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

Xu, Yong; Pei, Bin; Guo, Guobin

doi:10.1016/j.amc.2015.04.070

Cited by 26 publications

(10 citation statements)

References 29 publications

(39 reference statements)

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“…Now, we discuss a standard SDE under the non‐Lipschitz condition just like (C1) in R d . And we have proved that Equation has a unique solution under the condition (C1) . The standard SDE is defined as

\begin{matrix} aligned \\ right X_{ϵ} MathClass-open(t MathClass-close) & left = X MathClass-open(0 MathClass-close) + ϵ \int_{0}^{t} b MathClass-open(s, X_{ϵ} MathClass-open(s - MathClass-close) MathClass-close) ds + \sqrt{ϵ} \int_{0}^{t} σ MathClass-open(s, X_{ϵ} MathClass-open(s - MathClass-close) MathClass-close) dB MathClass-open(s MathClass-close) & right \\ right & left + \sqrt{ϵ} \int_{0}^{t} \int_{| x | < c} F MathClass-open(s, X_{ϵ} MathClass-open(s - MathClass-close), x MathClass-close) Ñ MathClass-open(ds, dx MathClass-close), \end{matrix}

where the coefficients have the same conditions as in Equation and ϵ ∈ (0, ϵ 0 ] is a positive parameter with ϵ 0 a fixed number.…”

Section: The Main Resultsmentioning

confidence: 98%

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

Pei

2014

Math. Meth. Appl. Sci.

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In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy 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.

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\begin{matrix} aligned \\ right X_{ϵ} MathClass-open(t MathClass-close) & left = X MathClass-open(0 MathClass-close) + ϵ \int_{0}^{t} b MathClass-open(s, X_{ϵ} MathClass-open(s - MathClass-close) MathClass-close) ds + \sqrt{ϵ} \int_{0}^{t} σ MathClass-open(s, X_{ϵ} MathClass-open(s - MathClass-close) MathClass-close) dB MathClass-open(s MathClass-close) & right \\ right & left + \sqrt{ϵ} \int_{0}^{t} \int_{| x | < c} F MathClass-open(s, X_{ϵ} MathClass-open(s - MathClass-close), x MathClass-close) Ñ MathClass-open(ds, dx MathClass-close), \end{matrix}

where the coefficients have the same conditions as in Equation and ϵ ∈ (0, ϵ 0 ] is a positive parameter with ϵ 0 a fixed number.…”

Section: The Main Resultsmentioning

confidence: 98%

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

Pei

2014

Math. Meth. Appl. Sci.

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“…using the continuity and noting h(s) ∈ L e , there exist a scalar λ ∈ (0, λ 0 ) such that (25) holds. Consequently, from (27), (29) and Theorem 3.1, one derives the following estimate…”

Section: Assumption 31mentioning

confidence: 98%

“…Recently, these systems received remarkable attention from the researchers. Many important results can be found in the literature concerning the stability [23][24][25], the periodic solution [26] and the existence and uniqueness [27] of these systems. However, the problem of the boundedness of stochastic integro-differential systems with Lévy noise is more complicated and still open.…”

Section: Introductionmentioning

confidence: 99%

Boundedness analysis of stochastic integro-differential systems with Lévy noise

2019

Journal of Taibah University for Science

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This paper is concerned with the pth moment globally exponential ultimate boundedness of stochastic integro-differential systems with Lévy noise. With the help of the Lyapunov function methods and the inequality techniques, several sufficient criteria on the exponential ultimate boundedness are presented for the systems. The results show that the boundedness was determined by the coefficients in the estimation of the Itô operator LV of the energy function V along the trajectories of the addressed systems. ARTICLE HISTORYwhere the scalar λ satisfies 0 < λ < λ 0 and is determined by (25). This concludes the proof of the result. Corollary 3.3:Suppose that the hypotheses of Theorem 3.1 hold. If 5 = 0, then system (1) is pth moment globally exponentially stable (p-GES).Proof: This result follows directly from Theorems 3.1.Corollary 3.4: Suppose that the Assumption 3.1 with K 3 = K 6 = K 9 = K 12 = 0 holds. Ifˆ 3 >ˆ 4 > 0, then system (1) is p-GES.Proof: This result follows directly from Theorems 3.2.

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“…Our assumptions on the coefficients b, c, k and σ will be specified below. SDE driven by Lévy processes are widely applied in many fields such as finance, insurance, biology and attracted more and more attention lately (see, [14,[17][18][19]). Applebaum [1] introduced systematically the definition and relative properties of Lévy processes, and studied the fundamental theory of stochastic integrals such as the Wiener-Lévy-type stochastic integrals and Itô formula, and constructed the framework for the solution and Lévy flow of stochastic ordinary differential equations driven by Lévy processes.…”

Section: Introductionmentioning

confidence: 99%

On a Wiener-Poisson equation with rapidly fluctuating coefficients: application to large deviations

Coulibaly¹,

Allaya²

2021

J. Nonlinear Sci. Appl.

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In this paper, we deal with a stochastic differential equation with fast oscillating coefficients and with respect to a Brownian motion and a Poisson random measure. The large deviation principle of solution is established, and the effect of the highly nonlinear and locally periodic coefficients is stated. Moreover, we derive an explicit expression for the action functional when the viscosity parameter ε is of order 1 while the homogenization parameter δ ε tends to zero.

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

Cited by 26 publications

References 29 publications

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

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

Boundedness analysis of stochastic integro-differential systems with Lévy noise

On a Wiener-Poisson equation with rapidly fluctuating coefficients: application to large deviations

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