A study of a class of stochastic differential equations with non-Lipschitzian coefficients

Abstract

We study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x − y| log 1 |x−y| . Finally a large deviation principle of Freidlin-Wentzell type is also established in the paper.

Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients

“…Hence, the result of this paper includes the result of [15]. What is more, only need continuity in Theorem 2; however, must be differential function in [8,15], and our proof is more directive.…”

“…In this section, we will prove the following pathwise uniqueness result; when the driving process is Brownian motion, such kind of properties was studied for non-Lipschitzian coefficients in [8]. …”

“…From the viewpoint of application, to what extent the condition on coefficients should be weakened is an important problem. Fang and Zhang discussed the pathwise uniqueness and the nonexplosion for a class of stochastic differential equations driven by Brownian motion with non-Lipschitz coefficients (see [8]). Using Zvonkin's transformation, Zhang studied the homeomorphic property of solutions of multidimensional stochastic differential equations with non-Lipschitz coefficients (see [9]).…”

The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

“…For a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of this equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [2,4,5]) or a given non-Lipschitz condition and the linear growth condition (see [22]). However, when the coefficients of the system (5) satisfy the local Lipschitz condition and the polynomial growth condition, the solution of the system (5) may explode at a finite time.…”

Stochastic suppression and stabilization of delay differential systems

“…In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of this equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [4,14]) or a given non-Lipschitz condition and the linear growth condition (References [3,15]). These show that the linear growth condition plays an important role to suppress the potential explosion of solutions and guarantee the existence of global solutions.…”