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On mixed fractional stochastic differential equations with discontinuous drift coefficient

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic analysis

Published online by Cambridge University Press:  01 December 2022

Ercan Sönmez
Ercan Sönmez*
Affiliation:
University of Klagenfurt
*
*Postal address: Department of Statistics, Universitätsstrase 65–67, 9020 Klagenfurt, Austria. Email: ercan.soenmez@aau.at
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Abstract

We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.

Keywords

Mixed stochastic differential equationdiscontinuous driftlong-range dependenceItô formulaabsolute continuity

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 65C30: Stochastic differential and integral equations 60H99: None of the above, but in this section
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 589 - 606
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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