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On solutions of backward stochastic differential equations with jumps and applications. (English) Zbl 0890.60049

Existence and uniqueness results are obtained for a backward stochastic differential equation with jumps, with bounded (random) stopping time as a terminal time and with non-Lipschitz coefficients. The coefficients are assumed to be jointly continuous and satisfying the sublinear growth condition and a weaker form of monotonicity condition. Also, a convergence result (a kind of continuous dependence of solutions on coefficients of the equation) is proved. In the second part of the paper, a Feynman-Kac type formula is derived by means of a generalized Itô lemma. By virtue of this formula a probabilistic interpretation of solutions to partial differential and integral equations is found and existence results are given.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Bensoussan, A., Lectures on stochastic control, (Mittler, S. K.; Moro, A., Nonlinear Filtering and Stochastic Control. Nonlinear Filtering and Stochastic Control, Lecture Notes in Mathematics, Vol. 972 (1982), Springer: Springer Berlin), 1-62 · Zbl 0505.93078
[2] Bismut, J. M., Theorie probabiliste du controle des diffusions, Mem Amer Math. Soc., 176, 1-30 (1973)
[3] Krylov, N. V., Controlled Diffusion Processes (1980), Springer: Springer Berlin · Zbl 0459.93002
[4] Ladyzenskaja, O. A.; Solonnikov, V. A.; Uralceva, N. N., Linear and Quasi-linear Equations of Parabolic Type, (Translations of Monographs, 23 (1968), AMS: AMS Providence, RI)
[5] Mao, X., Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process. Appl., 58, 281-292 (1995) · Zbl 0835.60049
[6] Marhno, S., The first boundary problem for integral-differential equations, Theory of Stochastic Processes, 4, 66-72 (1976), (in Russian)
[7] Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems & Control Lett., 14, 55-61 (1990) · Zbl 0692.93064
[8] Pardoux, E.; Peng, S., Backward doubly stochastic differential equations and systems of quasilinear stochastic partial differential equations, Probab. Theory Related Fields, 98, 2 (1994)
[9] Peng, S., A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979 (1990) · Zbl 0712.93067
[10] Peng, S., Probabilistic interpletation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 32, 61-74 (1991) · Zbl 0739.60060
[11] Peng, S., A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics and Stochastics Reports, 38, 119-134 (1992) · Zbl 0756.49015
[12] Peng, S., Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27, 125-144 (1993) · Zbl 0769.60054
[13] Situ, R., On strong solution, uniqueness, stability and comparison theorem for a stochastic system with Poisson jumps, (Lecture Notes in Control & Inform. Sci; vol. 75, Distributed Parameter System (1985), Springer: Springer Berlin), 352-381
[14] Situ, R., Theory and application of stochastic differential equations in China, (Contemporary Math., 118 (1991), Amer. Math. Soci: Amer. Math. Soci Providence, RI), 263-280
[15] Situ, R., A maximum principle for optimal controls of stochastic systems with random jumps, (Proc. National Conf. on Control Theory and its Applications. Proc. National Conf. on Control Theory and its Applications, Qingdao (1991)), 1-7
[16] Situ, R.; Chen, W. L., Existence of solutions and optimal control for reflecting stochastic differential equations with applications to population control theory, Stochastic Anal. Appl., 10, 45-106 (1992) · Zbl 0755.93084
[17] Tang, S.; Li, X., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32, 1447-1475 (1994) · Zbl 0922.49021
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