Stochastic Analysis in Discrete and Continuous Settings

Stochastic analysis can be viewed as a branch of infinite-dimensional analysis that stems from a combined use of analytic and probabilistic tools, and is developed in interaction with stochastic processes. In recent decades it has turned into a powerful approach to the treatment of numerous theoret- ical and applied problems ranging from existence and regularity criteria for densities (Malliavin calculus) to functional and deviation inequalities, math- ematical finance, anticipative extensions of stochastic calculus. The basic tools of stochastic analysis consist in a gradient and a divergence operator which are linked by an integration by parts formula. Such gradient operators can be defined by finite differences or by infinitesimal shifts of the paths of a given stochastic process. Whenever possible, the divergence operator is connected to the stochastic integral with respect to that same underlying process. In this way, deep connections can be established between the algebraic and geometric aspects of differentiation and integration by parts on the one hand, and their probabilistic counterpart on the other hand. Note that the term “stochastic analysis” is also used with somewhat different sig- nifications especially in engineering or applied probability; here we refer to stochastic analysis from a functional analytic point of view

In this chapter we introduce the tools of stochastic analysis in the simple framework of discrete time random walks. Our presentation relies on the use of finite difference gradient and divergence operators which are defined along with single and multiple stochastic integrals. The main applications of stochastic analysis to be considered in the following chapters, including func- tional inequalities and mathematical finance, are discussed in this elementary setting. Some technical difficulties involving measurability and integrability conditions, that are typical of the continuous-time case, are absent in the discrete time case.

This chapter is concerned with the basics of stochastic calculus in continuous time. In continuation of Chapter 1 we keep considering the point of view of normal martingales and structure equations, which provides a unified treat- ment of stochastic integration and calculus that applies to both continuous and discontinuous processes. In particular we cover the construction of single and multiple stochastic integrals with respect to normal martingales and we discuss other classical topics such as quadratic variations and the Itˆo formula

In this chapter we construct an abstract framework for stochastic analysis in continuous time with respect to a normal martingale (Mt)t?R+, using the construction of stochastic calculus presented in Section 2. In particular we identify some minimal properties that should be satisfied in order to connect a gradient and a divergence operator to stochastic integration, and to apply them to the predictable representation of random variables. Some applica- tions, such as logarithmic Sobolev and deviation inequalities, are formulated in this general setting. In the next chapters we will examine concrete exam- ples of operators that can be included in this framework, in particular when (Mt)t?R+ is a Brownian motion or a compensated Poisson process

In this chapter we present a first example of a pair of gradient and diver- gence operators satisfying the duality Assumption 3.1.1, the Clark formula Assumption 3.2.1 and the stability Assumption 3.2.10 of Section 3.1. This construction is based on annihilation and creation operators acting on multi- ple stochastic integrals with respect to a normal martingale. In the following chapters we will implement several constructions of such operators, respec- tively when the normal martingale (Mt)t?R+ is a Brownian motion or a compensated Poisson process. Other examples of operators satisfying the above assumptions will be built in the sequel by addition of a process with vanishing adapted projection to the gradient D, such as in Section 7.7 on the Poisson space

In this chapter we consider the particular case where the normal martin- gale (Mt)t?R+ is a standard Brownian motion. The general results stated in Chapters 3 and 4 are developed in this particular setting of a continuous martingale. Here, the gradient operator has the derivation property and can be interpreted as a derivative in the directions of Brownian paths, while the multiple stochastic integrals are connected to the Hermite polynomials. The connection is also made between the gradient and divergence operators and other transformations of Brownian motion, e.g. by time changes. We also de- scribe in more detail the specific forms of covariance identities and deviation inequalities that can be obtained on the Wiener space and on Riemannian path space.

In this chapter we give the definition of the Poisson measure on a space of configurations of a metric space X, and we construct an isomorphism between the Poisson measure on X and the Poisson process on R+. From this we obtain the probabilistic interpretation of the gradient D as a finite difference operator and the relation between Poisson multiple stochastic integrals and Charlier polynomials. Using the gradient and divergence operators we also derive an integration by parts characterization of Poisson measures, and other results such as deviation and concentration inequalities on the Poisson space.

We study a class of local gradient operators on Poisson space that have the derivation property. This allows us to give another example of a gra- dient operator that satisfies the hypotheses of Chapter 3, this time for a discontinuous process. In particular we obtain an anticipative extension of the compensated Poisson stochastic integral and other expressions for the Clark predictable representation formula. The fact that the gradient oper- ator satisfies the chain rule of derivation has important consequences for deviation inequalities, computation of chaos expansions, characterizations of Poisson measures, and sensitivity analysis. It also leads to the definition of an infinite dimensional geometry under Poisson measures.

Here we review some applications to mathematical finance of the tools in- troduced in the previous chapters. We construct a market model with jumps in which exponential normal martingales are used to model random prices. We obtain pricing and hedging formulas for contingent claims, extending the classical Black-Scholes theory to other complete markets with jumps.

This appendix shortly reviews some notions used in the preceding chapters. It does not aim at completeness and is addressed to the non-probabilistic reader, who is referred to standard texts, e.g. [67], [119] for more details.