Uniform in time estimates for the weak error of the Euler method for SDEs and a Pathwise Approach to Derivative Estimates for Diffusion Semigroups

Dan Crisan, Paul Dobson, Michela Ottobre

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Abstract

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii).

Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.
Original languageEnglish
Pages (from-to)3289-3330
Number of pages42
JournalTransactions of the American Mathematical Society
Volume374
Issue number5
Early online date26 Feb 2021
DOIs
Publication statusPublished - May 2021

Keywords

  • Derivative estimates
  • Euler method for SDEs
  • Markov semigroups
  • Stochastic differential equations

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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