Reflected BSDEs with jumps in time-dependent convex càdlàg domains. (English) Zbl 1454.60076
Summary: In the first part of the paper, we study the unique solvability of multidimensional reflected backward stochastic differential equations (RBSDEs) of Wiener-Poisson type with reflection in the inward spatial normal direction of a time-dependent adapted càdlàg convex set \(\mathcal{D}=\{D_t,t\in[0,T]\}\). The existence result is obtained by approximating the solutions of this class of RBSDEs by solutions of BSDEs with reflection in discretizations of \(\mathcal{D}\), while the uniqueness is established by using Itô’s formula. In the second part of the paper, we show that the solutions of our RBSDEs can be approximated via a non-standard penalization method.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H20 | Stochastic integral equations |
Keywords:
reflected backward stochastic differential equation; Poisson point process; time-dependent convex domain; penalization methodReferences:
| [1] | Aazizi, S.; El Mellali, T.; Fakhouri, I.; Ouknine, Y., Optimal switching problem and related system of BSDEs with left-Lipschitz coefficients and mixed reflections, Statist. Probab. Lett., 137, 70-78 (2018) · Zbl 1396.60043 |
| [2] | Aazizi, S.; Fakhouri, I., Optimal switching problem and system of reflected multi-dimensional FBSDEs with random terminal time, Bull. Sci. Math., 137, 4, 523-540 (2013) · Zbl 1281.60052 |
| [3] | Cvitanic, J.; Karatzas, I., Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24, 2024-2056 (1996) · Zbl 0876.60031 |
| [4] | Djehiche, B.; Hamadene, S.; Popier, A., A fnite horizon optimal multiple switching problem, SIAM J. Control Opt., 48, 2751-2770 (2010) · Zbl 1196.60069 |
| [5] | El Asri, B.; Fakhouri, I., Viscosity solutions for a system of PDEs and optimal switching, IMA J. Math. Control Inform., 34, 3, 937-960 (2017) · Zbl 1417.93154 |
| [6] | El Asri, B.; Hamadene, S., The fnite horizon optimal multi-modes switching problem: The viscosity solution approach, Appl. Math. Opt., 60, 213-235 (2009) · Zbl 1179.49006 |
| [7] | El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M.-C., Reflected solutions of backward SDEs and related obstacle problems for PDEs, Ann. Probab., 25, 702-737 (1997) · Zbl 0899.60047 |
| [8] | El Karoui, N.; Pardoux, É.; Quenez, M.-C., Reflected backward SDEs and American options, (Rogers, L. C.G.; Talay, D., Numerical Methods in Finance (1997), Cambridge University Press: Cambridge University Press Cambridg), 215-231 · Zbl 0898.90033 |
| [9] | Essaky, E. H., Reflected backward stochastic differential equation with jumps and RCLL obstacle, Bull. Sci. marth., 132, 690-710 (2008) · Zbl 1157.60057 |
| [10] | Essaky, E.-H.; Hassani, M., Generalised BSDE with two reflecting barriers and stochastic quadratic growth, J. Differential Equations, 254, 1500-1528 (2013) · Zbl 1278.60092 |
| [11] | Fakhouri, I.; Ouknine, Y., \( \mathbb{L}^2\)-Solutions for reflected BSDEs with jumps under monotonicity and general growth conditions: A penalization method, SeMA, 76, 37-63 (2019) · Zbl 1420.60067 |
| [12] | Fakhouri, I.; Ouknine, Y.; Ren, Y., Reflected backward stochastic differential equations with jumps in time-dependent random convex domains, Stochastics, 90, 2, 256-296 (2018) · Zbl 1492.60165 |
| [13] | Gegout-Petit, A.; Pardoux, É., Equations différentielles stochastiques rétrogrades réfléchies dans un convexe, Stoch. Stoch. Rep., 57, 111-128 (1996) · Zbl 0891.60050 |
| [14] | Hamadène, S., Reflected BSDEs with discontinuous barriers and application, Stoch. Stoch. Rep., 74, 3-4, 571-596 (2002) · Zbl 1015.60057 |
| [15] | Hamadène, S.; Hassani, M., BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game, Electron. J. Probab., 11, 121-145 (2006) · Zbl 1184.91038 |
| [16] | Hamadène, S.; Ouknine, Y., Reflected backward stochastic differential equations with jumps and random obstacle, Electron. J. Probab., 8, 2, 1-20 (2003) · Zbl 1015.60051 |
| [17] | Hamadène, S.; Ouknine, Y., Reflected backward SDEs with general jumps, Theory Probab. Appl., 60, 263-280 (2016) · Zbl 1341.60054 |
| [18] | Hamadène, S.; Wang, H., BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure and a related mixed zero-sum game, Stochastic Process. Appl., 119, 9, 2881-2912 (2009) · Zbl 1229.60083 |
| [19] | Hamadène, S.; Zhang, J., Switching problem and related system of reflected backward SDEs, Stochastic Process. Appl., 120, 4, 403-426 (2010) · Zbl 1191.60056 |
| [20] | Hu, Y.; Tang, S., Multidimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147, 89-121 (2010) · Zbl 1188.60029 |
| [21] | Klimsiak, T., Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form, Stochastic Process. Appl., 122, 1, 134-169 (2012) · Zbl 1242.60070 |
| [22] | Klimsiak, T., BSDEs with monotone generator and two irregular reflecting barriers, Bull. Sci. Math., 137, 3, 268-321 (2013) · Zbl 1311.60064 |
| [23] | Klimsiak, T., Reflected BSDEs on filtered probability spaces, Stoch. Process. Appl., 125, 4204-4241 (2015) · Zbl 1323.60079 |
| [24] | Klimsiak, T.; Rozkosz, A.; Slomiński, L., Reflected BSDEs in time-dependent convex regions, Stochastic Process. Appl., 125, 571-596 (2015) · Zbl 1322.60089 |
| [25] | Kruse, T.; Popier, A., BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics, 88, 4, 491-539 (2016) · Zbl 1337.60123 |
| [26] | Menaldi, J. L., Stochastic variational inequality for reflected diffusion, Indiana Univ. Math. J., 32, 733-744 (1983) · Zbl 0492.60057 |
| [27] | Nystrom, K.; Olofsson, M., Reflected BSDE of wiener-Poisson type in time-dependent domains, Stoch. Models, 32, 2, 275-300 (2016) · Zbl 1339.60075 |
| [28] | Ouknine, Y., Reflected backward stochastic differential equations with jumps, Stoch. Stoch. Rep., 65, 111-125 (1998) · Zbl 0918.60046 |
| [29] | Ph. Protter, Y., Stochastic Integration and Differential Equations (2005), Springer-Verlag: Springer-Verlag Berlin Heidelberg |
| [30] | Ramasubramanian, S., Reflected backward stochastic differetial equations in an orthant, Proc. Indian Acad. Sci., 112, 347-360 (2002) · Zbl 1007.60061 |
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