Strang Splitting Combined with Periodically Fitted Adams–Bashforth–Moulton Method for High-Precision Simulation of Multiplicative Noise SDEs with Periodic Drift

Many applications in finance and biology involve multiplicative noise geometric Brownian motion (GBM)-type stochastic differential equations (SDEs) whose drift carries a single dominant periodic component. Such structures arise in seasonal Black–Scholes option pricing, commodity derivatives with annual price cycles, and stochastic biological oscillators driven by a known frequency; the primary contribution of this paper is a high-precision numerical scheme validated on GBM-type test problems with periodic drift. This paper proposes a Strang operator splitting scheme within the Logarithmic Drift-Diffusion Splitting (LDDS) framework, which splits the SDE in y-space into a deterministic drift ODE sub-step (Step A) and an exactly solvable multiplicative diffusion sub-step (Step B). Step A employs the Periodically Fitted Adams–Bashforth–Moulton fourth-order predictor–corrector method (PABM4), which achieves zero local truncation error for trigonometric forcing terms by introducing additional shift terms and simultaneously imposing polynomial exactness conditions and trigonometric fitting conditions. When the Step A forcing belongs to the PABM4 exact function class Fω=span{1,t,t2,sinωt,cosωt}, the Strang+PABM4 scheme achieves floating-point precision saturation. We investigate three test problems: the cosine-drift GBM (Test Problem 1), the polynomial–trigonometric mixed drift GBM (Test Problem 2), and a dual-frequency drift applicability test (Test Problem 3). Monte Carlo strong error experiments (M=1000 paths) validate that Strang+PABM4 achieves saturation at machine precision (≈10−15) on Test Problems 1 and 2, improving precision by ≈102× over the best algebraically convergent reference. Test Problem 3 identifies the method’s applicability boundary: when the drift contains a second frequency outside Fω, Strang+PABM4 degrades gracefully to order ≈ 4 without catastrophic failure. The floating-point saturation of Strang+PABM4 is contingent on the drift belonging to F^ω; when this condition is violated, the method degrades gracefully to algebraic order ≈ 4, as demonstrated in Test Problem 3.