Malliavin Calculus for LÚvy Processes and Infinite-Dimensional Brownian Motion
Assuming only basic knowledge of probability theory and functional analysis, this book provides a self-contained introduction to Malliavin calculus and infinite-dimensional Brownian motion. In an effort to demystify a subject thought to be difficult, it exploits the framework of nonstandard analysis, which allows infinite-dimensional problems to be treated as finite-dimensional. The result is an intuitive, indeed enjoyable, development of both Malliavin calculus and nonstandard analysis. The main aspects of stochastic analysis and Malliavin calculus are incorporated into this simplifying framework. Topics covered include Brownian motion, Ornstein-Uhlenbeck processes both with values in abstract Wiener spaces, LÚvy processes, multiple stochastic integrals, chaos decomposition, Malliavin derivative, Clark-Ocone formula, Skorohod integral processes and Girsanov transformations. The careful exposition, which is neither too abstract nor too theoretical, makes this book accessible to graduate students, as well as to researchers interested in the techniques.
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0-algebra absolutely continuous abstract Wiener spaces according to Theorem algebra Borel set Brownian motion cadlag called Cauchy sequence chaos decomposition continuous functions continuous version Corollary cylinder set deﬁned denoted elementary embedding elements equation example exists a sequence exists an unlimited ﬁltration ﬁnite ﬁnite-dimensional ﬁrst ﬁx following result FrÚchet space function f G L2 Gaussian measure Hilbert space ifand image measure inequality infinite-dimensional inﬁnitely inﬁnitesimal internal extension internal function iterated integral Lebesgue measure LÚvy processes lifting F linear ll-ll locally in SL2 Loeb spaces Malliavin calculus Malliavin derivative Malliavin differentiable mapping martingale Moreover nearstandard Note O,oo orthogonal poly-saturated models polynomials probability measure Proof Fix proof of Theorem properties Proposition 8.4.1 random variables real numbers Recall S-continuous Section semi-norms Skorokhod integral square-integrable standard model stochastic subset sufﬁces to prove Suppose symmetric timeline true