Comparative Analysis of Numerical Methods for Stochastic Differential Equations

Thesis Abstract

Stochastic differential equations (SDEs) provide a foundational framework for modeling systems influenced by random fluctuations across finance, physics, and engineering, yet the practical implementation hinges on the numerical methods used to approximate solutions. This study addresses the comparative performance of prominent numerical schemes for SDEs, focusing on accuracy, stability, computational efficiency, and robustness across varying regimes of stochasticity and stiffness. The aim is to identify the most reliable methods for broad classes of SDEs encountered in applied contexts, and to delineate practical guidelines for method selection. Specific objectives include (i) implementing and benchmarking Euler–Maruyama, Milstein, stochastic Runge–Kutta, and semi-implicit and tamed schemes on canonical test problems with known analytical or highly accurate reference solutions; (ii) assessing strong and weak convergence rates under additive and multiplicative noise; (iii) evaluating stability properties and long-time behavior under stiff drift terms; (iv) examining computational cost in relation to accuracy, by profiling runtimes on high-performance computing infrastructure; (v) performing sensitivity analyses with respect to step size, noise intensity, and problem dimensionality; and (vi) synthesizing a decision framework for method selection tailored to problem characteristics and resource constraints. The methodological design adopts a controlled experimental, cross-sectional approach applied to a representative portfolio of SDEs, spanning one- and multi-dimensional systems with linear and nonlinear drift and diffusion terms. The population comprises standard benchmark SDEs (Ornstein–Uhlenbeck, geometric Brownian motion, Cox–Ingersoll–Ross, double-well potentials) augmented with higher-dimensional coupling and regime-switching features to simulate realistic dynamics. A deterministic grid of parameter values is constructed to emulate varying stochastic intensity and stiffness. The sample consists of multiple numerical trajectories generated for each method and problem instance, with a target of 1,000 trajectories per configuration to ensure stable statistical estimates. Data collection employs high-precision reference solutions obtained via adaptive, high-order stochastic integrators and, where possible, analytic solutions for validation. Instrumentation includes code libraries implemented in Python (NumPy, SciPy) and Julia (Differentials.jl), with reproducible software containers to ensure cross-platform consistency. Validation involves comparing numerical results to reference solutions through error metrics, including strong and weak errors, and stability indicators such as mean-square stability regions. Analytical techniques encompass both classical and modern quantitative methods. Strong error analysis is conducted through pointwise norm differences against reference trajectories, while weak error is assessed via moments and distributional characteristics using Kolmogorov–Smirnov statistics and moment matching. Statistical performance is evaluated using repeated-measures ANOVA and multivariate regression to quantify the influence of method, time step, and problem class on error and runtime, with post hoc pairwise comparisons adjusted by the Holm–Bonferroni method. Computational efficiency is measured by wall-clock time and memory usage, normalized by problem dimension and error tolerance. Sensitivity analyses utilize Sobol indices to apportion variance among inputs. Theoretical underpinnings draw on the strong and weak convergence theories of SDE numerics, with explicit references to the Itô calculus and stability criteria for stochastic numerical schemes. The study also interprets results through the lens of the Milstein–Kloeden framework and recent developments in taming and implicit methods for stiff SDEs. Expected findings indicate that Milstein and stochastic Runge–Kutta methods exhibit superior strong convergence for smooth diffusion terms, while Euler–Maruyama remains competitive for low-noise or low-dimensional problems due to favorable computational cost. Semi-implicit and taming approaches are anticipated to offer improved stability for stiff drift structures, with trade-offs in accuracy contingent on step-size regimes. Weak convergence is projected to favor higher-order schemes for distributional accuracy, particularly in multi-dimensional settings. The study will demonstrate that no single method universally dominates; instead, a pragmatic selection framework based on noise characteristics, stiffness, and computational resources is warranted. The contribution to knowledge lies in providing a rigorous, empirically grounded comparison database, validated guidelines, and a practical decision framework for method selection in applied SDE modeling. The main conclusion posits that method choice should be problem-aware, balancing convergence properties with efficiency, and that adaptive, hybrid strategies combining explicit and implicit components offer promising avenues. Recommendations include guidelines for practitioners on selecting schemes according to problem structure, recommended error tolerances for common applications, and directions for future work involving adaptive step-size control and parallelized solvers.