A limit theorem for solutions to BSDEs in the space of processes

Abstract

In this paper, under the most elementary conditions on stochastic differential equations (SDEs in short) and the most elementary conditions on backward stochastic differential equations (BSDEs in short) introduced by Peng, in the space of processes, a limit theorem for solutions to BSDEs with its terminal data being solutions of the SDEs is obtained, based on some recent results of Jiang in the space of random variables in [Jiang, L., 2005a. Converse comparison theorems for backward stochastic differential equations. Statist. Probab. Lett. 71, 173–183; Jiang, L., 2005b. Representation theorems for generators of backward stochastic differential equations. C.R. Acad. Sci. Paris 340 (Ser. I), 161–166; Jiang, L., 2005c. Representation theorems for generators of backward stochastic differential equations and their applications. Stochastic Process. Appl. 115 (12), 1883–1903; Jiang, L., 2005d. Nonlinear expectation— g -expectation theory and its applications in finance. Ph.D Thesis, ShanDong University, China; Jiang, L., 2006. Limit theorem and uniqueness theorem for backward stochastic differential equations. Sci. China Ser. A 49 (10), 1353–1362]. This result generalizes the known results on the limit theorem for solutions to BSDEs in [Jiang, L., 2005a. Converse comparison theorems for backward stochastic differential equations. Statist. Probab. Lett. 71, 173–183; Jiang, L., 2005b. Representation theorems for generators of backward stochastic differential equations. C.R. Acad. Sci. Paris 340 (Ser. I), 161–166; Jiang, L., 2005c. Representation theorems for generators of backward stochastic differential equations and their applications. Stochastic Process. Appl. 115 (12), 1883–1903; Jiang, L., 2005d. Nonlinear expectation— g -expectation theory and its applications in finance. Ph.D Thesis, ShanDong University, China; Jiang, L., 2006. Limit theorem and uniqueness theorem for backward stochastic differential equations. Sci. China Ser. A 49 (10), 1353–1362; Fan, S., 2007. A relationship between the conditional g -evaluation system and the generator g and its applications. Acta Math. Sin. (Engl. Ser.) 23 (8), 1427–1434; Fan, S., 2006. Jensen’s inequality for g -expectation on convex (concave) function. Chinese Ann. Math. Ser. A 27 (5), 635–644 (in Chinese)].