Implicit numerical methods for stiff stochastic differential equations and numerical simulations of stochastic models

2001 ◽  
Author(s):  
Tianhai Tian
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Stochastic Differential Equations ◽  
Stochastic Models

scholarly journals Milstein-type semi-implicit split-step numerical methods for nonlinear stochastic differential equations with locally Lipschitz drift terms

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 1-12 ◽  
Author(s):  
Burhaneddin Izgi ◽  
Coskun Cetin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Ginzburg Landau ◽  
Linear Growth Condition ◽  
Locally Lipschitz ◽  
Non Linear ◽  
Ginzburg Landau Equations

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of non-linear stochastic differential equations with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein, and truncated Milstein procedures on non-linear stochastic differential equations including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

On effects of discretization on estimators of drift parameters for diffusion processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimators ◽  
Statistical Properties ◽  
Stochastic Integrals

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

scholarly journals Note on the Sensitivity of Simulated Solutions of the Nonlinear Singularly Perturbed Dynamical System on the Used Different Rewriting into a System of First Order Differential Equations

2016 ◽  
Vol 24 (39) ◽  
pp. 51-55
Author(s):  
Vladimír Liška ◽  
Zuzana Šútova ◽  
Dušan Pavliak
Keyword(s):  
Dynamical System ◽  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Numerical Scheme ◽  
Singularly Perturbed ◽  
First Order ◽  
First Order Differential Equations

Abstract In this paper we analyze the sensitivity of solutions to a nonlinear singularly perturbed dynamical system based on different rewriting into a System of the First Order Differential Equations to a numerical scheme. Numerical simulations of the solutions use numerical methods implemented in MATLAB.

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