Fundamental equations with higher order Malliavin operators

Abstract

We consider three fundamental equations of the Malliavin calculus: the equation involving the Malliavin derivative, the Skorokhod integral and the Ornstein -Uhlenbeck operator of order k, k [ N. These equations provide a complete characterization of the domain and range of the aforementioned operators. Applying the chaos expansion method in white noise spaces we solve these equations and obtain an explicit form of the solutions in the space of Kondratiev generalized stochastic processes.

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

“…Therefore, we are able to represent the stochastic perturbation appearing in equation 1explicitly. Note also that δ(v) belongs to the Wiener chaos space of higher order than v, see also [25,42].…”

“…Classes of elliptic and evolution stochastic differential equations (SDEs) that involve operators of the Malliavin calculus within white noise framework were recently studed in [46,40,44,38,48,58]. In [42,43] it was proved that the Malliavin derivative indicates the rate of change in time between the ordinary product and the Wick product. In this paper, we consider stochastic optimal control problems with stochastic perturbations given in an integral form.…”

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

“…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. Moreover, the solutions to equations (1.1) are obtained in an explicit form, which is highly useful for computer modelling that involves polynomial chaos expansion simulation methods used in numerical stochastic analysis [9,31,49].…”

“…After a short revision of the results on uniqueness of the solutions to equations (1.1) (Theorem 3.1, Theorem 4.1, Theorem 5.1) obtained in [21] and [25] we proceed by proving some properties such as the duality relationship between the Malliavin derivative and the Skorokhod integral (Theorem 6.1) and the chain rule (Theorem 6.11) as well as many others such as the product rule (Theorem 6.6, Theorem 6.8), partial integration etc.…”

“…We also provide several illustrative examples to facilitate comprehension of our results. These examples can be considered as supplementary material to [21] and [25].…”