Approximate solutions of hybrid stochastic pantograph equations with Levy jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 퐿 2 sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory

Approximate solutions of hybrid stochastic pantograph equations with Levy jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 퐿 2 sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory

Approximate solutions of hybrid stochastic pantograph equations with Levy jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 퐿 2 sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory

The Hitting Times of A Stochastic Epidemic Model

In this paper, we focus on the hitting times of a stochastic epidemic model presented by \cite{Gray}. Under the help of the auxiliary stopping times, we investigate the asymptotic limits of the hitting times by the variations of calculus and the large deviation inequalities when the noise is sufficiently small. It can be shown that the relative position between the initial state and the hitting state determines the scope of the hitting times greatly

On almost sure stability of hybrid stochastic systems with mode-dependent interval delays

This note develops a criterion for almost sure stability of hybrid stochastic systems with mode-dependent interval time delays, which improves an existing result by exploiting the relation between the bounds of the time delays and the generator of the continuous-time Markov chain. The improved result shows that the presence of Markovian switching is quite involved in the stability analysis of delay systems. Numerical examples are given to verify the effectiveness

Almost sure exponential stabilisation of stochastic systems by state-feedback control

So far, a major part of the literature on the stabilisation issues of stochastic systems has been dedicated to mean square stability. This paper develops a new class of criteria for designing a controller to stabilise a stochastic system almost surely which is unable to be stabilised in mean-square sense. The results are expressed in terms of linear matrix inequalities (LMIs) which are easy to be checked in practice by using MATLAB Toolbox. Moreover, the control structure in this paper appears not only in the drift part but also in the diusion part of the underlying stochastic system

Delay geometric Brownian motion in financial option valuation

Motivated by influential work on complete stochastic volatility models, such as Hobson and Rogers [11], we introduce a model driven by a delay geometric Brownian motion (DGBM) which is described by the stochastic delay differential equation dSðtÞ ¼ mðSðt 2tÞÞSðtÞdt þ VðSðt 2tÞÞSðtÞdWðtÞ. We show that the equation has a unique positive solution under a very general condition, namely that the volatility function V is a continuous mapping from Rþ to itself. Moreover, we show that the delay effect is not too sensitive to time lag changes. The desirable robustness of the delay effect is demonstrated on several important financial derivatives as well as on the value process of the underlying asset. Finally, we introduce an Euler–Maruyama numerical scheme for our proposed model and show that this numerical method approximates option prices very well. All these features show that the proposedDGBMserves as a rich alternative in modelling financial instruments in a complete market framework

SMC design for robust H∞ control of uncertain stochastic delay systems

Recently, sliding mode control method has been extended to accommodate stochastic systems. However, the existing results employ an assumption that may be too restrictive for many stochastic systems. This paper aims to remove this assumption and present in terms of LMIs a sliding mode control design method for stochastic systems with state delay. In some cases, the proposed method provides a control scheme for finite-time stabilization of stochastic delay systems

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

Almost sure exponential stabilization by discrete-time stochastic feedback control

Given an unstable linear scalar differential equation x˙ (t) = αx(t) (α > 0), we will show that the discrete-time stochastic feedback control σx([t/τ ]τ )dB(t) can stabilize it. That is, we will show that the stochastically controlled system dx(t) = αx(t)dt +σx([t/τ ]τ )dB(t) is almost surely exponentially stable when σ2 > 2α and τ > 0 is sufficiently small, where B(t) is a Brownian motion and [t/τ ] is the integer part of t/τ . We will also discuss the nonlinear stabilization problem by a discrete- time stochastic feedback control. The reason why we consider the discrete-time stochastic feedback control is because that the state of the given system is in fact observed only at discrete times, say 0, τ, 2τ, • • • , for example, where τ > 0 is the duration between two consecutive observations. Accordingly, the stochastic feedback control should be designed based on these discrete-time observations, namely the stochastic feedback control should be of the form σx([t/τ ]τ )dB(t). From the point of control cost, it is cheaper if one only needs to observe the state less frequently. It is therefore useful to give a bound on τ from below as larger as better