[Paper Review] A note on convergence and stability of the truncated Milstein method for stochastic differential equations

Weijun Zhan, Yanan Jiang+1 more|arXiv (Cornell University)|Sep 17, 2018

Stochastic processes and financial applications21 references3 citations

TL;DR

This paper improves the truncated Milstein method for stochastic differential equations by relaxing the restrictive step size constraints present in prior work, using novel proof techniques to achieve strong convergence under super-linear growth conditions. It further establishes almost sure stability, enhancing the method's practical applicability for non-linear SDEs without requiring implicit schemes.

ABSTRACT

Some new techniques are employed to release significantly the requirements on the step size of the truncated Milstein method, which was originally developed in Guo, Liu, Mao and Yue (2018). The almost sure stability of the method is also investigated. Numerical simulations are presented to demonstrate the theoretical results.

Motivation & Objective

• Address the restrictive step size requirements in the original truncated Milstein method for SDEs with super-linearly growing coefficients.
• Improve the convergence properties of the truncated Milstein method by introducing new proof techniques to reduce dependence on small step sizes.
• Investigate and establish almost sure stability of the truncated Milstein method under the same conditions.
• Demonstrate the theoretical improvements through numerical simulations.

Proposed method

• Introduce a modified truncated Milstein method using a truncation function to control growth in drift and diffusion coefficients.
• Employ new analytical techniques in the proof of convergence, specifically tailored to relax the step size constraint compared to prior work.
• Use moment bounds and Taylor expansion with remainder terms to estimate the difference between exact and numerical solutions.
• Apply Itô's formula and stochastic integral estimates to control the remainder terms in the Taylor expansion of the solution.
• Establish almost sure stability by analyzing the long-term behavior of the numerical solution under the same assumptions.
• Utilize a continuous-time version of the numerical scheme to compare with the exact solution and derive convergence rates.

Experimental results

Research questions

• RQ1 Can the step size constraint in the truncated Milstein method be significantly relaxed while maintaining strong convergence?
• RQ2 What new proof techniques enable tighter control over the numerical error under super-linear growth conditions?
• RQ3 Does the truncated Milstein method preserve almost sure stability for SDEs with non-linear coefficients?
• RQ4 How do the theoretical convergence and stability results compare with numerical simulations?

Key findings

• The strong convergence rate of the truncated Milstein method is established under relaxed step size constraints, improving on earlier results that required very small steps.
• The method achieves a strong convergence rate of order one under the super-linear growth condition, matching the classical Milstein method but with broader applicability.
• Almost sure stability of the truncated Milstein method is proven under the same assumptions as convergence, ensuring long-term numerical reliability.
• Numerical simulations confirm the theoretical findings, showing stable and accurate behavior even with larger step sizes than previously allowed.
• The moment bound for the numerical solution is uniformly controlled over time, independent of the step size Δ ∈ (0, Δ*], ensuring robustness.
• The remainder terms in the Taylor expansion of the solution are shown to be O(Δ^p (h(Δ))^{2p}), which is critical for controlling the global error.