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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
KeywordsBrownian Motion Stochastic Analysis Stochastic Calculus Regularity Criterion Logarithmic Sobolev Inequality
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