A Course of Stochastic Analysis

In the first chapter, Kolmogorov’s axioms of the theory of probability are presented. The choice of the system of the axioms is explained in context of the Caratheodory theorem. Different probability spaces as well as probability distributions are introduced too. The famous Kolmogorov consistency theorem is formulated, and as a consequence, a brief construction of the Wiener measure is shown (see [1], [6], [7], [10], [15], [19], [40], and [45]).

In the second chapter random variables are introduced and investigated in the framework of axiomatic of Kolmogorov. It is shown a connection of probability distributions and distributions of random variables as well as their distribution functions. The notion of the Lebesgue integral is given in context of definition of moments of random variables (see [1], [7], [10], [15], [19], [21], [40], and [45]).

In the third chapter asymptotic properties of sequences of random variables are studied. Lemma of Fatou and the Lebesgue dominated convergence theorem are presented as permanent technical tools of stochastic analysis. It is also emphasized the role of a uniform integrability condition of families of random variables. Classical probabilistic inequalities of Chebyshev, Jensen and Cauchy-Schwartz are proved. It is shown how these inequalities work to investigate interconnections between different types of convergence of sequences of random variables. In particular, the large numbers law (LNL) is derived for the case of independent identically distributed random variables (see [1], [7], [10], [15], [19], [40], and [45]).

The chapter four is devoted in a systematic study of a weak convergence of sequences of random variables. It is shown the equivalence between a weak convergence and convergence in distribution. It is shown that a weak compactness and tightness for families of probability distributions (Prokhorov’s theorem) are equivalent. It is discussed a connection between characteristic functions and distributions of random variables. The method of characteristic functions is applied to prove the  Central Limit Theorem (CLT) for sums of independent identically distributed random variables (see [6], [15], [21], and [40]).

In this chapter a special attention is devoted to the absolute continuity  of measures. It is shown how this notion and the Radon-Nikodym theorem work to define conditional expectations. The list of properties of conditional expectations are given here. In particular, it is emphasized the optimality in the mean-square sense of conditional expectations (see [7], [15], [19], [40], [41] and [45]).

Chapter 6 is completely devoted to a discrete time stochastic analysis. It contains the key notions adapted to discrete time like stochastic basis, filtration, predictability, stopping times, martingales, sub- and supermartingales, local densities of probability measures, discrete stochastic integrals and stochastic exponents. It is stated the Doob decomposition for stochastic sequences, maximal inequalities, and other Doob’s theorems. The developed martingale technique is further applied to prove several asymptotical properties for martingales and submartingales (see [1], [7], [8], [10], [15], [40], and [45]).

In this chapter a characterization of sets of convergence of martingale is given in predictable terms. As a consequence, the strong LNL for square-integrable martingales is proved. This result is applied for derivation of strong consistency of the least-squared estimates in the framework of regression model with martingale errors. Moreover, the CLT for martingales is stated, and further this theorem together with the martingale LNL is applied to derive the asymptotic normality and strong consistency of martingale stochastic approximation procedures. A discrete version of the Girsanov theorem is given here with its further application for derivation of a discrete time Bachelier option pricing formula. In the last section, the notion of a martingale is extended in several directions: from asymptotic martingales and local martingales to martingale transforms and generalized martingales (see [4], [8], [12], [13], [15], [26], [30], [34], and [40]).

This chapter contains a general notion of random processes with continuous time. It is given in context of the Kolmogorov consistency theorem. The notion of a Wiener process with variety of its properties are also presented here. Its existence is stated by two ways: with the help of the Kolmogorov theorem as well as with the help of orthogonal functional systems. Besides the Wiener process as a basic process for many others, the Poisson process is also considered here. Stochastic integration with respect to Wiener process is developed for a class of progressively measurable functions. It leads to the Ito processes, the Ito formula, the Girsanov theorem and representation of martingales (see [5], [6], [14], [17], [21], [35], [41], and [44]).

The chapter presents stochastic differential equations (SDEs) and their connections with diffusion processes and partial differential equations (PDEs). The existence and uniqueness of solutions of SDEs are proved under Lipschitz’s conditions. Two important processes (Geometric Brownian Motion (GBM) and the Ornstein-Uhlenbeck process) are constructed on this theoretical base. The difference between ordinary differential equations and SDEs are discussed. As a part of this discussion, the existence of a solution (weak solution) of any SDE with measurable bounded drift coefficient and unit diffusion is proved with the help of the miracle Girsanov theorem. Moreover, it is shown by mean of the method of monotonic approximations that such a solution will be strong if the grift coefficient is a bounded piece-wise smooth function. Diffusion processes are defined as Markov processes for which their transition densities satisfy the asymptotic properties of Kolmogorov. The backward and forward equations of Kolmogorov are derived. A connection between SDEs and PDEs are stated with the help of the Feynman-Kac theorem. Absolute continuity of distributions of diffusion processes is studied with the help of the Girsanov theorem. A special attention is paid to the class of controlled diffusion processes for which the Hamilton-Jacobi-Bellman optimality equation is derived. It is shown how the theory of diffusion processes and SDEs are helpful in mathematical finance (Bachelier and Black-Scholes models) and in statistics of random processes (see [3], [5], [14], [17], [21], [22], [23], [24], [25], [30], [35], [39], [41], [42], and [44]).

Chapter 10 is devoted to a systematic exposition of a continuous time version of stochastic analysis under “usual conditions” with its standard notions like a stochastic basis, filtration, stopping times, random sets, predictable and optional sigma-algebras etc. It is shown how the discrete time martingale theory as well as a pure continuous time theory of diffusion processes are generalized for so-called cadlag processes. Using the predictable notion of a compensator the fundamental Doob-Meyer theorem is formulated for the class of sub- and supermartingales of class D. The full version of stochastic integration of predictable processes with respect to square-integrable martingale is developed. Moreover, different decompositions of such martingales are proved as well as the Kunuta-Watanabe inequality. It is shown how the theory can be extended with the help of localization procedures (local martingales, processes with locally integrable variation, semimartingales). The Ito formula is proved for semimartingales. SDEs with respect to semimartingales are studied including the existence and uniqueness of solutions of such equations with the Lipschitz coefficients (see [2], [8], [9], [16], [18], [20], [26], [33], [36], and [37]).

The main goal of this chapter is to show how the general theory developed before can be applied to mathematical finance and statistics of random processes. In the area of mathematical finance a semimartingale financial market model is introduced. Applying to this general model the technique of stochastic exponents the fundamental questions of arbitrage and completeness of such a market are studied. These results have a number of corollaries for modeling and option pricing (Black-Scholes model and formula, Cox-Ross-Rubinstein model and formula etc). In the area of statistics of random processes the technique developed above gives a possibility to introduce semimartingale models. It is shown that classical discrete time and continuous time models of stochastic approximation are embedded in a semimartingale scheme. Moreover, it is proved that semimartingale stochastic approximation procedures are strong consistent and asymptotically normal under very wide conditions. In case of semimartingale regression the structural least-squared estimates are strong consistent and their sequential versions satisfy the important  fixed accuracy property (see [3], [4], [11], [13], [18], [23], [30], [31], [32], [34], and [43]).

The list below contains problems which are related to all chapters of the book. Some of them are numerical and some others are pure theoretical, but in any case they are helped for both students and instructors. Students can improve their understanding and scope. Instructors can transform most of the problems for teaching and examination purposes. The following references might be useful to create detailed solutions (see [1], [5], [7], [10], [11], [13], [14], [15], [16], [17], [18], [21], [22], [23], [24], [27], [28], [29], [30], [31], [35], [37], [43], [44], and [45]).