Solving Unbounded Quadratic BSDEs by a Domination Method

Khaled Bahlali (1)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134

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 |H(t, y, z)| ≤ α t + β t |y| + θ t |z| + f (|y|)|z| 2 , 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| |z| p , e e |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.

Mots clés: Superlinear backward stochastic differential equations; Quadratic backward stochas-tic differential equations; unbounded solutions maximal and a minimal; AMS 2000 Classification subjects: 60H10; 60H20; 60H30; 91G10 Keywords : Superlinear backward stochastic differential equations; Quadratic backward stochastic differential equations

Domaines: Probabilités [math.PR]

Khaled Bahlali. Solving Unbounded Quadratic BSDEs by a Domination Method. The Graduate Journal of Mathematics, 2020. ⟨hal-01972711v2⟩