A Note on Existence of Global Solutions and Invariant Measures for Jump Sdes with Locally One-Sided Lipschitz Drift

• Autorzy: Majka Mateusz B.

Identyfikatory

DOI

10.37190/0208-4147.40.1.3

• Języki publikacji: EN

Abstrakty

EN

We extend some methods developed by Albeverio, Brzeźniak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.

Słowa kluczowe

EN

stochastic differential equations; invariant measures; jump processes

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2020
• Tom: Vol. 40, Fasc. 1
• Strony: 37--55
• Opis fizyczny: Bibliogr. 20 poz.

Twórcy

autor

Majka Mateusz B.

• Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Bibliografia

• [1] S. Albeverio, Z. Brzeźniak and J.-L. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, J. Math. Anal. Appl. 371 (2010), 309-322.
• [2] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Univ. Press, 2009.
• [3] A. Arapostathis, A. Biswas and L. Caffarelli, On a class of stochastic differential equations with jumps and its properties, arXiv:1401.6198v4 (2015).
• [4] K. Bichteler, J.-B. Gravereaux and J. Jacod, Malliavin Calculus for Processes with Jumps, Stochastics Monographs, 2. Gordon and Breach, New York, 1987.
• [5] Z. Brzeźniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl. 17 (2014), 283-310.
• [6] G. Da Prato, D. Gątarek and J. Zabczyk, Invariant measures for semilinear stochastic equations, Stochastic Anal. Appl. 10 (1992), 387-408.
• [7] I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics 4 (1980/81), 1-21.
• [8] R. Z. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.
• [9] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland and Kodansha, 1989.
• [10] T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl. 122 (2012), 2155-2184.
• [11] N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, J. Soviet Math. 16 (1981), 1233-1277.
• [12] W. Liu and J. M. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts, Electron. Comm. Probab. 16 (2011), 447-457.
• [13] Y. Ma, Transportation inequalities for stochastic differential equations with jumps, Stochastic Process. Appl. 120 (2010), 2-21.
• [14] M. B. Majka, Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling, Ann. Inst. Henri Poincaré Probab. Statist. 55 (2019), 2019-2057.
• [15] J.-L. Menaldi and M. Robin, Invariant measure for diffusions with jumps, Appl. Math. Optim. 40 (1999), 105-140.
• [16] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007.
• [17] J. Shao and C. Yuan, Transportation-cost inequalities for diffusions with jumps and its application to regime-switching processes, J. Math. Anal. Appl. 425 (2015), 632-654.
• [18] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Springer, New York, 2005.
• [19] J. Wang, Lp-Wasserstein distance for stochastic differential equations driven by Lévy processes, Bernoulli 22 (2016), 1598-1616.
• [20] L. Wu, Transportation inequalities for stochastic differential equations of pure jumps, Ann. Inst. Henri Poincaré Probab. Statist. 46 (2010), 465-479.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).