The existence and uniqueness of mild solutions to stochastic differential equations with L\xe9vy noise

Lasheng Wang

doi:10.1186/s13662-017-1224-0

Abstract

In this paper, we study a class of neutral stochastic differential equations (NSDEs) with the cylindrical Brownian motion and Lévy noises in an infinite-dimensional Hilbert space. The existence and uniqueness of the mild solutions to these stochastic differential equations are discussed under assumptions of linear growth on the coefficients. The results of Taniguchi (J. Math. Anal. Appl. 360:245-253, 2009) are generalized and improved as a special case of our theory.

Highlights

The stochastic neutral differential equations have attracted much attention because of their practical applications in many areas such as physics, population dynamics, electrical engineering, medicine biology, ecology and other areas of science and engineering [ – ]
It should be mentioned that only a few papers have discussed the existence and uniqueness of mild solutions of stochastic differential equations driven by Brownian motion and Lévy noises
Cao established the existence and uniqueness of mild solutions to semilinear backward stochastic evolution equations driven by the cylindrical Brownian motion and the Poisson point process in a Hilbert space with non-Lipschitzian coefficients by the successive approximation [ ]

Summary

Introduction

The stochastic neutral differential equations have attracted much attention because of their practical applications in many areas such as physics, population dynamics, electrical engineering, medicine biology, ecology and other areas of science and engineering [ – ]. It should be mentioned that only a few papers have discussed the existence and uniqueness of mild solutions of stochastic differential equations driven by Brownian motion and Lévy noises. Cao established the existence and uniqueness of mild solutions to semilinear backward stochastic evolution equations driven by the cylindrical Brownian motion and the Poisson point process in a Hilbert space with non-Lipschitzian coefficients by the successive approximation [ ]. Mao established the existence and uniqueness theorem of mild solutions to general neutral stochastic functional differential equations with infinite delay and Lévy jumps under the local Carathéodorytype conditions [ ]. Motivated by the aforementioned works, we aim to study the existence and uniqueness of mild solutions for SDEs with the cylindrical Brownian motion and Lévy noises in an infinite-dimensional Hilbert space of the form dx(t) = Tx(t) + A t, x(t) dt + g t, x(t) dWt +.

Examples

Conclusions