On the existence of solutions to BSDEs with generalized uniformly continuous generators (Q968481)

This paper deals with the existence of backward stochastic differential equations (BSDEs) in a Brownian setting, where the generator \(g(\omega,t,y,z) : \Omega \times [0,T] \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\) satisfies what the authors call a \textit{generalized uniform continuous} condition. It extends an existence result from \textit{S. Hamadène} [Bernoulli 9, No.~3, 517--534 (2003; Zbl 1048.60047)]. This \textit{generalized uniform continuous} condition is more precisely characterized by \(|g(t,y_1,z_1) - g(t,y_2,z_2)| \leq b(t) \phi(|y_1-y_2|) + c(t) \psi|z_1-z_2|\) a.s. where \(b(t)\) and \(c(t)\) are two positive deterministic functions satisfying \(\int_0^T \big( b(t) + c(t)^2 \big) dt < \infty\) and \(\phi, \psi: \mathbb{R}^+ \to \mathbb{R}^+\) are two continuous, subadditive, nondecreasing functions of linear growth with \(\phi(0)=0\), \(\psi(0) = 0\). Key tool to achieve the existence result is a monotonic iteration technique which has been previously employed in \textit{G. Jia} [C. R., Math., Acad. Sci. Paris 342, No.~9, 685--688 (2006; Zbl 1119.60046), Stat. Probab. Lett. 78, No.~3, 231--237 (2008; Zbl 1134.60041)] (in the context of BSDEs with discontinuous generators) and \textit{S. Zheng} and \textit{S. Zhou} [Stat. Probab. Lett. 78, No.~5, 528--536 (2008; Zbl 1134.60041)] (in the context of reflected BSDEs) and \textit{M. N'zi} and \textit{J.-M. Owo} [Stat. Probab. Lett. 79, No.~7, 920--926 (2009; Zbl 1168.60353) (in the context of backward doubly SDEs) to prove the existence of the respective equations. The authors then construct a monotone sequence of BSDEs. Herein, the comparison principle is employed to prove monotonicity. Moreover, the authors provide uniform \(L^2\)-bounds for the BSDE sequence. This finally allows to pass to the limit of the BSDEs sequence, thus settling the existence problem.