Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Abstract

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma

Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations

“…Moreover, the stopping time is essential for the application of the semimartingale convergence theory in our approach. Benefiting from the random variable stepsize, the sufficient conditions for the almost sure stability of the EM method obtained in this paper are much weaker than those established in [14] and [23]. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.…”

“…The ability to reproduce the almost sure stability is one important characteristic of numerical methods. Many papers have studied the numerical reproduction of the almost sure stability by adopting the semimartingale convergence theory, for example [2,14,19,20,23,24,25] and the references therein. However, in most of the papers the stepsize is either fixed or nonrandom variable.…”

“…For SDEs with non-global Lipschitz coefficient, the EulerMaruyama method may not preserve this properties with any stepsize (see for example Lemma 3.1 in [8]). To tackle this, the methods with implicit structure are often employed as the alternatives [13,19,23]. Compared with the classical explicit methods, those implicit methods can reproduce larger ranges of SDEs with less restrictions on the stepsize.…”

A Note on the Almost Sure Exponential Stability of the Milstein Method for Stochastic Delay Differential Equations With Jumps

“…Since most of these equations cannot be solved explicitly, numerical approximations become an important tool in studying the properties of these stochastic systems (see [1], [3], [10], [17], [20] and [26]). Since stability is one of the major concerns in systems analysis, the stability of numerical methods becomes also one of the main tools to examine the stability of the exact solution of the stochastic systems (see [18], [21] and [24]).…”

Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

“…For the nonlinear SDE without the linear growth condition, they answered (Q1) using the backward Euler method. The research in this area has since then been developed by many authors, e.g., [4,17,25,26], but all of these authors have addressed (Q1) …”