Doubly reflected BSDEs with quadratic growth and random terminal horizon

Abstract

In this paper, we study one-dimensional backward stochastic differential equations featuring two reflecting barriers. When the terminal time is not necessarily bounded or finite and the generator $f(t, y, z)$ exhibits quadratic growth in $z$, we prove existence and uniqueness of solutions. In the Markovian case, we establish the link with an obstacle problem for quadratic elliptic partial differential equation with Dirichlet boundary conditions.

Keywords

Double reflected BSDE / Random terminal time / Quadratic growth / Elliptic partial differential equations / Dirichlet boundary conditions

References

[1] Bahlali, K., Eddahbi, M. and Ouknine, Y., Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô’s-Krylov’s formula and existence results, The Annals of Probability, 2017, 45(4): 2377-2397.

[2] Bahlali, K., Hamadène, S. and Mezerdi, B., Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stochastic Processes and their Applications, 2005, 115(7): 1107-1129.

[3] Bally, V. and Matoussi, A., Weak solutions for SPDEs and backward doubly stochastic differential equations, Journal of Theoretical Probability, 2001, 14: 125-164.

[4] Barles, G., Buckdahn, R. and Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stochastics, 1997, 60(1-2): 57-83.

[5] Barles, G. and Lesigne, E., BSDE and PDE, Pitman Research Notes in Mathematics Series, 1997: 47-82.

[6] Beznea, L. and Cîmpean, I., Quasimartingales associated to Markov processes, Transactions of the American Mathematical Society, 2018, 370(11): 7761-7787.

[7] Briand, P. and Hu, Y., Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, Journal of Functional Analysis, 1998, 155(2): 455-494.

[8] Chung, K. L., A Course in Probability Theory, Academic Press, 3rd edn., 2001.

[9] Confortola, F. and Briand, P., Quadratic BSDEs with random terminal time and elliptic pdes in infinite dimension, Electronic Journal of Probability, 2008, 13: 1529-1561.

[10] Crandall, M. G. and Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 1983, 277(1): 1-42.

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

[12] Darling, R. W. and Pardoux, E., Backwards SDE with random terminal time and applications to semilinear elliptic PDE, The Annals of Probability, 1997, 25(3): 1135-1159.

[13] Dellacherie, C. and Meyer, P.-A., Probabilités et Potentiel, Hermann, 1975.

[14] 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.

[15] Essaky, E. and Hassani, M., Generalized BSDE with 2-reflecting barriers and stochastic quadratic growth, Journal of Differential Equations, 2013, 254(3): 1500-1528.

[16] Essaky, E., Hassani, M. and Rhazlane, C., Doubly reflected BSDEs with stochastic quadratic growth: Around the predictable obstacles, Stochastic Processes and their Applications, 2023, 163: 473-497.

[17] Hamadène, S. and Hassani, M., BSDEs with two reflecting barriers: The general result, Probability Theory and Related Fields, 2005, 132: 237-264.

[18] Hamadène, S. and Hdhiri, I., Backward stochastic differential equations with two distinct reflecting barriers and quadratic growth generator, Journal of Applied Mathematics and Stochastic Analysis, 2006, 2006: 095818.

[19] Hamadène, S., Lepeltier, J. P. and Matoussi, A., Double Barrier Backward SDEs with Continuous Coefficient, Pitman Research Notes in Mathematics Series, 1997: 161-176.

[20] Kazamaki, N., Continuous Exponential Martingales and BMO, Springer, 2006.

[21] Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 2000, 28(2): 558-602.

[22] Lepeltier, J.-P. and San Martín, J., Backward SDEs with two barriers and continuous coefficient: An existence result, Journal of Applied Probability, 2004, 41(1): 162-175.

[23] Madoui, I., Eddahbi, M. and Khelfallah, N., Quadratic BSDEs with jumps and related pides, Stochastics, 2022, 94(3): 386-414.

[24] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems & 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, Springer, Berlin, Heidelberg, 2002, 176: 200-217.

[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] Pardoux, E. and Răşcanu, A., Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability 69, 1st edn., Springer, 2014.

[28] Pardoux, E. and Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probability Theory and Related Fields, 1999, 114: 123-150.

[29] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 1991, 37(1-2): 61-74.

[30] Protter, P. E., Stochastic Integration and Differential Equations, 2nd edn., Springer, 2004.

[31] Royer, M., BSDEs with a random terminal time driven by a monotone generator and their links with PDEs, Stochastics and Stochastic Reports, 2004, 76(4): 281-307.

[32] Saisho, Y., Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probability Theory and Related Fields, 1987, 74(3): 455-477.

[33] Yosida, K., Functional Analysis, Grundlehren der Mathematischen Wissenschaften, 6th edn., Springer-Verlag, 1980.