In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, significant contributors such as S. Kusuoka, D. Stroock, J-M. Bismut, Shinzo Watanabe, I. Shigekawa, and so on completed the foundations for the field. Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well. The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.
Overview and history Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to stochastic partial differential equations.
Gaussian probability space
Consider a Wiener functional F {\displaystyle F} (a functional from the classical Wiener space) and consider the task of finding a derivative for it. The natural idea would be to use the Gateaux derivative
D g F [ f ] := lim τ → 0 F [ f + τ g ] − F [ f ] τ {\displaystyle D_{g}F[f]:=\lim _{\tau \to 0}{\frac {F[f+\tau g]-F[f]}{\tau }}} , however this does not always exist. Therefore it does make sense to find a new differential calculus for such spaces by limiting the directions. The toy model of Malliavin calculus is an irreducible Gaussian probability space X = ( Ω , F , P , H ) {\displaystyle X=(\Omega ,{\mathcal {F}},P,{\mathcal {H}})} . This is a (complete) probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} together with a closed subspace H ⊂ L 2 ( Ω , F , P ) {\displaystyle {\mathcal {H}}\subset L^{2}(\Omega ,{\mathcal {F}},P)} such that all H ∈ H {\displaystyle H\in {\mathcal {H}}} are mean zero Gaussian variables and F = σ ( H : H ∈ H ) {\displaystyle {\mathcal {F}}=\sigma (H:H\in {\mathcal {H}})} . If one chooses a basis for H {\displaystyle {\mathcal {H}}} then one calls X {\displaystyle X} a numerical model. On the other hand, for any separable Hilbert space G {\displaystyle {\mathcal {G}}} exists a canonical irreducible Gaussian probability space Seg ( G ) {\displaystyle \operatorname {Seg} ({\mathcal {G}})} named the Segal model (named after Irving Segal) having G {\displaystyle {\mathcal {G}}} as its Gaussian subspace. In this case for a g ∈ G {\displaystyle g\in {\mathcal {G}}} one notates the associated random variable in Seg ( G ) {\displaystyle \operatorname {Seg} ({\mathcal {G}})} as W ( g ) {\displaystyle W(g)} . Properties of a Gaussian probability space that do not depend on the particular choice of basis are called intrinsic and such that do depend on the choice extrensic. We denote the countably infinite product of real spaces as R N = ∏ i = 1 ∞ R {\displaystyle \mathbb {R} ^{\mathbb {N} }=\prod \limits _{i=1}^{\infty }\mathbb {R} } . Recall the modern version of the Cameron-Martin theorem
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