On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations

In this paper, under a local Lipschitz condition and a monotonicity condition, the problems on the existence and uniqueness theorem as well as the almost surely asymptotic behavior for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are discussed by using the Lyapunov function and some stochastic analysis techniques. Two integral lemmas are firstly established to overcome the difficulty stemming from the coexistence of the stochastic perturbation and the time-varying delay. Then, without any redundant restrictive condition on the time-varying delay, by utilizing the integral inequality, the exponential stability in pth(p ≥ 1)-moment for such equations is investigated. By employing the nonnegative semi-martingale convergence theorem, the almost sure exponential stability is analyzed. Finally, two examples are given to show the usefulness of the results obtained.National Natural Science Foundation of ChinaNatural Science Foundation of Jiangxi Province of ChinaFoundation of Jiangxi Provincial Educations of ChinaMinisterio de Economía y Competitividad (MINECO). EspañaJunta de Andalucí

Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations

By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately

Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions

We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of Lyapunov function to study the stochastic open loop stabilizability in probability and the local and global asymptotic stabilizability (or asymptotic controllability). Finally we illustrate the theory with some examples.Comment: 22 page

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models

Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods