Infinite time interval BSDEs and the convergence of g-martingales
Journal of the Australian Mathematical Society
In this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and
... n both finance and economics. 2000 Mathematics subject classification: primary 60H10, 60G48. The adapted solution for a linear BSDE which appears as the adjoint process for a stochastic control problem was first introduced by Bismut in 1973, then by Bensoussan and others, while the first result for the existence and uniqueness of an adapted solution to a nonlinear BSDE with finite time interval and Lipschitzian coefficient was obtained by Pardoux and Peng  . Later many researchers developed the theory and its applications in a series of papers (see for example Darling , Hu and Peng , Lepeltier and Martin , Pardoux , Peng [24, 25], Karoui, Peng and Quenez  and the references therein) under some other assumptions on coefficients but for fixed terminal time. From these papers, the basic theorem is that, for a fixed terminal time T > 0, under the suitable assumptions on terminal value £, coefficient g and driving process M, the following BSDE has a solution pair (y" z,) in the interval [0, T]: (0.1) y, = t-+ f g(y s , z s , s)ds-I z s dM s , 0 < t < T. Jt Jt More recently, Peng  introduced the notions of ^-expectations and g-martingales via the above finite time interval BSDEs driven by a Brownian motion process. In Chen and Peng [1, 2] and Peng  , some properties of ^-martingales (such as upcrossing inequality, stopping sampling theorem and decomposition theorem for gmartingales) are discussed. As a supplement, in this paper, we discuss the convergence of ^-martingales. One difficulty of this problem is how to study the existence and uniqueness of BSDE (0.1) when T = oo. In fact, such a problem has been investigated by Peng , Pardoux , Darling and Pardoux , Pardoux and Zhang  and other researchers under the assumption that terminal value f = 0 or Ee pT \%\ 2 < oo for some constant p > 0 and random terminal time T. A natural question is under which conditions on g, does BSDE (0.1) still have a unique solution pair for any given square integrable £ when T = oo? Obviously, the assumptions on g in the papers mentioned above do not solve this question. In this paper, we first give a sufficient condition on coefficient g under which for any square integrable random variable £, BSDE (0.1) still has a unique solution pair when T -oo. Furthermore, we explain such a condition usually is necessary. We also give an example to show that our conditions on g allow the coefficients to be unbounded, thus our result still extends Pardoux and Peng's result even for finite time horizon BSDE. Using these results, we show the convergence of g-martingales. Finally, we discuss some applications of g-expectations and g-martingales. This paper is organized as follows. In Section 1, we consider a class of infinite time interval BSDEs: existence, uniqueness and convergence. In Section 2 we recall the notions of ^-expectations and ^-martingales introduced by Peng via BSDE. In Section 3, we show the convergence of g-martingales. In Section 4, we apply our results to economic theory and the pricing of contingent claims in incomplete security markets.