Representations and regularities for solutions to BSDEs with reflections

In this paper we study a class of backward stochastic differential equations with reflections (BSDER, for short). Three types of discretization procedures are introduced in the spirit of the so-called Bermuda Options in finance, so as to first establish a Feynman–Kac type formula for the martingale integrand of the BSDER, and then to derive the continuity of the paths of the martingale integrand, as well as the C 1 -regularity of the solution to a corresponding obstacle problem. We also introduce a new notion of regularity for a stochastic process, which we call the “ L 2 -modulus regularity”. Such a regularity is different from the usual path regularity in the literature, and we show that such regularity of the martingale integrand produces exactly the rate of convergence of a numerical scheme for BSDERs. Both numerical scheme and its rate of convergence are novel.