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Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups


Authors: D. Crisan, P. Dobson and M. Ottobre
Journal: Trans. Amer. Math. Soc. 374 (2021), 3289-3330
MSC (2020): Primary 65C20, 65C30, 60H10, 65G99, 47D07, 60J60
DOI: https://doi.org/10.1090/tran/8301
Published electronically: February 26, 2021
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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.


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Additional Information

D. Crisan
Affiliation: Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: d.crisan@imperial.ac.uk

P. Dobson
Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
Email: p.dobson@tudelft.nu

M. Ottobre
Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
Email: m.ottobre@hw.ac.uk

DOI: https://doi.org/10.1090/tran/8301
Keywords: Stochastic differential equations, Euler method for SDEs, Markov semigroups, derivative estimates.
Received by editor(s): May 9, 2019
Received by editor(s) in revised form: July 20, 2020
Published electronically: February 26, 2021
Additional Notes: The work of the first author was partially supported by a UC3M-Santander Chair of Excellence grant held at the Universidad Carlos III de Madrid.
The second author was supported by the Maxwell Institute Graduate School in Analysis and its Applications (MIGSAA), a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot–Watt University and the University of Edinburgh.
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