Numerical methods for stochastic differential equations

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Models used in engineering, biological or financial applications inherently include parameter uncertainties, or other kinds of random variations, that can be expressed mathematically using stochastic differential equations. Since very few types of stochastic differential equations can be solved analytically, it is important to find efficient methods to approximate the solutions numerically. The focus of my project is the development of new numerical methods that inherit the qualitative properties of the original stochastic differential equations. I am also interested in the study of the error of the proposed numerical schemes, and the analysis of the properties of the stochastic processes defined by the numerical schemes.Stochastic Hamiltonian systems and Langevin type equations are used in many models from classical mechanics, the dynamics of particle accelerators, chemistry, and biology, and they appear also in the numerical study of nonlinear stochastic partial differential equations.Stochastic Hamiltonian systems preserve the symplectic structure, so it is important to construct numerical schemes with similar properties. We have developed a systematic method to construct new symplectic numerical schemes for stochastic Hamiltonian systems. Numerical simulations show that symplectic schemes are more accurate than non-symplectic methods for long-time simulations. There is no theoretical proof of this fact in the stochastic case, and it is challenging to extend the approach used in the deterministic case. One of the goals of this project is to study the error associated with the symplectic schemes for stochastic Hamiltonian systems. Classical assumptions for the study of numerical solution of stochastic differential equations require globally Lipschitz coefficients. These conditions are not met for stochastic non-linear oscillators and other highly non-linear systems arising from financial mathematics or bio-mathematics. Relaxing these assumptions is a subtle problem because some numerical schemes might not be convergent. Since I investigate a stochastic Hamiltonian system with locally Lipschitz coefficients and a fully implicit scheme, the study of the error is a challenging problem. In Monte Carlo simulations, explicit numerical schemes are preferable because they require less computing time than implicit schemes, but, unless we consider special stochastic Hamiltonian systems, symplectic schemes are implicit. In the deterministic case, exponentially fitted methods are constructed for differential equations with periodic or oscillating solutions. I intend to extend this approach in the stochastic case and to develop explicit symplectic Runge-Kutta-Nyström methods for stochastic oscillators. This proposal includes both challenging theoretical analyses and applications, so it will contribute to the knowledge transfer from academia to industry.