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Numerical Methods for SDEs under the Local Lipschitz Condition

The aims of this PhD is to develop the truncated EM method are to study the strong convergence in finite-time for SDEs under the generalised Khasminskii condition and its convergence rate and to investigate the numerical stability of the nonlinear SDEs.

Number of places

One

Opens

27 February 2018

Eligibility

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics, Statistics, Mathematics and Statistics or a closely related discipline with a high mathematical content. 

Project Details

Up to 2002, most of the existing strong convergence theory for numerical methods requires the coefficients of the SDEs to be globally Lipschitz continuous [1]. However, most SDE models in real life do not obey the global Lipschitz condition. It was in this spirit that Higham, Mao and Stuart in 2002 published a very influential paper [2] (Google citation 319) which opened a new chapter in the study of numerical solutions of SDEs to study the strong convergence question for numerical approximations under the local Lipschitz condition.

Since the classical explicit Euler-Maruyama (EM) method has its simple algebraic structure, cheap computational cost and acceptable convergence rate under the global Lipschitz condition, it has been attracting lots of attention. Although it was showed that the strong divergence in finite time of the EM method for SDEs under the local Lipschitz condition, some modified EM methods have recently been developed these SDEs. For example, the tamed EM method was developed in 2012 to approximate SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient. The stopped EM method was developed in 2013. Recently, Mao [3] initiated a significantly new method, called the truncated EM method, for the nonlinear SDEs.

The aim of this PhD is to develop the truncated EM method. The detailed objectives are:

(1) To study the strong convergence of the truncated EM method in finite-time for SDEs under the generalised Khasminskii condition and its convergence rate.

(2) To use the truncated EM method to investigate the stability of the nonlinear SDEs. Namely to study if the numerical method is stochastically stable when the underlying SDE is stochastically stable and to study if we can infer that the underlying SDE is stochastically stable when the numerical method is stochastically stable for small stepsize.

References

[1] Mao X., Stochastic Differential Equations and Applications}, 2nd Edtion, Elsevier, 2007.

[2] Higham D., Mao X., Stuart A., Strong convergence of Euler-type methods for nonlinear stochastic differential equations}, SIAM J. Numer. Anal. 40(3) (2003), 1041--1063. [3] Mao X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370--384.

Further information

For more information about the project, please contact Professor Mao