# A domination method for solving unbounded quadratic BSDEs

Khaled Bahlali – GJM, Volume 5, Special Issue (2020), 20-36.

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 $\vert H(t,y,z)\vert\leq \alpha_t
+ \beta_t\vert y\vert + \theta_t\vert z\vert + f(|y|)\vert z\vert^{2}$
, where $\alpha_t$, $\beta_t$, $\theta_t$ are positive processes and the function $f$ is positive, continuous and increasing (or even only positive and locally bounded) on $\mathbb R$. This is done with unbounded terminal value. In particular, we cover the classical case ($f$ constant) obtained in [11,13,26,28], the case $f$ polynomial obtained in ([24]) and also the cases where the generator has super linear growth (not covered by the previous papers) such as $e^{|y|^k} |z|^p$ ($k \geq 0$, $0\leq p < 2$), $e^{e^{|y|}} |z|^2$ and so on. In contrast to the works [11,13,24,26,28], we get the existence of a maximal and a minimal solution and we also cover the BSDEs with at most linear growth (take $f=0$), and in particular some results obtained in [25,14,23]. We moreover establish the existence and uniqueness of solutions to BSDEs driven by $f(y)|z|^2$ when $f$ is merely locally integrable on $\mathbb R$.