Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps

Qiao, Huijie; Zhang, Xicheng

doi:10.1016/j.spa.2007.12.006

Cited by 23 publications

(21 citation statements)

References 8 publications

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“…This equation arises in nonlinear filtering and has been considered recently in [11,8,9] (see also the monograph [12]). The characterisation theorem for path-independent property of Girsanov density for the above equation with non-degenerated σ was established in [10].…”

Section: Introductionmentioning

confidence: 99%

Characterising the path-independent property of the Girsanov density for degenerated stochastic differential equations

2018

Statistics & Probability Letters

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In this paper, we derive a characterisation theorem for the path-independent property of the density of the Girsanov transformation for degenerated stochastic differential equations (SDEs), extending the characterisation theorem of [13] for the nondegenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalises the corresponding characterisation theorem established in [10].

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Section: Introductionmentioning

confidence: 99%

Characterising the path-independent property of the Girsanov density for degenerated stochastic differential equations

2018

Statistics & Probability Letters

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“…12 By the same method to that in Qiao (2014, Proposition 2.4), one can prove the above result. 13 To apply the Girsanov transformation, we assume further the following (H b,σ ,λ ) 16 where σ (s, X s ) −1 stands for the inverse of σ (s, X s ).…”

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confidence: 99%

“…Assume that there exists a C 1,2 -function v(t, x) satisfying (4)- (6). For the composition 16 process v(t, X t ), the Itô formula admits us to get (8). Combining (4)-(6) with (8), we have…”

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Characterizing the path-independence of the Girsanov transformation for non-Lipschitz SDEs with jumps

Qiao

2016

Statistics & Probability Letters

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Available online xxxx MSC: 60H10 35Q53Keywords: Non-Lipschitz stochastic differential equations with jumps The Girsanov transformation Semi-linear partial integro-differential equation of parabolic type a b s t r a c tIn the paper, by virtue of the Girsanov transformation, we derive a link of a class of (timeinhomogeneous) non-Lipschitz stochastic differential equations (SDEs) with jumps to a class of semi-linear partial integro-differential equations (PIDEs) of parabolic type, in such a manner that these obtained PIDEs characterize the path-independence property of the density process of Girsanov transformation for the non-Lipschitz SDEs with jumps.

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“…These conditions were further relaxed in Lan and Wu (2014) using Euler's approximation method. Further studies on jump type SDEs with non-Lipschitz coefficients can be found in Priola (2012, 2015), Qiao (2014), Qiao and Zhang (2008), among others.This paper aims to establish sufficient non-Lipschitz conditions for pathwise uniqueness for multidimensional SDEs with jumps. Two sets of sufficient non-Lipschitz conditions (Assumptions 2.3 and 2.5) for pathwise uniqueness are provided; both of them only require the deal with Feller property while Theorem 5.2 and Proposition 5.4 establish strong Feller property.…”

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confidence: 99%

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

Zhu

2019

Journal of Differential Equations

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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...

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Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps

Cited by 23 publications

References 8 publications

Characterising the path-independent property of the Girsanov density for degenerated stochastic differential equations

Characterising the path-independent property of the Girsanov density for degenerated stochastic differential equations

Characterizing the path-independence of the Girsanov transformation for non-Lipschitz SDEs with jumps

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

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