Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs
This work addresses a computational challenge in financial mathematics for pricing options with barriers, offering an incremental improvement in error analysis for DRBSDEs.
The paper tackles the numerical approximation of doubly reflected backward stochastic differential equations (DRBSDEs) with two obstacles by developing a two-grid penalty scheme that combines penalization and time discretization, achieving an explicit error bound of O(Δt^{1/2}) under specific tuning rules and validating it with numerical experiments showing O(n^{-1/2}) grid-refinement errors.
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 \(λ\) and discretized by an implicit Euler scheme with step \(Ît\). A key difficulty is that the forward approximation used to evaluate the obstacles generates an error term that is amplified by \(λ\). 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 \(\tilde{Ît}\) and projecting onto the backward grid (two-grid scheme). Under structural assumptions motivated by financial barriers we sharpen penalization rates and obtain a uniform \(O(λ^{-1})\) bound for the value process. We derive an explicit error bound in \((Ît,\tilde{Ît},λ)\) and tuning rules; for \(Z\)-independent drivers, \(λ\asymp Ît^{-1/2}\) with \(\tilde{Ît}=O(Ît/λ^2)\) yields the target \(O(Ît^{1/2})\) rate. Nonsmooth barriers/payoffs are handled via a multivariate Itô--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.