Generalized BSDEs driven by RCLL martingales with stochastic monotone coefficients
Volume 11, Issue 1 (2024), pp. 109–128
Pub. online: 5 December 2023
Type: Research Article
Open Access
Open Access
Received
4 July 2023
4 July 2023
Revised
4 November 2023
4 November 2023
Accepted
22 November 2023
22 November 2023
Published
5 December 2023
5 December 2023
Abstract
A solution is given to generalized backward stochastic differential equations driven by a real-valued RCLL martingale on an arbitrary filtered probability space. The existence and uniqueness of a solution are proved via the Yosida approximation method when the generators are only stochastic monotone with respect to the y-variable and stochastic Lipschitz with respect to the z-variable, with different linear growth conditions.
References
Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Int. J. Probab. Stoch. Process. 60(1-2), 57–83 (1997). MR1436432. https://doi.org/10.1080/17442509708834099
Bender, C., Kohlmann, M.: BSDEs with stochastic Lipschitz condition. Technical report, CoFE Discussion Paper (2000). https://www.econstor.eu/bitstream/10419/85163/1/dp00-08.pdf
Bismut, J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44(2), 384–404 (1973). MR0329726. https://doi.org/10.1016/0022-247X(73)90066-8
Carbone, R., Ferrario, B., Santacroce, M.: Backward stochastic differential equations driven by càdlàg martingales. Theory Probab. Appl. 52(2), 304–314 (2008). MR2742510. https://doi.org/10.1137/S0040585X97983055
El Karoui, N., Huang, S.: A general result of existence and uniqueness of backward stochastic differential equations. In: Backward Stochastic Differential Equations (Paris, 1995–1996). Pitman Research Notes in Mathematics Series, vol. 364, pp. 27–38 (1997). MR1752673
El Otmani, M.: Generalized BSDE driven by a Lévy process. J. Appl. Math. Stoch. Anal. 2006, 85407 (2006). MR2253532. https://doi.org/10.1155/JAMSA/2006/85407
Hu, Y.: On the solution of forward–backward sdes with monotone and continuous coefficients. Nonlinear Anal. 42(1), 1–12 (2000). MR1769248. https://doi.org/10.1016/S0362-546X(98)00315-0
Hu, Y., Peng, S.: Solution of forward-backward stochastic differential equations. Probab. Theory Relat. Fields 103, 273–283 (1995). MR1355060. https://doi.org/10.1007/BF01204218
Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, vol. 288. Springer, Berlin (2013). MR0959133. https://doi.org/10.1007/978-3-662-02514-7
Nie, T., Rutkowski, M.: Bsdes driven by multidimensional martingales and their applications to markets with funding costs. Theory Probab. Appl. 60(4), 604–630 (2016). MR3583450. https://doi.org/10.1137/S0040585X97T987880
Nie, T., Rutkowski, M.: Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales. Probab. Uncertain. Quant. Risk 6(4), 319–342 (2021). MR4399893. https://doi.org/10.3934/puqr.2021016
Pardoux, E.: Generalized discontinuous backward stochastic differential equations. In: Backward Stochastic Differential Equations (Paris, 1995–1996). Pitman Research Notes in Mathematics Series, vol. 364, pp. 207–219 (1997). MR1752684. https://doi.org/10.1016/S0377-0427(97)00124-6
Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990). MR1037747. https://doi.org/10.1016/0167-6911(90)90082-6
Pardoux, E., Râs,canu, A.: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, vol. 69. Springer, Switzerland (2014). MR3308895. https://doi.org/10.1007/978-3-319-05714-9
Pardoux, E., Zhang, S.: Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields 110, 535–558 (1998). MR1626963. https://doi.org/10.1007/s004400050158
Protter, P.E.: Stochastic Integration and Differential Equations. Applications of Mathematics, 2nd edn. Springer, Berlin (2004). MR2020294
Situ, R.: On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66(2), 209–236 (1997). MR1440399. https://doi.org/10.1016/S0304-4149(96)00120-2
Tang, S., Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994). MR1288257. https://doi.org/10.1137/S0363012992233858
