TY  - JOUR
U1  - Zeitschriftenartikel, wissenschaftlich - begutachtet (reviewed)
A1  - Herzog, Bodo
T1  - Adopting Feynman-Kac formula in stochastic differential equations with (sub-)fractional Brownian motion
JF  - Mathematics
N2  - 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.
KW  - fractional calculus
KW  - Feynman-Kac formula
KW  - sub-fractional processes
KW  - fractional Brownian motion
KW  - SDE
KW  - fractional-PDE
KW  - Cauchy problem
Y1  - 2022
UN  - https://nbn-resolving.org/urn:nbn:de:bsz:rt2-opus4-39340
SN  - 2227-7390
SS  - 2227-7390
U6  - https://doi.org/10.3390/math10030340
DO  - https://doi.org/10.3390/math10030340
VL  - 10
IS  - 3
SP  - 13
S1  - 13
PB  - MDPI
CY  - Basel
ER  -