Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions

TL;DR

This paper studies mixed stochastic differential equations driven by Brownian motion and a H"older continuous process, proving existence, uniqueness, and convergence of solutions, and analyzing their moments under certain conditions.

Contribution

It introduces new results on the solvability and stability of mixed SDEs involving long-range dependence and establishes moment existence under exponential moment assumptions.

Findings

Proved unique solvability of mixed SDEs with H"older continuous processes.

Derived estimates for differences between solutions with different processes.

Established a limit theorem related to solutions of mixed SDEs.

Abstract

For a mixed stochastic differential equation involving standard Brownian motion and an almost surely H\"older continuous process $Z$ with H\"older exponent $\gamma>1/2$, we establish a new result on its unique solvability. We also establish an estimate for difference of solutions to such equations with different processes $Z$ and deduce a corresponding limit theorem. As a by-product, we obtain a result on existence of moments of a solution to a mixed equation under an assumption that $Z$ has certain exponential moments.