tk and jx − yj2 ρ(t; x; y) = (2πt)−1=2 exp − : 2t Definition 2.2. The quadratic variation of a stochastic process X(t) on the time interval [0; t] is defined as n X [X](t) = lim jX(tn) − X(tn )j2 ; n!1 i i−1 i=1 if the sum converges stochastically, where the limit is taken over all shrinking partitions n n of [0; t], with δn = maxi(ti+1 − ti ) ! 0, as n ! 0. In the case of Brownian motion, the sum above converges in L2 to [W ](t) = t, and since the paths are continuous, the total variation of this process is infinite. Therefore, it is not possible to define the usual Lebesgue-Stieltjes integral with respect to Brownian motion for processes of the same kind. The It^ointegral overcomes this issue. 2.1.2 Construction of the It^ointegral + Consider the Brownian motion on an interval [0;T ] ⊆ R , that can be also infinite. 2 Let fFt; 0 ≤ t ≤ T g denote the natural filtration of the Brownian motion, L (Ω) = L2(Ω; F;P ), and L2([0;T ] × Ω) = L2([0;T ] × Ω; B([0;T ]) ⊗ F; λ × P ), the set of square integrable stochastic processes on [0;T ]. From here, t is always in [0;T ]. Definition 2.3. A stochastic process u(t) is adapted, if u(t) is Ft measurable for all t. 2 2 La([0;T ] × Ω) ⊂ L ([0;T ] × Ω) is the subspace of square-integrable, adapted processes. A process is an elementary adapted process, if there exists a partition 0 ≤ t1 < ··· < tn+1, and a set of random variables F1;:::;Fn, such that every Fi is an Fti measurable and square Pn 2 integrable random variable for all i, and u(t) = i=1 Fi1(ti;ti+1](t). E ⊂ La([0;T ] × Ω) is the set of elementary adapted processes. Definition 2.4. The It^ointegral (or stochastic integral) of an elementary adapted process Pn u(t) = i=1 Fi1(ti;ti+1](t) with respect to the Brownian motion W (t) is n Z T X u(t)dW (t) = Fi(W (ti+1) − W (ti)) (2.1) 0 i=1 CHAPTER 2. THEORY OF MALLIAVIN CALCULUS 5 The integral of an elementary adapted process is a function in L2(Ω), that is, a square integrable random variable. The integral is linear, and has the following properties: Z T E( u(t)dW (t)) = 0 (2.2) 0 Z T Z T E(j u(t)dW (t)j2) = E( u(t)2dt) (2.3) 0 0 R T 2 From (2.3) it follows that the linear operator u(t) 7! 0 u(t)dW (t) is an E! L (Ω) 2 isometry.
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