A Short Review of the Malliavin Calculus in Hilbert Spaces
So far, the Malliavin calculus has been rarely used in articles covering the numerical analysis of stochastic processes, in particular for SPDEs, and we find it appropriate to provide a gentle and self-contained introduction into this theory.
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A Short Review of the Malliavin Calculus in Hilbert Spaces
So far, the Malliavin calculus has been rarely used in articles covering the numerical analysis of stochastic processes, in particular for SPDEs, and we find it appropriate to provide a gentle and self-contained introduction into this theory. In order to give a probabilistic proof of Hörmander’s hypoelliptic theorem P. Malliavin [55] developed 1976 what he called the stochastic calculus of variations. This method has been expanded by several authors and is nowadays called the Malliavin calculus. For readers which are completely new to this field we refer, for example, to the monograph by D. Nualart [58] which also contains an exhaustive list of references. In the first two sections we focus on the Malliavin calculus for Hilbert space valued stochastic processes and cylindrical Wiener processes. Most of the presented results are found in [54], which in turn is based on [32]. Since we were not able to locate a reference in the literature, we also give a detailed proof of a chain rule.
4.1 The Derivative Operator Let .˝; F ; P/ be a complete probability space with normal filtration .Ft /t 2Œ0;T , t 2 Œ0; T . For separable Hilbert spaces .H; .; /; kk/ and .U; .; /U ; kkU / we consider an adapted cylindrical Q-Wiener process W , where the covariance operator satisfies Assumption 2.15. In order to avoid technical issues in the introduction of the Malliavin derivative we assume that F and the filtration .Ft /t 2Œ0;T are generated by the Wiener process W . By C k .H1 I H2 /, k 0, we denote the set of all continuous mappings 'W H1 ! H2 , which are k-times continuously Fréchet-differentiable. Since below the letter D is reserved for the Malliavin derivative operator, we write ' 0 W H1 ! L.H1 ; H2 / and ' 00 W H1 ! L.H1 ; L.H1 ; H2 // for the first and second order Fréchet derivative.
R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Mathematics 2093, DOI 10.1007/978-3-319-02231-4__4, © Springer International Publishing Switzerland 2014
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4 A Short Review of the Malliavin Calculus in Hilbert Spaces
Further for ' 2 C 2 .H I R/, by using the representation theorem, we understand the first derivative ' 0 .h/ at h 2 H as an element in H . Similarly, the second derivative ' 00 .h/ at h 2 H is identified with a self-adjoint linear operator in L.H /, that is h d2 i '.h/ .h1 ; h2 / D ' 00 .h/h1 ; h2 ; 2 dh
for all h; h1 ; h2 2 H:
The set Cpk .H1 I H2 / contains all mappings ' 2 C k .H1 I H2 /, where ' and all its derivatives are at most polynomially growing. Similarly, the set Cbk .H1 I H2 / consists of all mappings ' 2 C k .H1 I H2 / such that ' and all its derivatives are bounded. For a deterministic mapping ˚ 2 L2 .Œ0; T I L2 .U0 ; R// let us define Z W .˚/ WD
T
˚.t/ dW .t/;
0
where the integral on the right hand side is the usual Itô-integral with respect to the cylindrical Wiener process W as in Sect. 2.2. We now introduce the first important operator of the Malliavin calculus, the s
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