Infinite time interval BSDEs and the convergence of g-martingales

Abstract

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 economics.2000 Mathematics subject classification: primary 60H10, 60G48. Keywords and phrases: infinite time interval BSDEs, g-expectation, g-martingale, upcrossing inequality of g-martingale, convergence of g-martingale.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 [20]. Later many researchers developed the theory and its applications in a series of papers (see for example Darling [5] Karoui, Peng and Quenez [8] 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,)

Lpsolutions of finite and infinite time interval BSDEs with non-Lipschitz coefficients

“…In order to prove Theorem 3.2, we need first to establish the following Proposition 3.3, which is just Theorem 1.2 of Chen and Wang [5] when p = 2. …”

“…In this section, we will introduce some examples, corollaries and remarks to show that Theorem 3.2 of this paper is a generalization of the main results in Pardoux and Peng [11], Mao [9], Chen [4], Chen and Wang [5], Wang and Wang [13] and Wang and Huang [12]. Firstly, by Remark 3.1 and Theorem 3.2, the following corollary is immediate, which generalizes the main results in Mao [9], Wang and Wang [13] and Wang and Huang [12].…”

“…Moreover, Chen [4] and Chen and Wang [5] proposed the following non-uniformly Lipschitz condition for the generator g of multidimensional BSDEs:…”

“…We establish a general existence and uniqueness theorem of solutions in L p (p > 1) to finite time and infinite interval BSDEs (see Theorem 3.2 in Section 3), which includes the corresponding results in Pardoux and Peng [11], Mao [9], Chen [4], Constantin [6], Wang and Wang [13], Chen and Wang [5] and Wang and Huang [12] as its particular cases. The paper is organized as follows.…”

Backward doubly stochastic differential equations with infinite time horizon

“…However, their results essentially require the terminal values to be decay in infinite horizon. Later Chen and Wang [4] were the first to show a kind of sufficient conditions on coefficients, under which for any square integrable random variables ξ as terminal values, BSDEs still have a unique pair of solutions for infinite horizon case. This result is pivotal for discussing the convergence of g-martingales which were introduced by Peng [13].…”