Conditional Reflected Backward Stochastic Differential Equations

In this paper, we introduce a new type of reflected backward stochastic differential equations (reflected BSDEs), referred to as conditional reflected BSDEs, in which the conditional expectation of the Y-component of the solution with respect to a given sub-σ-algebra appears in both the generator and the obstacle. Under a Lipschitz condition on the loss function with sufficiently small Lipschitz constants, we establish the existence and uniqueness of the solution for both the case of Lipschitz generator and the case of quadratic generator, bounded terminal value and bounded loss function. When the generator has quadratic growth and the terminal value and loss function are unbounded, we establish the uniqueness with the help of θ-method. Finally, a penalization approximation scheme holds under an additional monotonicity structure imposed on the generator and the loss function, without any restriction on the Lipschitz constants of the loss function.