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

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- by D. Crisan, P. Dobson and M. Ottobre PDF
- Trans. Amer. Math. Soc.
**374**(2021), 3289-3330 Request permission

## 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.

## References

- Dominique Bakry, Ivan Gentil, and Michel Ledoux,
*Analysis and geometry of Markov diffusion operators*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348, Springer, Cham, 2014. MR**3155209**, DOI 10.1007/978-3-319-00227-9 - T. Cass, D. Crisan, P. Dobson, and M. Ottobre,
*Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes*, arXiv preprint, arXiv:1805.01350, 2018. - Dan Crisan and François Delarue,
*Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations*, J. Funct. Anal.**263**(2012), no. 10, 3024–3101. MR**2973334**, DOI 10.1016/j.jfa.2012.07.015 - D. Crisan, P. Dobson, and M. Ottobre,
*Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups*, arXiv version. - Dan Crisan, Christian Litterer, and Terry Lyons,
*Kusuoka-Stroock gradient bounds for the solution of the filtering equation*, J. Funct. Anal.**268**(2015), no. 7, 1928–1971. MR**3315583**, DOI 10.1016/j.jfa.2014.12.009 - D. Crisan, K. Manolarakis, and C. Nee,
*Cubature methods and applications*, Paris-Princeton Lectures on Mathematical Finance 2013, Lecture Notes in Math., vol. 2081, Springer, Cham, 2013, pp. 203–316. MR**3183925**, DOI 10.1007/978-3-319-00413-6_{4} - D. Crisan and M. Ottobre,
*Pointwise gradient bounds for degenerate semigroups (of UFG type)*, Proc. A.**472**(2016), no. 2195, 20160442, 23. MR**3592268**, DOI 10.1098/rspa.2016.0442 - P. Dobson.
*A pathwise approach to the Bakry-Emery theory for derivative estimates for Markov Semigroups*, work in progress. - Monroe D. Donsker and SR Srinivasa Varadhan.
*Asymptotic evaluation of certain Markov process expectations for large time, I*, Communications on Pure and Applied Mathematics 28.1 (1975): 1-47. - M. D. Donsker and S. R. S. Varadhan,
*Asymptotic evaluation of certain Markov process expectations for large time, II*, Communications on Pure and Applied Mathematics 28.2 (1975): 279-301. - M. D. Donsker and S. R. S. Varadhan,
*Asymptotic evaluation of certain Markov process expectations for large time. III*, Comm. Pure Appl. Math.**29**(1976), no. 4, 389–461. MR**428471**, DOI 10.1002/cpa.3160290405 - M. D. Donsker and S. R. S. Varadhan,
*Asymptotic evaluation of certain Markov process expectations for large time. IV*, Comm. Pure Appl. Math.**36**(1983), no. 2, 183–212. MR**690656**, DOI 10.1002/cpa.3160360204 - Federica Dragoni, Vasilis Kontis, and Bogusław Zegarliński,
*Ergodicity of Markov semigroups with Hörmander type generators in infinite dimensions*, Potential Anal.**37**(2012), no. 3, 199–227. MR**2969300**, DOI 10.1007/s11118-011-9253-x - Ioannis Karatzas and Steven E. Shreve,
*Brownian motion and stochastic calculus*, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR**1121940**, DOI 10.1007/978-1-4612-0949-2 - Peter E. Kloeden and Eckhard Platen,
*Numerical solution of stochastic differential equations*, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR**1214374**, DOI 10.1007/978-3-662-12616-5 - V. Kontis, M. Ottobre, and B. Zegarlinski,
*Markov semigroups with hypocoercive-type generator in infinite dimensions: ergodicity and smoothing*, J. Funct. Anal.**270**(2016), no. 9, 3173–3223. MR**3475455**, DOI 10.1016/j.jfa.2016.02.005 - H. Kunita,
*Stochastic differential equations and stochastic flows of diffeomorphisms*, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 143–303. MR**876080**, DOI 10.1007/BFb0099433 - Shigeo Kusuoka,
