Paper
Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs
Authors
Wonjae Lee, Hyunbin Park
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 \(λ\) 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.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.09757v1</id>\n <title>Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs</title>\n <updated>2026-03-10T14:57:00Z</updated>\n <link href='https://arxiv.org/abs/2603.09757v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.09757v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.PR'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <published>2026-03-10T14:57:00Z</published>\n <arxiv:primary_category term='math.PR'/>\n <author>\n <name>Wonjae Lee</name>\n </author>\n <author>\n <name>Hyunbin Park</name>\n </author>\n </entry>"
}