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

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  • Bodo Herzog

    (Economics Department, ESB Business School, Reutlingen University, 72762 Reutlingen, Germany
    Reutlingen Research Institute (RRI), 72762 Reutlingen, Germany
    Institute of Finance and Economics (IFE), Reutlingen University, 72762 Reutlingen, Germany)

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 { B t H , t ≥ 0 } and sub-fractional Brownian motions { ξ t H , t ≥ 0 } with Hurst parameter H ∈ ( 1 2 , 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.

Suggested Citation

  • Bodo Herzog, 2022. "Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion," Mathematics, MDPI, vol. 10(3), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:3:p:340-:d:731547
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    Cited by:

    1. Bodo Herzog, 2023. "Fractional Stochastic Search Algorithms: Modelling Complex Systems via AI," Mathematics, MDPI, vol. 11(9), pages 1-11, April.

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