*Malliavin calculus revisited*, J. Math. Sci. Univ. Tokyo**10**(2003), no. 2, 261–277. MR**1987133** - Shigeo Kusuoka and Daniel Stroock,
*Applications of the Malliavin calculus. I*, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 271–306. MR**780762**, DOI 10.1016/S0924-6509(08)70397-0 - S. Kusuoka and D. Stroock,
*Applications of the Malliavin calculus. II*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**32**(1985), no. 1, 1–76. MR**783181** - S. Kusuoka and D. Stroock,
*Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator*, Ann. of Math. (2)**127**(1988), no. 1, 165–189. MR**924675**, DOI 10.2307/1971418 - S. Kusuoka and D. Stroock,
*Applications of the Malliavin calculus. III*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**34**(1987), no. 2, 391–442. MR**914028** - Shigeo Kusuoka,
*Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus*, Advances in mathematical economics. Vol. 6, Adv. Math. Econ., vol. 6, Springer, Tokyo, 2004, pp. 69–83. MR**2079333**, DOI 10.1007/978-4-431-68450-3_{4} - Cónall Kelly and Gabriel J. Lord,
*Adaptive time-stepping strategies for nonlinear stochastic systems*, IMA J. Numer. Anal.**38**(2018), no. 3, 1523–1549. MR**3829168**, DOI 10.1093/imanum/drx036 - Damien Lamberton and Gilles Pagès,
*Recursive computation of the invariant distribution of a diffusion: the case of a weakly mean reverting drift*, Stoch. Dyn.**3**(2003), no. 4, 435–451. MR**2030742**, DOI 10.1142/S0219493703000838 - Alessandra Lunardi,
*On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures*, Trans. Amer. Math. Soc.**349**(1997), no. 1, 155–169. MR**1389786**, DOI 10.1090/S0002-9947-97-01802-3 - J. C. Mattingly, A. M. Stuart, and D. J. Higham,
*Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise*, Stochastic Process. Appl.**101**(2002), no. 2, 185–232. MR**1931266**, DOI 10.1016/S0304-4149(02)00150-3 - C. Nee,
*Sharp gradient bounds for the diffusion semigroup*, PhD Thesis, Imperial College London, 2011. - David Nualart,
*The Malliavin calculus and related topics*, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR**1344217**, DOI 10.1007/978-1-4757-2437-0 - M. Ottobre,
*Asymptotic analysis for Markovian models in non-equilibrium statistical mechanics*, PhD Thesis, Imperial College London, 2012. - Enrico Priola and Feng-Yu Wang,
*Gradient estimates for diffusion semigroups with singular coefficients*, J. Funct. Anal.**236**(2006), no. 1, 244–264. MR**2227134**, DOI 10.1016/j.jfa.2005.12.010 - D. Talay,
*Second-order discretization schemes of stochastic differential systems for the computation of the invariant law*, Stochastics**29**(1990), no. 1, 13–36. - Denis Talay and Luciano Tubaro,
*Expansion of the global error for numerical schemes solving stochastic differential equations*, Stochastic Anal. Appl.**8**(1990), no. 4, 483–509 (1991). MR**1091544**, DOI 10.1080/07362999008809220 - Cédric Villani,
*Hypocoercivity*, Mem. Amer. Math. Soc.**202**(2009), no. 950, iv+141. MR**2562709**, DOI 10.1090/S0065-9266-09-00567-5

## Additional Information

**D. Crisan**- Affiliation: Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
- MR Author ID: 305379
- Email: d.crisan@imperial.ac.uk
**P. Dobson**- Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
- MR Author ID: 1384275
- Email: p.dobson@tudelft.nu
**M. Ottobre**- Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
- MR Author ID: 935458
- Email: m.ottobre@hw.ac.uk
- 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. - © Copyright 2021 American Mathematical Society
- 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
- MathSciNet review: 4237949