The truncated EM method for stochastic differential equations with Poisson jumps

In this paper, we use the truncated Euler–Maruyama (EM) method to study the finite time strong convergence for SDEs with Poisson jumps under the Khasminskii-type condition. We establish the finite time L r (r≥2)-convergence order when the drift and diffusion coefficients satisfy the super-linear growth condition and the jump coefficient satisfies the linear growth condition. The result shows that the optimal L r - convergence order is close to 1. This is significantly different from the result on SDEs without jumps. When all the three coefficients of SDEs are allowing to grow super-linearly, the L r (0<r<2)-convergence results are also investigated and the optimal L r - convergence order is shown to be not greater than 1∕4. Moreover, we prove that the truncated EM method preserves nicely the mean square exponential stability and asymptotic boundedness of the underlying SDEs with Poisson jumps. Several examples are given to illustrate our results

Convergence of the split-step θ-method for stochastic age-dependent population equations with Markovian switching and variable delay

We present a stochastic age-dependent population model that accounts for Markovian switching and variable delay. By using the approximate value at the nearest grid-point on the left of the delayed argument to estimate the delay function, we propose a class of split-step θ -method for solving stochastic delay age-dependent population equations (SDAPEs) with Markovian switch- ing. We show that the numerical method is convergent under the given conditions. Numerical examples are provided to illustrate our results

Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method

Consider a stochastic differential delay equation driven by G-Brownian motion (G-SDDE) dx(t) = f(x(t), x(t − τ))dt + g(x(t), x(t − τ))dB(t) + h(x(t), x(t − τ))dhBi(t). Under the global Lipschitz condition for the G-SDDE, we show that the G-SDDE is exponentially stable in mean square if and only if for sufficiently small step size, the Euler-Maruyama (EM) method is exponentially stable in mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the underlying G-SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is provided to illustrate our results

Exponential stability of highly nonlinear neutral pantograph stochastic differential equations

In this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations(NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M-matrix are used to study exponential stability and boundedness of the hybrid NPSDEs

Stabilisation in distribution by delay feedback control for stochastic differential equations with Markovian switching and Lévy noise

This paper is devoted to the stability in distribution of stochastic differential equations with Markovian switching and Lévy noise by delay feedback control. By constructing efficient Lyapunov functional and linear delay feedback controls, the stability in distribution of stochastic differential equations with Markovian switching and Lévy noise is accomplished with the coefficients satisfying globally Lipschitz continuous. Moreover, the design methods of feedback control under two structures of state feedback and output injection are discussed. Finally, a numerical experiment and new algorithm are provided to sustain the new results

Tamed EM schemes for neutral stochastic differential delay equations with superlinear diffusion coefficients

In this article, we propose two types of explicit tamed Euler-Maruyama (EM) schemes for neutral stochastic differential delay equations with super linearly growing drift and diffusion coefficients. The first type is convergent in the Lq sense under the local Lipschitz plus Khasminskii-type conditions. The second type is of order half in the mean-square sense under the Khasminskii-type, global monotonicity and polynomial growth conditions. Moreover, it is proved that the partially tamed EM scheme has the property of mean-square exponential stability. Numerical examples are provided to illustrate the theoretical findings

Asymptotic stability in distribution of highly nonlinear stochastic differential equations with G-Brownian motion

Following the analysis on the stability in distribution of stochastic differential equations discussed in Fei, Fei & Mao (2023) [11], this article further investigates the stability in distribution of highly nonlinear stochastic differential equations driven by G-Brownian motion (G-HNSDEs). To this end, by employing the theory on sublinear expectations, the stability in distribution of G-HNSDEs is analysed. Moreover, a sufficient criterion of the stability in distribution of G-HNSDEs is provided for convenient use

The truncated EM method for jump-diffusion SDDEs with super-linearly growing diffusion and jump coefficients

This work is concerned with the convergence and stability of the truncated Euler-Maruyama (EM) method for super-linear stochastic differential delay equations (SDDEs) with time-variable delay and Poisson jumps. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present two types of the truncated EM method for such jump-diffusion SDDEs with time-variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The first type is proved to have a strong convergence order which is arbitrarily close to 1=2 in mean-square sense, under the Khasminskii-type, global monotonicity with U function and polynomial growth conditions. The second type is convergent in qth (q < 2) moment under the local Lipschitz plus generalized Khasminskii-type conditions. In addition, we show that the partially truncated EM method preserves the mean-square and H1 stabilities of the true solutions. Lastly, we carry out some numerical experiments to support the theoretical results

Generalized Ait-Sahalia-type interest rate model with Poisson jumps and convergence of the numerical approximation

In this paper, we consider a generalized Ait-Sahalia interest rate model with Poisson jumps in finance. The analytical properties including positivity, boundedness and pathwise asymptotic estimations of the solution to this model are investigated. Moreover, we prove that the EulerMaruyama (EM) numerical solution converges to the true solution of the model in probability. Finally, we apply the EM solution to compute some financial quantities. A numerical example is provided to demonstrate the effectiveness of our results

Positivity-preserving truncated Euler-Maruyama method for generalised Ait-Sahalia-type interest model

The well-known Ait-Sahalia-type interest model, arising in mathematical finance, has some typical features: polynomial drift that blows up at the origin, highly nonlinear diffusion, and positive solution. The known explicit numerical methods including truncated/tamed Euler-Maruyama (EM) applied to it do not preserve its positivity. The main interest of this work is to investigate the numerical conservation of positivity of the solution of generalised Ait-Sahalia-type model. By modifying the truncated EM method to generate positive sequences of numerical approximations, we obtain the rate of convergence of the numerical algorithm not only at time T but also over the time interval [0,T]. Numerical experiments confirm the theoretical results