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

Abstract

This paper considers multidimensional jump type stochastic differential equations with super linear and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient local non-Lipschitz conditions for pathwise uniqueness. The non confluence property for solutions is investigated. Feller and strong Feller properties under local non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman-Kac formula for Lévy type operators.where W is a standard d-dimensional Brownian motion, and N is a Poisson random measure on [0, ∞) × U with intensity dt ν(du) and compensated Poisson random measure N. It is well-known that if the coefficients b, σ and c of (1.1) satisfy the linear growth and local Lipschitz conditions, then (1.1) admits a non-exploding strong solution and the solution is pathwise unique; see, for example, (Ikeda and Watanabe, 1989, Theorem IV.9.1) for details.The linear growth condition is a standard assumption in the literature; it guarantees that the solution X to (1.1) does not explode in finite time with probability one. But such a condition is often too restrictive in practice. For example, in many mathematical ecological models (such as those in Khasminskii and Klebaner (2001), Mao et al. (2002), Zhu and Yin (2009)), the coefficients do not satisfy the linear growth condition; yet non-explosion is still guaranteed thanks to the special structures of the underlying SDEs in these papers. For general multidimensional SDEs without jumps, the relaxation of linear growth condition can be found in Fang and Zhang (2005) and Lan and Wu (2014). For jump type SDEs, can we relax the usual linear growth condition as well? In this paper, we provide a sufficient condition in Theorem 2.2 for non-explosion for solutions to (1.1) when the coefficients have super linear growth in a neighborhood of ∞.Concerning the pathwise uniqueness, the usual argument is to use the (local) Lipschitz condition and Gronwall's inequality to demonstrate that the L 2 distance E[| X(t) − X(t)| 2 ] between two solutions X, X vanishes if they have the same initial condition; see, for example, the proof of Ikeda and Watanabe (1989, Theorem IV.9.1). The paper Yamada and Watanabe (1971) relaxes the local Lipschitz condition to Hölder condition for one-dimensional SDEs without jumps. Since then, the problem of existence and pathwise uniqueness of solutions to SDEs with non-Lipschitz conditions has attracted growing attention. To name just a few, Bass (2003) presents a sharp condition for existence and pathwise uniqueness for a one-dimensional SDE with a symmetric stable driving noise; Fu and Li (2010) and Li and Mytnik (2011) provide sufficient conditions for existence and pathwise uniqueness for one-dimensional jump type SDEs with non-Lipschitz conditions; a crucial assum...

Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties

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Characterization of the most probable transition paths of stochastic dynamical systems with stable Lévy noise

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Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

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