Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients

Abstract

In the paper, we consider the no-explosion condition and pathwise uniqueness for SDEs driven by a Poisson random measure with coefficients that are super-linear and non-Lipschitz. We give a comparison theorem in the one-dimensional case under some additional condition. Moreover, we study the non-contact property and the continuity with respect to the space variable of the stochastic flow. As an application, we will show that there exists a unique strong solution for SDEs with coefficients like x log |x|. (Yuchao Dong).The purpose of this paper is to study the no-explosion condition and pathwise uniqueness for the solutions of (1.1) with super-linear and non-Lipschitz coefficients including the functions like x log |x|. Some other properties of the solutions, including comparison principle, non-contact property and continuity with respect to x of the stochastic flow, are also proved under some additional assumptions. As an application, we prove that the following SDE:

A general continuous-state nonlinear branching process

“…Other than the above mentioned results, we are not aware of any previous results on hitting probability and coming down from infinity for solutions to SDEs of type (1.3). There is some literature on nonexplosion of solutions to general SDE with jumps; see Dong (2016) for a recent result. But we do not find any systematic discussions on the explosion/nonexplosion dichotomy and the coming down from infinity property of the solutions.…”

Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence, Feller and strong Feller properties, and exponential ergodicity

“…The rest of the proof is quite standard: one can apply Itô's formula and the optional sampling theorem to the process {e −κt Φ(X(t)), t ≥ 0} to argue that P{lim n→∞ τ n = ∞} = 1. Indeed similar arguments can be found in, e.g., the proofs of Theorem 2.1 of Meyn and Tweedie (1993), Theorem A of Fang and Zhang (2005), and Theorem 2.1 of Dong (2018). We shall omit the details here.…”

“…We refer to Fang and Zhang (2005) and Lan and Wu (2014) for sufficient conditions for non confluence for SDEs without jumps. The recent paper Dong (2018) contains some sufficient conditions for non confluence for jump SDEs. The key assumption in Dong (2018) is on the jumps: for each u ∈ U, the function x → x + c(x, u) is homeomorphic and that its inverse satisfies the linear growth and Lipschitz conditions.…”

“…The recent paper Dong (2018) contains some sufficient conditions for non confluence for jump SDEs. The key assumption in Dong (2018) is on the jumps: for each u ∈ U, the function x → x + c(x, u) is homeomorphic and that its inverse satisfies the linear growth and Lipschitz conditions. Such conditions are quite strong and not easy to verify in practice.…”

“…These conditions are quite different from our corresponding conditions and they do not seem to imply one another. In addition, the methodologies in Dong (2018) and this paper have different flavors, even though certain technical aspects are similar.…”