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Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs

Wonjae Lee, Hyungbin 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.

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 and discretized by an implicit Euler scheme with step . A key difficulty is that the forward approximation used to evaluate the obstacles generates an error term that is amplified by . 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 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 -independent drivers, 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.
Paper Structure (14 sections, 14 theorems, 188 equations, 2 figures, 2 tables)

This paper contains 14 sections, 14 theorems, 188 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Doubly reflected BSDEs: setting and preliminaries
  3. Penalty error
  4. Numerical scheme and error analysis
  5. Numerical scheme
  6. A priori estimates
  7. Square error bound
  8. Absolute error bound for Z-independent driver
  9. Experimental results
  10. Problem setup and DRBSDE.
  11. Experimental procedure and parameter sweeps.
  12. Results.
  13. Conclusion
  14. Acknowledgments

Key Result

Theorem 2.2

Under Assumption assumptions, there exists a unique solution $(X,Y,Z,A,K)\in\mathbb{S}^2(\mathbb{R}^d)\times\mathbb{S}^2(\mathbb{R})\times\mathbb{H}^2(\mathbb{R}^d)\times\mathcal{A}^2\times \mathcal{A}^2$ to the BSDE BSDE.

Figures (2)

  • Figure 1: Relative error as a function of the coarse grid size $n$ under the theoretically motivated choice $\lambda=2000\,n^{1/2}$. The dashed curve represents the fitted profile $Cn^{-1/2}$ with $C=3.82\times 10^{-1}$. The measured error closely follows the fitted $n^{-1/2}$ trend for the first four grid levels and then falls slightly below it, in agreement with the theoretical prediction. The plot is generated from the values reported in Table \ref{['tab:grid_sweep']}.
  • Figure 2: Relative error as a function of the penalty parameter $\lambda$, with $\lambda=2000\,n^a$ for several values of $a$. Over the tested range, the relative error decreases monotonically as $\lambda$ increases. This indicates that the experiment remains in a regime where increasing penalization improves the approximation, while the asymptotic balancing effect predicted by the theory is not yet visibly resolved. The plot is generated from the values reported in Table \ref{['tab:penalty_sweep']}.

Theorems & Definitions (25)

  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.6
  • Example 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • ...and 15 more