[02169] Recent advances on numerical methods for stochastic ordinary differential equations
Session Time & Room : 1E (Aug.21, 17:40-19:20) @E506
Type : Proposal of Minisymposium
Abstract : Stochastic differential equations appear nowadays as a modeling tool in many branches of science and industry as finance, biology, and mean field theory, etc. Numerical methods play a key role in understanding and exploring the dynamics of stochastic differential equations. Some new challenges arise from real-world applications, for example, a stochastic model with non-globally Lipschitz diffusion, singular initial value problems with white noises, mean-field interactions, or the positivity preserving property. The aim of this minisymposium is to bring the researchers in these fields together to discuss recent advances and influence more collaborations.
Organizer(s) : Qian Guo, Wanrong Cao, Hongjiong Tian, Liangjian Hu
[03230] Deterministic implicit two-step Milstein methods for stochastic differential equations
Format : Talk at Waseda University
Author(s) :
Hongjiong Tian (Shanghai Normal University)
Quanwei Ren (Henan University of Technology)
Tianhai Tian (Monash University)
Abstract : We propose a class of deterministic implicit two-step Milstein methods for solving Itô stochastic differential equations. Theoretical analysis is conducted for the convergence and stability properties of the proposed methods. We derive sufficient conditions such that these methods have the mean-square(M-S) convergence of order one, as well as sufficient and necessary conditions for linear M-S stability of the implicit two-step Milstein methods. Stability analysis shows that our proposed implicit two-step Milstein methods have much better stability property than those of the corresponding two-step explicit or semi-implicit Milstein methods. Numerical results are presented to confirm our theoretical analysis results.
[03262] A Positivity Preserving Lamperti Transformed Euler-Maruyama Method for Solving the Stochastic Lotka-Volterra Competition Model
Format : Talk at Waseda University
Author(s) :
Yan Li (Southeast university)
Wanrong Cao (Southeast University)
Abstract : A new positivity preserving numerical scheme is presented for a class of d-dimensional stochastic Lotka-Volterra competitive models, which are characterized by super-linear coefficients and positive solutions. The scheme, dubbed the Lamperti transformed Euler-Maruyama method, approximates the exact solution by integrating a Lamperti-type transformation with an explicit Euler-Maruyama method that has the benefit of being explicit and straightforward to implement. Even though the coefficients of the transformed models grow exponentially and do not satisfy the general monotonicity condition, based on the exponential integrability of the solution, it is proved that the proposed numerical method is of 1/2-order strong convergence. In particular, when matrix A of the model is a diagonal matrix, the first-order strong convergence is also obtained. Without any step size constraints, the method can preserve long-time dynamical properties such as extinction and pth moment exponential asymptotic stability. Numerical examples are given to support our theoretical conclusions.
[03158] Numerical methods for stochastic singular initial value problems
Format : Online Talk on Zoom
Author(s) :
Nan Deng (Southeast University)
Wanrong Cao (Southeast University)
Guofei Pang (Southeast University)
Abstract : In this work, we investigate the strong convergence of the Euler-Maruyama method for second-order stochastic singular initial value problems with additive white noise. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution is studied. Moreover, under some suitable assumptions, it is proved that the Euler-Maruyama scheme is of $(1/2-\epsilon)$ order convergence in mean-square sense, where $\epsilon$ is an arbitrary small positive number, which is different from the consensus that the Euler-Maruyama method is convergent with first order in strong sense when solving stochastic differential equations with additive white noise. While, it is found that if the diffusion coefficient vanishes at the origin, the convergent order in mean-square sense will be raised to $1-\epsilon$. Our theoretical findings are well verified by numerical examples.
[03037] Convergence rate in L^p sense of tamed EM scheme for highly nonlinear neutral multiple-delay stochastic McKean-Vlasov equations
Format : Online Talk on Zoom
Author(s) :
Shuaibin Gao (Shanghai Normal University)
Qian Guo (Shanghai Normal University)
Junhao Hu (South-Central University For Nationalities)
Chenggui Yuan (Swansea University)
Abstract : This paper focuses on the numerical scheme of highly nonlinear neutral multiple-delay stochastic McKean-Vlasov equation (NMSMVE) by virtue of the stochastic particle method. First, under general assumptions, the results about propagation of chaos in $\mathcal{L}^p$ sense are shown. Then the tamed Euler-Maruyama scheme to the corresponding particle system is established and the convergence rate in $\mathcal{L}^p$ sense is obtained. Furthermore, combining these two results gives the convergence error between the objective NMSMVE and numerical approximation, which is related to the particle number and step size. Finally, two numerical examples are provided to support the finding.
