Inverse Stochastic Transfer Principle

In [10] a “direct” stochastic transfer principle was introduced, which represented multiple integrals with respect to fractional Brownian motion in terms of multiple integrals with respect to standard Brownian motion. The method employed in [10] involved an operator Γ (n) H , mapping a class of functions LH to L 2. However, the operator does not map LH onto L 2. Hence Γ (n) H is not invertible. The non-invertibility arises from the fact that Γ (n) H is defined in terms of n-fold Riemann-Liouville fractional integrals and it is well known that functions f ∈ Lp(a, b) cannot always be represented as fractional integrals of functions φ ∈ Lp(a, b). This paper establishes the inverse stochastic transfer principle. Our purpose is to explicitly define an operator, which, acting on a certain class of functions, gives the transfer principle inverting that in [10]. As a result, multiple stochastic integrals with respect to standard Brownian motion are represented in terms of multiple stochastic integrals with respect to persistent fractional Brownian motion. In order to do this, we first prove a characterization of a class of functions for which there exists φ, defined on a compact set in Rn, such that f is equal to the n-fold Riemann-Liouville fractional integral of φ. We establish a method for computing φ and conclude the paper with an example of using the inverse transfer principle in filtering.