[Paper Review] Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Xiaojie Wang|arXiv (Cornell University)|Jul 30, 2020

Stochastic processes and financial applications46 references4 citations

TL;DR

This paper introduces a family of double implicit Milstein-type methods for stochastic differential equations (SDEs) with non-globally Lipschitz coefficients, using two parameters (θ, η) to implicitly treat both drift and diffusion terms. The method achieves mean-square convergence rate of order one under relaxed conditions, with optimal convergence and positivity preservation demonstrated on three financial SDE models, including the Heston 3/2 and Ait-Sahalia interest rate models.

ABSTRACT

A class of implicit Milstein type methods is introduced and analyzed in the present article for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters $θ, η\in [0, 1]$ into both the drift and diffusion parts, the new schemes are indeed a kind of drift-diffusion double implicit methods. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily analyze mean-square convergence rates of the schemes, without relying on a priori high-order moment estimates of numerical approximations. Putting further globally polynomial growth condition, we successfully recover the expected mean-square convergence rate of order one for the considered schemes with $θ\in [ frac12, 1], η\in [0, 1]$. Also, some of the proposed schemes are applied to solve three SDE models evolving in the positive domain $(0, \infty)$. More specifically, the particular drift-diffusion implicit Milstein method ($ θ= η= 1 $) is utilized to approximate the Heston $ frac32$-volatility model and the stochastic Lotka-Volterra competition model. The semi-implicit Milstein method ($θ=1, η= 0$) is used to solve the Ait-Sahalia interest rate model. Thanks to the previously obtained error bounds, we reveal the optimal mean-square convergence rate of the positivity preserving schemes under more relaxed conditions, compared with existing relevant results in the literature. Numerical examples are also reported to confirm the previous findings.

Motivation & Objective

• To develop robust numerical schemes for SDEs with non-globally Lipschitz coefficients, where standard Euler-Maruyama methods fail due to divergence.
• To address the lack of mean-square convergence guarantees for explicit schemes in stiff or super-linear SDEs, especially in financial models requiring positivity.
• To design implicit Milstein-type methods that preserve positivity and achieve optimal convergence rates without relying on high-order moment estimates of numerical approximations.
• To establish convergence theory under weaker conditions than existing works, particularly for positivity-preserving schemes.
• To validate the theoretical results through numerical experiments on three benchmark SDE models: Heston 3/2, stochastic Lotka-Volterra, and Ait-Sahalia interest rate models.

Proposed method

• A double implicit Milstein-type scheme is proposed, parameterized by θ, η ∈ [0,1], implicitly treating both drift f(Y) and diffusion g(Y) terms.
• The scheme incorporates increments ΔW_n and iterated Itô integrals I_{j1,j2}^{tn,tn+1} to capture second-order effects, improving accuracy over Euler-Maruyama.
• The method uses a semi-implicit formulation with θ and η controlling the implicitness in drift and diffusion, respectively, enabling stability and positivity.
• Error analysis is conducted via bounds involving only the exact solution, avoiding reliance on a priori moment estimates of numerical approximations.
• Theoretical convergence is established under a globally polynomial growth condition on coefficients, enabling recovery of order-one mean-square convergence for θ ∈ [1/2,1], η ∈ [0,1].
• Explicit solvability is demonstrated for specific cases (e.g., θ=η=1 and θ=1, η=0), enabling practical implementation on positive-domain SDEs.

Experimental results

Research questions

• RQ1 Can implicit Milstein-type schemes achieve order-one mean-square convergence for SDEs with non-globally Lipschitz coefficients without requiring a priori moment bounds on numerical approximations?
• RQ2 What conditions on the coefficients and method parameters (θ, η) ensure both convergence and positivity preservation in SDEs with super-linear growth?
• RQ3 How do the proposed double implicit schemes compare to existing explicit or singly implicit schemes in terms of convergence rate and stability for stiff SDEs?
• RQ4 Can the theoretical convergence rate of order one be rigorously recovered and numerically confirmed for positivity-preserving schemes under relaxed assumptions?
• RQ5 What is the performance of the scheme on real financial models such as the Heston 3/2 volatility model and the Ait-Sahalia interest rate model?

Key findings

• The proposed double implicit Milstein method with θ=η=1 achieves a mean-square convergence rate of order one for the Heston 3/2 volatility model under relaxed conditions.
• For the stochastic Lotka-Volterra competition model, the same scheme maintains positivity and achieves a mean-square convergence rate of 0.9923, confirming theoretical predictions.
• The semi-implicit Milstein scheme (θ=1, η=0) applied to the Ait-Sahalia interest rate model achieves a convergence rate of 0.9798 in the standard case and 1.0129 in the critical case, both consistent with order-one convergence.
• Numerical simulations confirm that all schemes preserve positivity across 10,000 sample paths for the Heston 3/2 and stochastic LV models.
• The error analysis framework avoids reliance on a priori high-order moment estimates, enabling simpler convergence proofs for non-Lipschitz SDEs.
• The results represent the first optimal convergence rate for positivity-preserving schemes in such models under less restrictive assumptions than previous literature.