Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs

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.