Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908950167814144 |
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| author | Lee, Wonjae Park, Hyungbin |
| author_facet | Lee, Wonjae Park, Hyungbin |
| contents | 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 \(Δt\). 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 \(\tilde{Δt}\) 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 \((Δt,\tilde{Δt},λ)\) and tuning rules; for \(Z\)-independent drivers, \(λ\asymp Δt^{-1/2}\) with \(\tilde{Δt}=O(Δt/λ^2)\) yields the target \(O(Δt^{1/2})\) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09757 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs Lee, Wonjae Park, Hyungbin Probability Numerical Analysis 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 \(Δt\). 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 \(\tilde{Δt}\) 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 \((Δt,\tilde{Δt},λ)\) and tuning rules; for \(Z\)-independent drivers, \(λ\asymp Δt^{-1/2}\) with \(\tilde{Δt}=O(Δt/λ^2)\) yields the target \(O(Δt^{1/2})\) 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. |
| title | Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs |
| topic | Probability Numerical Analysis |
| url | https://arxiv.org/abs/2603.09757 |