Backward Stochastic Differential Equations with Two Distinct Reflecting Barriersand Quadratic Growth Generator

The adaptation is related to the natural filtration of the Brownian motion (Bt)t≤T . In 1990, Pardoux and Peng introduced the notion of nonlinear backward stochastic differential equation (BSDE), namely (1.1), and gave existence and uniqueness result in their founder paper [26]. Since then the interest in BSDEs has kept growing steadily and there have been several works on that subject. The main reason is that BSDEs are encountered in many fields of mathematics such as finance [6, 7, 31], stochastic games and optimal control [4, 10–12, 14, 15], partial differential equations and homogeneization [25, 27–29]. Further, other settings of BSDEs have been introduced. In [5], El-Karoui et al. consider one-barrier reflected BSDEs, that is, the situation where the process Y of (1.1) is forced to stay above a given barrier (Lt)t≤T . In (1.1), they add a nondecreasing continuous process (Kt)t≤T which allows us to have Y ≥ L. Their motivations are linked, on the one hand, to the pricing of American options and, on the other hand, to viscosity solutions