Averaging principles for forward-backward multivalued stochastic systems and applications to systems of nonlinear parabolic partial differential equations

Abstract

This work concerns a type of stochastic systems in which the forward equations are general stochastic differential equations and the backward equations are stochastic variational inequalities. We first prove an averaging principle for general stochastic differential equations in the 𝐿^2𝑝 (𝑝≥1) sense. In addition, a convergence rate for 𝑝=1 is presented. Combining general stochastic differential equations with backward stochastic variational inequalities, we then establish another averaging principle for backward stochastic variational inequalities in the 𝐿² sense using a time discretization method. Finally, we apply our result to nonlinear parabolic partial differential equations to obtain their averaging principles.

Keywords

Averaging principles / Backward stochastic variational inequalities / Averaging principles for nonlinear parabolic partial differential equations

Acknowledgements

This work was supported by the NSF of China (Grant No.12071071) and the Jiangsu Provincial Scientific Research Center of Applied Mathematics (Grant No. BK20233002).

The author is very grateful to Professor Ying Hu for valuable discussions. Besides, the author thanks two reviewers since their suggestions and comments allowed her to improve the results and the presentation of this paper.

References

[1] Bensoussan, A., Lions, J.-L. and Papanicolaou, G., Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam, 1978.

[2] Brézis, H., Operateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, North Holland Mathematics Studies, Vol. 5, North-Holland Publishing Co., Amsterdam, 1973.

[3] Buckdahn, R. and Hu, Y., Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures, Nonlinear Analysis: Theory, Methods & Applications, 1998, 32(5): 609-619.

[4] Buckdahn, R., Hu, Y. and Peng, S., Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs, Nonlinear Differential Equations and Applications NoDEA, 1999, 6: 395-411.

[5] Buckdahn, R. and Ichihara, N., Limit theorem for controlled backward SDEs and homogenization of Hamilton-Jacobi-Bellman equations, Applied Mathematics and Optimization, 2005, 51: 1-33.

[6] Cépa, E., Équations différentielles stochastiques multivoques, In: AzémaJ., EmeryM., MeyerP. A. and Yor, M. (eds.) Séminaire De Probabilités XXIX, Lecture Notes in Mathematics, Vol. 1613, Springer, Berlin, Heidelberg, 1995.

[7] Cépa, E., Probleme de Skorohod multivoque, The Annals of Probability, 1998, 26(2): 500-532.

[8] Chassagneux, J.-F., Nadtochiy, S. and Richou, A., Reflected BSDEs in non-convex domains, Probability Theory and Related Fields, 2022, 183: 1237-1284.

[9] Cvitanic, J. and Karatzas, I., Backward stochastic differential equations with reflection and Dynkin games, The Annals of Probability, 1996, 24(4): 2024-2056.

[10] Delarue, F., Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients, The Annals of Probability, 2004, 32(3B): 2305-2361.

[11] Fakhouri, I., Ouknine, Y. and Ren, Y., Reflected backward stochastic differential equations with jumps in time-dependent random convex domains, Stochastics, 2018, 90(2): 256-296.

[12] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C., Reflected solutions of backward SDE’s and related obstacle problems for PDE’s, The Annals of Probability, 1997, 25(2): 702-737.

[13] Essaky, E. H. and Ouknine, Y., Homogenization of multivalued partial differential equations via reflected backward stochastic differential equations, Stochastic Analysis and Applications, 2004, 22(1): 81-98.

[14] Gegout-Petit, A. and Pardoux, E., Equations différentielles stochastiques rétrogrades réfléchies dans un convexe, Stochastics and Stochastic Reports, 1996, 57(1-2): 111-128.

[15] Guo, Z., Xu, Y., Wang, W. and Hu, J., Averaging principle for stochastic differential equations with monotone condition, Applied Mathematics Letters, 2022, 125: 107705

[16] Hu, M., Jiang, L. and Wang, F., An averaging principle for nonlinear parabolic PDEs via FBSDEs driven by G-Brownian motion, Journal of Mathematical Analysis and Applications, 2022, 508(2): 125893

[17] Hu, Y. and Peng, S., A stability theorem of backward stochastic differential equations and its application, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 1997, 324(9): 1059-1064.

[18] Huang, Z., Basis of Stochastic Analysis, 2nd edn., Science Press, Beijing, 2001. (in Chinese)

[19] Klimsiak, T., Rozkosz, A. and Słomiński, L., Reflected BSDEs in time-dependent convex regions, Stochastic Processes and their Applications, 2015, 125(2): 571-596.

[20] Maticiuc, L., Pardoux, E., Răşcanu, A. and Zălinescu, A., Viscosity solutions for systems of parabolic variational inequalities, Bernoulli, 2010, 16(1): 258-273.

[21] N’Goran, L., and N’Zi, M., Averaging principle for multivalued stochastic differential equations, Random Operators and Stochastic Equations, 2001, 9: 399-407.

[22] N’Zi, M., and Ouknine, Y., Equations differentielles stochastiques retrogrades multivoques, Probability and Mathematical Statistics, 1997, 17(2): 259-275.

[23] Pardoux, E., Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: A probabilistic approach, Journal of Functional Analysis, 1999, 167(2): 498-520.

[24] Pardoux, E. and Peng, S., Adapted solutions of backward stochastic equations, System and Control Letters, 1990, 14(1): 55-61.

[25] Pardoux, E. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, In: RozovskiiB. L. and SowersR. B. (eds.), Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Information Sciences, Vol. 176, Springer, Berlin, Heidelberg, 1992.

[26] Pardoux, E. and Răşcanu, A., Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Processes and their Applications, 1998, 76(2): 191-215.

[27] Qiao, H., Infinite horizon BSDEs with dissipative coefficients in Hilbert spaces and applications, Journal of Mathematical Analysis and Applications, 2009, 355(2): 725-738.

[28] Qiao, H. and Gong, J., Backward multivalued McKean-Vlasov SDEs and associated variational inequalities, Discrete and Continuous Dynamical Systems-S, 2023, 16(5): 819-845.

[29] Shen, G., Song, J. and Wu, J. -L., Stochastic averaging principle for distribution dependent stochastic differential equations, Applied Mathematics Letters, 2002, 125: 107761

[30] Xu, J. and Liu, J., An averaging principle for multivalued stochastic differential equations, Stochastic Analysis and Applications, 2014, 32(6): 962-974.

[31] Zhang, X., Skorohod problem and multivalued stochastic evolution equations in Banach spaces, Bulletin Des Sciences Mathématiques, 2007, 131(2): 175-217.