Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs

Abstract

We study penalization coupled with time discretization for decoupled Markovian doubly reflected BSDEs with obstacles \(p_b(t,X_t)\le Y_t\le p_w(t,X_t)\). The DRBSDE is approximated by a penalized BSDE with parameter \(\lambda\) and discretized by an implicit Euler scheme with step \(\Delta t\). A key difficulty is that the forward approximation used to evaluate the obstacles generates an error term that is amplified by \(\lambda\). In the single-obstacle case this amplification can be removed by the shift \(Y-p_b(t,X)\), but no analogous transformation eliminates both obstacles simultaneously; this motivates simulating the forward SDE on a finer grid \(\tilde{\Delta t}\) and projecting onto the backward grid (two-grid scheme). Under structural assumptions motivated by financial barriers we sharpen penalization rates and obtain a uniform \(O(\lambda^{-1})\) bound for the value process. We derive an explicit error bound in \((\Delta t,\tilde{\Delta t},\lambda)\) and tuning rules; for \(Z\)-independent drivers, \(\lambda\asymp \Delta t^{-1/2}\) with \(\tilde{\Delta t}=O(\Delta t/\lambda^2)\) yields the target \(O(\Delta t^{1/2})\) rate. Nonsmooth barriers/payoffs are handled via a multivariate It\^o--Tanaka and local-time-on-surfaces argument. We also provide numerical experiments for a one-dimensional game put under the Black--Scholes model. The observed grid-refinement errors are consistent with the predicted \(O(n^{-1/2})\) behavior, while the penalty sweep indicates that the tested regime remains pre-asymptotic with respect to the penalty parameter.