Abstract
The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.
Highlights
We assume that the parameter κ is constant. This second-order partial differential equation (PDE) has a stochastic representation for η = 0, according to the Feynman–Kac formula [2,3]
For the Cauchy problem, we generalize the stochastic representation of Feynman–Kac by utilizing fractional Brownian motion with Hurst parameter H > 1/2
This article studies the relationships of the Cauchy problem (1) and relates them to fractional partial-differential equations, as well as to the stochastic representations by the Feynman–Kac formula with a generalized fractional and sub-fractional Brownian motion with Hurst parameter H > 1/2
Summary
Consider the Cauchy problem [1] of the following parabolic partial differential equation (PDE) on Rd. We assume that the parameter κ is constant This second-order PDE has a stochastic representation for η = 0, according to the Feynman–Kac formula [2,3]. For the Cauchy problem, we generalize the stochastic representation of Feynman–Kac by utilizing fractional Brownian motion (fBM) with Hurst parameter H > 1/2. Our purpose is to construct and prove a general link of the Cauchy problem with the Feynman–Kac equation via Itô’s formula for fBM and sub-fBM. Feynman–Kac representation to a fractional Brownian motion { BtH } and sub-fBM {ξ tH }.
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