2020/07/30 by Xiaojie Wang, Wang, Xiaojie · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #60H15 #60H35 #65C30 #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications #demographic modeling and climate adaptation
paper · pdf · doi:10.48550/arxiv.2007.15733
openalex publication_date 2020/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈ [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∈ [\tfrac12, 1], η∈ [0, 1]. Also, some of the proposed schemes are applied to solve three SDE models evolving in the positive domain (0, ∞). More specifically, the particular drift-diffusion implicit Milstein method ( θ= η= 1 ) is utilized to approximate the Heston \tfrac32-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.