non-Lipschitz Stochastic Differential Equations Research Articles

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations

Simulation of Non-Lipschitz Stochastic Differential Equations Driven by $\alpha$-Stable Noise: A Method Based on Deterministic Homogenization

Simulation of Non-Lipschitz Stochastic Differential Equations Driven by $\alpha$-Stable Noise: A Method Based on Deterministic Homogenization

Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations

In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods.

Characterizing the path-independence of the Girsanov transformation for non-Lipschitz SDEs with jumps

In the paper, by virtue of the Girsanov transformation, we derive a link of a class of (time-inhomogeneous) non-Lipschitz stochastic differential equations (SDEs) with jumps to a class of semi-linear partial integro-differential equations (PIDEs) of parabolic type, in such a manner that these obtained PIDEs characterize the path-independence property of the density process of Girsanov transformation for the non-Lipschitz SDEs with jumps.

On the non-Lipschitz stochastic differential equations driven by fractional Brownian motion

In this paper, we use a successive approximation method to prove the existence and uniqueness theorems of solutions to non-Lipschitz stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with the Hurst parameter $H\in(\frac{1}{2},1)$ . The non-Lipschitz condition which is motivated by a wider range of applications is much weaker than the Lipschitz one. Due to the fact that the stochastic integral with respect to fBm is no longer a martingale, we definitely lost good inequalities such as the Burkholder-Davis-Gundy inequality which is crucial for SDEs driven by Brownian motion. This point motivates us to carry out the present study.

Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise

In this paper, successive approximation method is applied to investigate the existence and uniqueness of solutions to stochastic differential equations (SDEs) driven by Lévy noise under non-Lipschitz condition which is a much weaker condition than Lipschitz one. The stability of solutions to non-Lipschitz SDEs driven by Lévy noise is also considered, and the stochastic stability is obtained in the sense of mean square.

The cocycle property of stochastic differential equations driven by G-Brownian motion

In this paper, solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion: $$X_t = x + \int_0^t {b(s,\omega ,X_s )ds} + \int_0^t {h(s,\omega ,X_s )d\left\langle B \right\rangle _s } + \int_0^t {\sigma (s,\omega ,X_s )dB_s } $$ are constructed. It is shown that they have the cocycle property. Moreover, under some special non-Lipschitz conditions, they are bi-continuous with respect to t, x.

Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non-Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Levy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non-Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.

Homeomorphism flows for non-lipschitz SDEs driven by lévy processes

In this article, homeomorphism flows for non-Lipschitz stochastic differential equations driven by Lévy processes are studied.

Exponential ergodicity of non-Lipschitz stochastic differential equations

Using the coupling method and Girsanov’s theorem, we study the strong Feller property and irreducibility for the transition probabilities of stochastic differential equations with non-Lipschitz and monotone coefficients. Then, the exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained under fewer assumptions.

Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps

In this paper we study the continuity property as well as the homeomorphism property for the solutions of multidimensional stochastic differential equations with jumps and non-Lipschitz coefficients with respect to the initial values.