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Adopting Feynman-Kac formula in stochastic differential equations with (sub-)fractional Brownian motion

  • 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 and sub-fractional Brownian motions 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.

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Metadaten
Author of HS ReutlingenHerzog, Bodo
URN:urn:nbn:de:bsz:rt2-opus4-39340
DOI:https://doi.org/10.3390/math10030340
ISSN:2227-7390
Erschienen in:Mathematics
Publisher:MDPI
Place of publication:Basel
Document Type:Journal article
Language:English
Publication year:2022
Tag:Cauchy problem; Feynman-Kac formula; SDE; fractional Brownian motion; fractional calculus; fractional-PDE; sub-fractional processes
Volume:10
Issue:3
Page Number:13
Article Number:340
DDC classes:510 Mathematik
Open access?:Ja
Licence (German):License Logo  Creative Commons - CC BY - Namensnennung 4.0 International