Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient

Existence and uniqueness result for a backward stochastic differential equation whose generator is Lipschitz continuous in y and uniformly continuous in z

“…Enlightened by these results and Bahlali et al [2], this paper extends the result in Lepetier and San Martin [13] by eliminating the condition that (g(t, 0, 0)) t∈[0,T ] is a bounded process. Furthermore, we prove that if g is Lipschitz continuous in y and uniformly continuous in z, and (g(t, 0, 0)) t∈[0,T ] is square integrable, then for each square integrable terminal condition ξ , the corresponding BSDE (1) has a unique solution.…”

“…In this section, inspired by Lepetier and San Martin [13] and Bahlali et al [2], we will extend the result obtained in Lepetier and San Martin [13] (see Theorem 1 in the following). Before that, let us present three important results on BSDEs, which will be useful in the proof of Theorem 1.…”

“…Proof The main idea comes from Lepetier and San Martin [13] and Bahlali et al [2]. Let g n be defined as in Lemma 3, and also consider h(ω, t, y, z) = −C(f t (ω) + |y| + |z|) where C and f t (ω) are taken from (H3).…”

Backward stochastic differential equations with two distinct reflecting barriers and quadratic growth generator

“…Let us point out that in our setting, contrary to those of some other works on the same subject (see, e.g., [1,3,13]), we require neither that Z ∈ Ᏼ 2,d nor K ± ∈ 2 . The main reason is that in many applications where those equations rise up, such as stochastic games or mathematical finance, we do not need such properties for Z and K ± .…”

“…A solution for that equation is a quadruple of adapted processes (Y ,Z,K + ,K Under Mokobodski's condition on the barriers, this equation has been considered by Bahlali et al in [1]. They show the existence of a solution.…”