Royer, Manuela Backward stochastic differential equations with jumps and related nonlinear expectations. (English) Zbl 1110.60062 Stochastic Processes Appl. 116, No. 10, 1358-1376 (2006). Consider real-valued backward stochastic differential equations with jumps together with their applications to nonlinear expectations, where the underlying filtration is generated by a Brownian motion and a Poisson random measure. The author studies comparison theorems and monotonicity of solutions of those equations. Additivity, inverse theorems, martingale-properties, and decomposition theorems are investigated among further properties of its solutions. The notions of \(f\)-expectations and nonlinear expectations are introduced. This paper can be understood in conjunction with the work of E. Pardoux and S. G. Peng [Syst. Control Lett. 14 , No. 1, 55–61 (1990; Zbl 0692.93064)]. Reviewer: Henri Schurz (Carbondale) Cited in 3 ReviewsCited in 104 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) Keywords:monotonicity; comparison theorems; martingale properties; inverse theorems; decomposition theorems; additivity Citations:Zbl 0692.93064 PDFBibTeX XMLCite \textit{M. Royer}, Stochastic Processes Appl. 116, No. 10, 1358--1376 (2006; Zbl 1110.60062) Full Text: DOI References: [1] Barles, G.; Buckdahn, R.; Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stoch. Stoch. Rep., 60, 1-2, 57-83 (1997) · Zbl 0878.60036 [2] Chen, Z.; Peng, S., Continuous properties of \(g\)-martingales, Chinese Ann. Math. Ser. B, 22, 1, 115-128 (2001) · Zbl 0980.60084 [3] Coquet, F.; Hu, Y.; Mémin, J.; Peng, S., Filtration-consistent nonlinear expectations and related \(g\)-expectations, Probab. Theory Related Fields, 123, 1-27 (2002) · Zbl 1007.60057 [4] Dellacherie, C.; Meyer, P.-A., Probabilités et Potentiel (1975), Hermann: Hermann Paris, (Chapitres I à IV) [5] El Karoui, N.; Kapoudjian, C.; Pardoux, E.; Peng, S.; Quenez, M.-C., Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, Ann. Probab., 25, 2, 702-737 (1997) · Zbl 0899.60047 [6] El Karoui, N.; Peng, S.; Quenez, M.-C., Backward stochastic differential equations in finance, Math. Finance, 7, 1, 1-71 (1997) · Zbl 0884.90035 [7] Elliott, R. J., Stochastic calculus and applications, (Appl. Math., vol. 18 (1982), Springer-Verlag: Springer-Verlag New York) · Zbl 0503.60062 [8] Jacod, J.; Shiryaev, A. N., Limit theorems for stochastic processes, (Grundlheren Math. Wiss., vol. 288 (1987), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York) · Zbl 0830.60025 [9] Pardoux, E., Generalized discontinuous backward stochastic differential equations, (El Karoui, N.; Mazliak, L., Backward Stochastic Differential Equations. Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser., vol. 364 (1997), Longman: Longman Harlow), 207-219 · Zbl 0886.60053 [10] Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14, 1, 55-61 (1990) · Zbl 0692.93064 [11] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37, 1-2, 61-74 (1991) · Zbl 0739.60060 [12] Peng, S., Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27, 2, 125-144 (1993) · Zbl 0769.60054 [13] Peng, S., Backward SDE and related \(g\)-expectation, (El Karoui, N.; Mazliak, L., Backward Stochastic Differential Equations. Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser., vol. 364 (1997), Longman: Longman Harlow), 141-159 · Zbl 0892.60066 [14] Peng, S., Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type, Probab. Theory Related Fields, 113, 4, 23-30 (1999) · Zbl 0953.60059 [15] Tang, S.; Li, X., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32, 5, 1447-1475 (1994) · Zbl 0922.49021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.