Talk Title: Study of higher order split-step methods for stiff Stochastic differential equations

Speaker: Mr. Samar Bahadur Singh; Research: Prof. Soumyendu Raha (SERC) & Guides Prof. A. K. Nandakumaran (Maths)

Abstract

Stochastic differential equations (SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation.

In many cases, we require more accurate methods than the Milstein and Euler methods, for example, if one wants to capture extreme asset price movements or simply needs more accurate in a scenario simulation, then one may use strong method which has higher order than the Euler and Milstein method . In many applications, drift or diffusion coefficients of SDEs does not satisfy the global Lipschitz condition for example Cox􀀀Ingersoll􀀀Ross (CIR) model. The study of CIR model shows that the CIR model has non-negative solutions over the simulation interval. In general, a higher order method is useful when the non-global Lipschitz continuity of the diffusion coefficients can lead to loss of the order of accuracy of the numerical approximation.

We obtain more accurate strong Taylor schemes by including into the scheme further multiple stochastic integral from the Wagner-Platen expansion. Each 1 of these multiple stochastic integrals contain additional information about the sample path of the driving Wiener process . Numerical stability can be defined as the ability of a numerical method to control the propagation of initial and round off errors. Such errors occur naturally in any simulation, but numerical methods differ in their ability to dampen them. Since numerical stability determines whether or not a numerical method generates reasonable results at all, it is clearly more important than the order of convergence of the numerical method. Hence, we can say that higher order numerical methods with better stability properties have the potential of improving the speed of computation for stochastic models of biological, economical, chemical problem, as it permit larger step-sizes while maintaining the accuracy of the numerical simulation.

In this thesis the common theme is to construct explicit numerical methods with strong order 1:0 and 1:5 for solving Itˆo SDEs. The five-stage Milstein (FSM) methods, split-step forward Milstein (SSFM) methods and M-stage split-step strong Taylor (M-SSST) methods are constructed for solving SDEs.

The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1:0, and
M􀀀 SSST methods are convergent with strong order 1:5. Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1:5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods.

In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations (SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler (PCE) methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order = 12 in the mean-square sense. The conditions under which the PCE methods are MS-stable and 2 GMS-stable are less restrictive as compared to the conditions for the Euler method.