Quadratic transportation inequalities for SDEs with measurable drift

Bahlali, Khaled; Mouchtabih, Soufiane; Tangpi, Ludovic

doi:10.1090/proc/15477

Quadratic transportation inequalities for SDEs with measurable drift
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by Khaled Bahlali, Soufiane Mouchtabih and Ludovic Tangpi PDF

Proc. Amer. Math. Soc. 149 (2021), 3583-3596 Request permission

Abstract:

Let $X$ be the solution of a stochastic differential equation in Euclidean space driven by standard Brownian motion, with measurable drift and Sobolev diffusion coefficient. In our main result we show that when the drift is measurable and the diffusion coefficient belongs to an appropriate Sobolev space, the law of $X$ satisfies Talagrand’s inequality with respect to the uniform distance.

References

Khaled Bahlali, Flows of homeomorphisms of stochastic differential equations with measurable drift, Stochastics Stochastics Rep. 67 (1999), no. 1-2, 53–82. MR 1717807, DOI 10.1080/17442509908834203
Adrian D. Banner, Robert Fernholz, and Ioannis Karatzas, Atlas models of equity markets, Ann. Appl. Probab. 15 (2005), no. 4, 2296–2330. MR 2187296, DOI 10.1214/105051605000000449
Daniel Bartl and Ludovic Tangpi, Functional inequalities for forward and backward diffusions, Electron. J. Probab. 25 (2020), Paper No. 94, 22. MR 4136474, DOI 10.1214/20-ejp495
Sergey G. Bobkov, Ivan Gentil, and Michel Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9) 80 (2001), no. 7, 669–696. MR 1846020, DOI 10.1016/S0021-7824(01)01208-9
Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration inequalities, Oxford University Press, Oxford, 2013. A nonasymptotic theory of independence; With a foreword by Michel Ledoux. MR 3185193, DOI 10.1093/acprof:oso/9780199535255.001.0001
Gerard Brunick and Steven Shreve, Mimicking an Itô process by a solution of a stochastic differential equation, Ann. Appl. Probab. 23 (2013), no. 4, 1584–1628. MR 3098443, DOI 10.1214/12-aap881
P. Cattiaux and A. Guillin, Semi log-concave Markov diffusions, Séminaire de Probabilités XLVI, Lecture Notes in Math., vol. 2123, Springer, Cham, 2014, pp. 231–292. MR 3330820, DOI 10.1007/978-3-319-11970-0_{9}
Dan Crisan, Thomas G. Kurtz, and Yoonjung Lee, Conditional distributions, exchangeable particle systems, and stochastic partial differential equations, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 3, 946–974 (English, with English and French summaries). MR 3224295, DOI 10.1214/13-AIHP543
François Delarue, Daniel Lacker, and Kavita Ramanan, From the master equation to mean field game limit theory: large deviations and concentration of measure, Ann. Probab. 48 (2020), no. 1, 211–263. MR 4079435, DOI 10.1214/19-AOP1359
H. Djellout, A. Guillin, and L. Wu, Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab. 32 (2004), no. 3B, 2702–2732. MR 2078555, DOI 10.1214/009117904000000531
Devdatt P. Dubhashi and Alessandro Panconesi, Concentration of measure for the analysis of randomized algorithms, Cambridge University Press, Cambridge, 2009. MR 2547432, DOI 10.1017/CBO9780511581274
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
E. Robert Fernholz, Stochastic portfolio theory, Applications of Mathematics (New York), vol. 48, Springer-Verlag, New York, 2002. Stochastic Modelling and Applied Probability. MR 1894767, DOI 10.1007/978-1-4757-3699-1
D. Feyel and A. S. Üstünel, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space, Probab. Theory Related Fields 128 (2004), no. 3, 347–385. MR 2036490, DOI 10.1007/s00440-003-0307-x
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
Fritz John, On quasi-isometric mappings. I, Comm. Pure Appl. Math. 21 (1968), 77–110. MR 222666, DOI 10.1002/cpa.3160210107
N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2005), no. 2, 154–196. MR 2117951, DOI 10.1007/s00440-004-0361-z
N. V. Krylov, Controlled diffusion processes, Stochastic Modelling and Applied Probability, vol. 14, Springer-Verlag, Berlin, 2009. Translated from the 1977 Russian original by A. B. Aries; Reprint of the 1980 edition. MR 2723141
Daniel Lacker, Liquidity, risk measures, and concentration of measure, Math. Oper. Res. 43 (2018), no. 3, 813–837. MR 3846074, DOI 10.1287/moor.2017.0885
Laurière, M. and L. Tangpi. Backward propagation of chaos, Preprint, 2019.
Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347, DOI 10.1090/surv/089
K. Marton, Bounding $\overline d$-distance by informational divergence: a method to prove measure concentration, Ann. Probab. 24 (1996), no. 2, 857–866. MR 1404531, DOI 10.1214/aop/1039639365
Pascal Massart, Concentration inequalities and model selection, Lecture Notes in Mathematics, vol. 1896, Springer, Berlin, 2007. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003; With a foreword by Jean Picard. MR 2319879
V. D. Milman, Asymptotic properties of functions of several variables that are defined on homogeneous spaces, Dokl. Akad. Nauk SSSR 199 (1971), 1247–1250 (Russian); English transl., Soviet Math. Dokl. 12 (1971), 1277–1281. MR 0303566
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361–400. MR 1760620, DOI 10.1006/jfan.1999.3557
Soumik Pal, Concentration for multidimensional diffusions and their boundary local times, Probab. Theory Related Fields 154 (2012), no. 1-2, 225–254. MR 2981423, DOI 10.1007/s00440-011-0368-1
Soumik Pal and Andrey Sarantsev, A note on transportation cost inequalities for diffusions with reflections, Electron. Commun. Probab. 24 (2019), Paper No. 21, 11. MR 3940196, DOI 10.1214/19-ECP223
Soumik Pal and Mykhaylo Shkolnikov, Concentration of measure for Brownian particle systems interacting through their ranks, Ann. Appl. Probab. 24 (2014), no. 4, 1482–1508. MR 3211002, DOI 10.1214/13-AAP954
Sebastian Riedel, Transportation-cost inequalities for diffusions driven by Gaussian processes, Electron. J. Probab. 22 (2017), Paper No. 24, 26. MR 3622894, DOI 10.1214/17-EJP40
Bruno Saussereau, Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion, Bernoulli 18 (2012), no. 1, 1–23. MR 2888696, DOI 10.3150/10-BEJ324
M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6 (1996), no. 3, 587–600. MR 1392331, DOI 10.1007/BF02249265
Ludovic Tangpi, Concentration of dynamic risk measures in a Brownian filtration, Stochastic Process. Appl. 129 (2019), no. 5, 1477–1491. MR 3944773, DOI 10.1016/j.spa.2018.05.008
Ali Suleyman Üstünel, Transportation cost inequalities for diffusions under uniform distance, Stochastic analysis and related topics, Springer Proc. Math. Stat., vol. 22, Springer, Heidelberg, 2012, pp. 203–214. MR 3236093, DOI 10.1007/978-3-642-29982-7_{9}
A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480 (Russian). MR 568986
Xicheng Zhang, Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients, Electron. J. Probab. 16 (2011), no. 38, 1096–1116. MR 2820071, DOI 10.1214/EJP.v16-887
A. K. Zvonkin, A transformation of the phase space of a diffusion process that will remove the drift, Mat. Sb. (N.S.) 93(135) (1974), 129–149, 152 (Russian). MR 0336813, DOI 10.1070/SM1974v022n01ABEH001689