Chaos expansion methods for stochastic differential equations involving the Malliavin derivative, Part I

Abstract

We consider Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise spaces, all represented through the corresponding orthogonal basis of the Hilbert space of random variables with finite second moments, given by the Hermite and the Charlier polynomials. There exist unitary mappings between the Gaussian and Poissonian white noise spaces. We investigate the relationship of the Malliavin derivative, the Skorokhod integral, the Ornstein-Uhlenbeck operator and their fractional counterparts on a general white noise space.

Fundamental equations with higher order Malliavin operators

“…In [1,3,14,15,19,24] the Malliavin derivative and the Skorokhod integral are defined on a subspace of ðLÞ 2 so that the resulting process after application of these operators always remains in ðLÞ 2 . In [9,10,12] we allowed values in ðSÞ 21 and thus obtained a larger domain for all operators.…”

Chaos expansion methods in Malliavin calculus: A survey of recent results

“…We will recall of these classical results and denote the corresponding domains with a "zero" in order to retain a nice symmetry between test and generalized processes. In [19,20,23,24] we allowed values in (S) −1 and thus obtained larger domains for all operators. These domains will be denoted by a "minus" sign to reflect the fact that they correspond to generalized processes.…”

“…In the classical setting, the domain of these operators is a strict subset of the set of processes with finite second moments [7,27,36] leading to Sobolev type normed spaces. A more general characterization of the domain of these operators in Kondratiev generalized function spaces has been derived in [19,23,24], while in [25] we considered their domains within Kondratiev test function spaces. The three equations in (1.1), that have been considered in [21] and [25] provide a full characterization of the range of all three operators.…”

The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

“…Following the idea of the construction of S (R) as an inductive limit space over L 2 (R) with appropriate weights, one can define stochastic generalized random variable spaces over L 2 (Ω) by adding certain weights in the convergence condition of the series expansion. Several spaces of this type, weighted by a sequence q = (q α ) α∈I , denoted by (Q) −ρ , for ρ ∈ [0, 1] were described in [41]. Thus a Gel' fand triplet…”

“…The most common weights and spaces appearing in applications are q α = (2N) α which correspond to the Kondratiev spaces of stochastic test functions (S) ρ and stochastic generalized functions (S) −ρ , and exponential weights q α = e (2N) α linked with the exponential growth spaces of stochastic test functions exp(S) ρ and stochastic generalized functions exp(S) −ρ . Note that, following ideas from financial mathematics, fractional white noise spaces could be constructed by replacing Brownian motion with fractional Brownian motion [25,41], or more general with Lévy processes .…”