We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The exponential mean-square stability of the θ-method for neutral stochastic delay differential equations NSDDEs with jumps is considered. With some monotone conditions, the trivial solution of the equation is proved to be exponentially mean-square stable. If the drift coefficient and the parameters satisfy more strengthened conditions, for the constrained stepsize, it is shown that the θ-method can preserve the exponential mean-square stability of the trivial solution for θ ∈ [0, 1]. Since θ-method covers the commonly used Euler–Maruyama EM method and the backward Euler–Maruyama BEM method, the results are valid for the above two methods. Moreover, they can adapt to the NSDDEs and the stochastic delay differential equations SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

Construction of positivity preserving numerical method for jumpdiffusion option pricing models

Asymptotic stability of balanced methods for stochastic jump-diffusion differential equations

Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

This paper deals with the balanced methods which are implicit methods for stochastic differential equations with Poisson-driven jumps. It is shown that the balanced methods give a strong convergence rate of at least 1/2 and can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability analysed. Some numerical experiments are given to demonstrate the conclusions.

Convergence and stability of the balanced methods for stochastic differential equations with jumps

A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.

/pdf/mean-square-convergence-of-drift-implicit-one-step-methods-e27uoozb8e.pdf

Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

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Convergence and stability of the balanced methods for stochastic differential equations with jumps

Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion