×

zbMATH — the first resource for mathematics

Sub-exponential convergence to equilibrium for Gaussian driven stochastic differential equations with semi-contractive drift. (English) Zbl 1462.60046
Summary: The convergence to the stationary regime is studied for stochastic differential equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.
MSC:
60G15 Gaussian processes
60G22 Fractional processes, including fractional Brownian motion
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Vladimir I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. · Zbl 0913.60035
[2] Patrick Cheridito, Gaussian moving averages, semimartingales and option pricing, Stochastic Process. Appl. 109 (2004), no. 1, 47-68. · Zbl 1075.60025
[3] Alexander Cherny, Brownian moving averages have conditional full support, Ann. Appl. Probab. 18 (2008), no. 5, 1825-1830. · Zbl 1151.91490
[4] Serge Cohen and Fabien Panloup, Approximation of stationary solutions of Gaussian driven stochastic differential equations, Stochastic Process. Appl. 121 (2011), no. 12, 2776-2801. · Zbl 1228.60076
[5] Fabienne Comte and Nicolas Marie, Nonparametric estimation in fractional SDE, Stat. Inference Stoch. Process. 22 (2019), no. 3, 359-382. · Zbl 1433.62097
[6] Harald Cramér, On some classes of nonstationary stochastic processes, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 57-78. · Zbl 0931.41017
[7] Aurélien Deya, Fabien Panloup, and Samy Tindel, Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise, Ann. Probab. 47 (2019), no. 1, 464-518. · Zbl 1440.60028
[8] Joaquin Fontbona and Fabien Panloup, Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 2, 503-538. · Zbl 1367.60067
[9] E. G. Gladyšev, On multi-dimensional stationary random processes, Teor. Veroyatnost. i Primenen. 3 (1958), 458-462.
[10] Martin Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab. 33 (2005), no. 2, 703-758. · Zbl 1071.60045
[11] Martin Hairer and Arturo Ohashi, Ergodic theory for SDEs with extrinsic memory, Ann. Probab. 35 (2007), no. 5, 1950-1977. · Zbl 1129.60052
[12] Martin Hairer and Natesh S. Pillai, Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths, Ann. Probab. 41 (2013), no. 4, 2544-2598. · Zbl 1288.60068
[13] Yaozhong Hu, David Nualart, and Hongjuan Zhou, Drift parameter estimation for nonlinear stochastic differential equations driven by fractional Brownian motion, Stochastics 91 (2019), no. 8, 1067-1091.
[14] Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. · Zbl 0887.60009
[15] Michel Ledoux, Isoperimetry and Gaussian analysis, Lectures on probability theory and statistics (Saint-Flour, 1994), Lecture Notes in Math., vol. 1648, Springer, Berlin, 1996, pp. 165-294. · Zbl 0874.60005
[16] Wenbo V. Li and Qi-Man Shao, Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, Handbook of Statist., vol. 19, North-Holland, Amsterdam, 2001, pp. 533-597. · Zbl 0987.60053
[17] P. Masani, On helixes in Hilbert space. I, Teor. Verojatnost. i Primenen. 17 (1972), 3-20. · Zbl 0283.60032
[18] Sean Meyn and Richard L. Tweedie, Markov chains and stochastic stability, second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W. Glynn. · Zbl 1165.60001
[19] Dragoslav S. Mitrinovic and Jovan D. Keckic, The Cauchy method of residues. Vol. 2, Mathematics and its Applications, vol. 259, Kluwer Academic Publishers Group, Dordrecht, 1993, Theory and applications. · Zbl 0786.30001
[20] Andreas Neuenkirch and Samy Tindel, A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise, Stat. Inference Stoch. Process. 17 (2014), no. 1, 99-120. · Zbl 1333.62199
[21] Michel Talagrand, New Gaussian estimates for enlarged balls, Geom. Funct. Anal. 3 (1993), no. 5, 502-526. · Zbl 0815.46021
[22] Alexandre B. Tsybakov, Introduction to nonparametric estimation, Springer Series in Statistics, Springer, New York, 2009, Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. · Zbl 1176.62032
[23] Maylis Varvenne, Rate of convergence to equilibrium for discrete-time stochastic dynamics with memory, Bernoulli 25 (2019), no. · Zbl 1431.62422
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.