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.
| Author of HS Reutlingen | Herzog, 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): |  Creative Commons - CC BY - Namensnennung 4.0 International |
