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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 |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 and locally bounded on R. This is done with unbounded terminal value. We cover the classical QBSDEs where the function f is constant 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 [12, 14, 23, 25, 27], we get the existence of a solution, moreover we cover the BSDEs with at most linear growth (take f = 0), in particular we extend the results of [24] and [26]. The existence and uniqueness of solutions are also established for BSDEs driven by f (y)|z| 2 when f is merely locally integrable on R.
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https://hal-univ-tln.archives-ouvertes.fr/hal-01972711
Contributor : Khaled Bahlali <>
Submitted on : Monday, January 7, 2019 - 7:36:40 PM
Last modification on : Saturday, March 30, 2019 - 2:05:36 AM
Long-term archiving on: : Monday, April 8, 2019 - 6:59:33 PM

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  • HAL Id : hal-01972711, version 1

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Khaled Bahlali. Solving Unbounded Quadratic BSDEs by a Domination Method *. 2019. ⟨hal-01972711v1⟩

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