NDLI: Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

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Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

Content Provider	MDPI
Author	Herzog, Bodo
Copyright Year	2022
Description	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 ${BtH,t\geq0}$ and sub-fractional Brownian motions ${ξtH,t\geq0}$ with Hurst parameter $H\in(12,1)$ . 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.
Starting Page	340
e-ISSN	22277390
DOI	10.3390/math10030340
Journal	Mathematics
Issue Number	3
Volume Number	10
Language	English
Publisher	MDPI
Publisher Date	2022-01-23
Access Restriction	Open
Subject Keyword	Mathematics Mathematical Physics Cauchy Problem Fractional-pde Sde Fractional Brownian Motion Sub-fractional Processes Feynman–kac Formula Fractional Calculus
Content Type	Text
Resource Type	Article

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