Click here to view fulltext PDF

Permanent link:
https://www.ias.ac.in/article/fulltext/pmsc/126/03/0433-0460

Backward stochastic differential equation, $\epsilon^g$-evaluation, $g$-expectation, $\epsilon_g$-martingale

In this paper, by using $L_p$ ($1 \lt p \leq 2$) weak convergence method on backward stochastic differential equations (BSDEs) with non-uniformly Lipschitz coefficients, we obtain the limit theorem of $g$-supersolutions. As applications of this theorem, we study the decomposition theorem of $\epsilon_g$-supermartingale, the nonlinear decomposition theorem of Doob-Meyer’s type and so on. Furthermore, by using the decomposition theorem of $\epsilon_g$-supermartingale, we provide some useful characterizations of an $\epsilon^g$-evaluation by the generating function $g(t; ω; y; z)$ without the assumption that $g$ is continuous with respect to $t$. Our results generalize the known results in Ph. Briand et al., Electronic Commun. Probab. {\bf 5} (2000) 101–117; L Jiang, Ann. Appl. Probab. {\bf 18} (2008) 245–258; S Peng, Probab. Theory Relat. Fields {\bf 113} (1999) 473–499; S Peng, Modelling derivatives pricing with their generating functions (2006) http://arxiv.org/abs/math/0605599 and E Rosazza Gianin, Insur. Math. Econ. {\bf 39} (2006) 19–34.

ZHAOJUN ZONG¹ FENG HU¹

1. School of Statistics, Qufu Normal University, Qufu, Shandong 273165, China

Published

5 August 2016

Manuscript received

28 October 2013

Manuscript revised

12 June 2015

Accepted

13 July 2015

Early published

Unedited version published online

Final version published online

23 July 2016