Stability-Preserving Stochastic Perturbations

The stability of equilibria is a topic of importance in mathematical models in many branches of science. Since systems generally do not start in equilibrium states, the equilibria of chief scientific interest are those which are attracting, in the sense that the system converges to these equilibria over time. Perhaps the simplest mathematical model which captures this is a deterministic differential equation, and we can study the long-time behaviour of the solution of the equation.

Such simple models, however, may underplay the role that randomness may play in the evolution of the system. The character of the stochastic disturbance may be of many types: one type, which this considers, is when the intensity of the random term is independent of the state of the system, but varies over time. This models persistent stochastic shocks. The question now is: which (possibly fading) time-varying intensities of the shock preserve the stability of the system, and which render the equilibrium unstable. The book seeks to answer this question for one-dimensional linear (and possibly) nonlinear differential equations in a very comprehensive manner. The book involves stochastic calculus (mainly stochastic differential equations), and also some ideas from the theory of differential equations which we will study. According to the writers’ interest, one can ask whether numerical simulations of the stochastic equation preserve the asymptotic behaviour of the original continuous-time problem. The research outcome is most relevant to Probability and Finance (as well as Simulation for Finance, if numerical studies are involved) as well as Ito calculus.

This book is concerned with the study of linear differential and difference equations which are subjected to an outside force. Any equation without any External force is considered to be “stable “in a sense we make precise below.

The question we address is: what are precisely the conditions on this external force for which this stability is preserved, and if these conditions are not satisfied, what then is the long–time behaviour of the solution?

The stochastic differential equations (SDEs) driven by Brownian motions have been attracting lots of attentions. When some unexpected events happen, some jumps may be needed to model the effects of those events. For example, the out break of a new disease can cause the National Economy have some instability in the country. To take both the continuous and discontinuous random effects into consideration, SDEs driven by both Brownian motions and Poisson jumps are often employed as a generalisation of the SDEs only driven by Brownian motions. Despite the wide applications, the explicit solutions to SDEs are hardly found. Therefore, to construct some efficient numerical methods is of great importance.

In general, we are able to characterise whether the solution has retained its stability, or whether the solution remain under control, but is not convergent, or whether the solution become unbounded.