A new approximation method for solving stochastic differential equations (2407.19350v2)

Abstract: We present a novel solution method for It^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main properties of the stochastic numerical methods, e.g. convergence, consistency, and stability are analyzed. We test the proposed method in SDE problem, demonstrating promising results.

• The paper presents a novel quadratic polynomial method that improves the approximation of stochastic differential equations by enhancing stability and error control.
• It approximates drift terms over three nodal points, offering a more efficient alternative to traditional Euler–Maruyama and Milstein schemes.
• Empirical results using L1, L2, and L∞ error metrics confirm significant performance gains and robust convergence relative to conventional methods.

An Approximation Method for Stochastic Differential Equations Using Quadratic Polynomials

This paper introduces a novel approximation method for solving stochastic differential equations (SDEs), integral to modeling systems embedded with inherent randomness, as observed across domains such as molecular biology and economics. Traditional methods for SDEs often leverage time-discrete approximations, such as the Euler–Maruyama and Milstein schemes. This research advances beyond these conventional approaches by utilizing quadratic polynomials to approximate solutions, potentially offering enhancements in convergence, stability, and computational efficiency.

Methodology Overview

The proposed method is built upon the numerical subdivision of the time interval [0, T] into smaller, equal sub-intervals. The quadratic polynomial approach specifically targets solutions for Ito stochastic differential equations and geometric Brownian motion. By approximating the drift term with quadratic polynomials over three nodal points, the method provides a structure for estimating values at sequential time steps, optimizing the balance between accuracy and computational cost. The equations delineate the application of the quadratic interpolating functions in deriving the approximation, showcasing a systematic technique to capture the solution dynamics across the defined interval.

Analytical Framework: Stability and Consistency

The paper rigorously analyzes the stability, consistency, and convergence properties of the method. The mean-square stability is established under specific conditions, providing assurances for the robustness of the derived numerical solutions against perturbations. The method's consistency, evaluated using a local error metric between successive time points, is demonstrated to hold in the mean-square sense. Such an analysis substantiates the method's reliability in estimating SDE solutions with a quantifiable and constrained error margin.

Remarkably, the conditional convergence of the proposed methodology follows the stochastic variant of the Lax-Richtmyer theorem, aligning the innovative approach with established numerical analysis paradigms.

Numerical Results

The empirical evaluation underscores the efficacy of the quadratic polynomial-based scheme, using L1, L2, and L∞ error metrics to compare numerical outputs against exact solutions. The method shows convincing performance improvements over the implicit Euler–Maruyama and Milstein methods, evident from reduced error levels across varied test cases. Notably, under the specified conditions for drift and volatility parameters, the method delineates superior stability regions and robustness, further validating the theoretical predictions.

Implications and Future Directions

The development of this approximation method holds practical significance for computational fields requiring robust and efficient stochastic model simulations. The enhanced stability and accuracy achieved through the quadratic polynomials suggest a promising direction for future explorations, particularly for more complex SDE forms and applications with high-dimensional stochastic inputs.

Future work may extend this methodology to encompass broader classes of SDEs and explore adaptive strategies that dynamically adjust the time-step size or polynomial degree based on local solution features. Such advancements could further optimize computational resources and extend the applicability of stochastic simulations in real-world scenarios.

The proposed method exemplifies an important step in refining numerical strategies for SDEs, and its foundational insights and performance analysis contribute valuably to the computational mathematics and engineering communities.