[论文解读] Differentiating through Stochastic Differential Equations: A Primer

Rishi Leburu, Levon Nurbekyan+1 more|arXiv (Cornell University)|Jan 13, 2026

Risk and Portfolio Optimization被引用 0

一句话总结

本 primer 介绍两种互补的方法来对 SDE 进行微分——先离散化再优化和先优化再离散化——适用于 Itô 与 Stratonovich 动力学，包含详细推导和一个 Black–Scholes 的例子。

ABSTRACT

Dynamical systems are essential to model various phenomena in physics, finance, economics, and are also of current interest in machine learning. A central modeling task is investigating parameter sensitivity, whether tuning atmospheric coefficients, computing financial Greeks, or optimizing neural networks. These sensitivities are mathematically expressed as derivatives of an objective function with respect to parameters of interest and are rarely available analytically, necessitating numerical methods for approximating them. While the literature for differentiation of deterministic systems is well-covered, the treatment of stochastic systems, such as stochastic differential equations (SDEs), in most curricula is less comprehensive than the subtleties arising from the interplay of noise and discretization require. This paper provides a primer on numerical differentiation of SDEs organized as a two-tale narrative. Tale 1 demonstrates differentiating through discretized SDEs, known the discretize-optimize approach, is reliable for both Itô and Stratonovich calculus. Tale 2 examines the optimize-discretize approach, investigating the continuous limit of backward equations from Tale 1 corresponding to the desired gradients. Our aim is to equip readers with a clear guide on the numerical differentiation of SDEs: computing gradients correctly in both Itô and Stratonovich settings, understanding when discretize-optimize and optimize-discretize agree or diverge, and developing intuition for reasoning about stochastic differentiation beyond the cases explicitly covered.

研究动机与目标

• 在物理、金融与机器学习领域，说明对随机微分方程的数值微分以获得敏感性与参数的需求。
• 在对 SDE 进行微分时，澄清来自噪声与离散化的细微差别。
• 提供易懂、适合课堂的关于 SDE 目标函数梯度计算实现的指导。
• 通过两则叙事将确定性常微分方程（ODE）微分技术桥接到随机设置。

提出的方法

• Discretize-then-optimize: 用 Euler-Maruyama（Itô）或 Heun（Stratonovich）对前向 SDE 离散化，并通过自动微分对离散目标函数求导。
• Derive pathwise gradients by differentiating the discrete objective with respect to initial state, enabling backward (adjoint) or forward differentiation; show backward adjoint recursions for Itô and Stratonovich.
• Show that discrete adjoints for Itô converge to a continuous adjoint in deterministic limits and discuss how to handle running costs and additional parameters via state augmentation.
• Use Stratonovich-optimized differentiation with Heun discretization to obtain correct continuous limits and derive the corresponding discrete adjoint recursion.
• Augment state with parameters or running-cost accumulators to compute sensitivities with respect to parameters in the SDEs (e.g., θ) or running costs (Y).
• Validate numerically on the Black–Scholes model by comparing discrete adjoint gradients to analytical Greeks and observing convergence behavior.

实验结果

研究问题

• RQ1 是否可以通过对 Itô 与 Stratonovich 形式的离散化 SDE 方案进行微分，从而计算 SDE 目标的梯度？
• RQ2 当时间步长趋于零时，离散的伴随（adjoint）是否收敛为明确的连续伴随过程，且离散化-优先与优化-离散化在此极限下如何比较？
• RQ3 离散化选择（Itô 的 Euler-Maruyama 与 Stratonovich 的 Heun）如何影响梯度估计的准确性与收敛性？
• RQ4 如何通过状态扩增及向后伴随将参数与运行成本并入梯度计算？

主要发现

• Discretize-then-optimize 当对梯度按随机路径求平均时，能够为 Itô 与 Stratonovich SDE 提供正确的梯度。
• Backward (adjoint) recursions 比对完全 Jacobian 传播的计算成本更低，支持高效的基于路径的梯度估计。
• 对于 Stratonovich SDE，Heun 离散化提供与前向兼容的离散化，保持 Stratonovich 极限且得到可处理的离散伴随。
• 在 Black–Scholes 的数值验证中，随着时间步 Δt 的减小，梯度误差在 Euler-Maruyama 离散化下呈 O(sqrt(Δt)) 递减，但蒙特卡洛噪声在较小 Δt 时导致平台期。
• 将状态扩展为包含参数或运行成本，可以实现对初始条件和参数（θ、运行成本 Y）的同时微分。
• Optimize-discretize 方法通常对 Itô SDE 不一定无偏，因为随机路径不光滑，但 discretize-optimize 仍然是获取离散目标确切梯度的安全直接方法。