A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations

2022/06/02 by Hutzenthaler, Martin, Nguyen, Tuan Anh
#34K50 #60E15 #65C30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2206.01049

Abstract

We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to Itô processes with coefficients which depend on earlier time points such as stochastic delay equations or Euler-type approximations of stochastic differential equations. We apply our stochastic Gronwall lemma with path-suprema to stochastic functional differential equations and prove a strong convergence rate for coefficient functions which depend on path-suprema.

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