Solving Unbounded Quadratic BSDEs by a Domination Method

Abstract

We introduce a domination argument which asserts that: if we can dominate the parameters of a quadratic backward stochastic differential equation (QBSDE) with continuous generator from above and from below by those of two BSDEs having ordered solutions, then also the original QBSDE admits at least one solution. This result is presented in a general framework: we do not impose any integrability condition on none of the terminal data of the three involved BSDEs, we do not require any constraint on the growth nor continuity of the two dominating generators. As a consequence, we establish the existence of a maximal and a minimal solution to BSDEs whose coefficient H is continuous and satisfies, where α t , β t , θ t are positive processes and the function f is positive, continuous and increasing (or even only positive and locally bounded) on R. This is done with unbounded terminal value. We cover the classical QBSDEs where the function f is constant ([10], [12], [23], [25]) and when f (y) = y p ([21]) and also the cases where the generator has super linear growth such as y|z|, e |y| k |z| p , e e |y| |z| 2 , (k ≥ 0, 0 ≤ p < 2) and so on. In contrast to the works [10,12,21,23,25], we get the existence of a a maximal and a minimal solution and we cover the BSDEs with at most linear growth (take f = 0). In particular, we cover and extend the results of [22] and [24]. Furthermore, we establish the existence and uniqueness of solutions to BSDEs driven by f (y)|z| 2 when f is merely locally integrable on R.

One-dimensional reflected BSDEs and BSDEs with quadratic growth generators

“…Motivated by the work of [1], we introduce a domination method, which can be used to obtain some existence and comparison results for BSDE(g, ξ, L t ) and BSDE(g, ξ).…”

“…When the function f is global integral on R, the existence and uniqueness of solutions was firstly studied by Bahlali et al [2] for L 2 integrable terminal variables, and then by Yang [28] for L p (p ≥ 1) integrable terminal variables. Furthermore, Bahlali et al [1] considered the case that the function f is local integral on R, and Bahlali and Tangpi [3] considered the case that f (y) = 1 y . Recently, Zheng et al [29] studied the BSDE with generator f (y)|z| 2 , where the function f is defined on an open interval D and locally integrable.…”

“…Note that when f is a constant and D = R, the related studies on quadratic RBSDEs have been done in [4], [19] and [21], etc, in which the proofs mainly depend on exponential transformation, estimates and approximation techniques. Inspired by [1] and [2], our proof mainly uses a transformation u f and a domination method. u f is an exponential transformation in the case that f is a constant.…”

“…It can be used to remove the quadratic term f (y)|z| 2 of generators. The domination method was firstly used by [2] in a special case, and then generalized by [1]. It is based on a general existence result of RBSDEs with double obstacles studied by [13], and can be used to obtain the existence of solutions to BSDEs (see [1], [2]).…”

“…The domination method was firstly used by [2] in a special case, and then generalized by [1]. It is based on a general existence result of RBSDEs with double obstacles studied by [13], and can be used to obtain the existence of solutions to BSDEs (see [1], [2]). To deal with our problem, we use a transformation u f defined on the open interval D with u f (α) = 0, for some constant α in D (u f in [2] is defined on R with u f (0) = 0), and introduce a domination method for RBSDEs with one obstacle, whose proof is slightly different from [1] and [2].…”