arXiv 2603.097572026-03math.PRcs.NAmath.NA

双重反射BSDEs的双重网格罚项近似方案

Two-grid Penalty Approximation Scheme for Doubly Reflected BSDEs

Seoul National University（首尔大学）

Wonjae Lee, Hyungbin Park

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本文研究了双重反射BSDEs的罚项与时间离散化结合方法，提出双重网格方案以处理障碍项误差，并通过数值实验验证了误差界与惩罚参数的关系。

摘要：

我们研究了罚项与时间离散化相结合用于解耦马尔可夫双重反射BSDEs（DRBSDEs）的方法，其中障碍项满足 $p_b(t,X_t)\le Y_t\le p_w(t,X_t)$。DRBSDE通过参数 $λ$ 的罚项BSDE和步长 $Δt$ 的隐式欧拉方案进行近似。关键困难在于用于评估障碍项的前向近似产生一个被 $λ$ 放大的误差项。在单障碍情况下，通过 $Y-p_b(t,X)$ 的位移可消除放大，但没有类似的变换能同时消除两个障碍；这促使在更细的网格 $\tilde{Δt}$ 上模拟前向SDE并投影到后向网格（双重网格方案）。基于金融障碍的结构假设，我们加强了罚项速率并获得值过程的统一 $O(λ^{-1})$ 有界性。我们推导了误差界在 $(Δt,\tilde{Δt},λ)$ 上的显式表达式并提供了调参规则；对于 $Z$-无关驱动器，$λ\asymp Δt^{-1/2}$ 与 $\tilde{Δt}=O(Δt/λ^2)$ 可获得目标 $O(Δt^{1/2})$ 速率。非光滑障碍/支付通过多变量Itô-Tanaka和局部时间在曲面的论证处理。我们还为一维Black-Scholes模型下的游戏期权提供了数值实验。观察到的网格细化误差与预测的 $O(n^{-1/2})$ 行为一致，而惩罚扫描表明测试区域仍相对于惩罚参数处于预渐进行态。

英文摘要：

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.

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