The mild sufficient conditions for exponential ergodicity of a Markov process, defined as the solution to SDE with a jump noise, are given. These conditions include three principal claims: recurrence condition R, topological irreducibility condition S and non-degeneracy condition N, the latter formulated in the terms of a certain random subspace of \Re^m, associated with the initial equation. The examples are given, showing that, in general, none of three principal claims can be removed without losing ergodicity of the process. The key point in the approach, developed in the paper, is that the local Doeblin condition can be derived from N and S via the stratification method and criterium for the convergence in variations of the family of induced measures on \Re^m.
Under Lyapunov and monotone conditions, the exponential ergodicity in the induced Wasserstein quasi-distance is proved for a class of fully non-dissipative McKean-Vlasov SDEs, which strengthen some recent results established under dissipative conditions in long distance. Moreover, when the SDE is order-preserving, the exponential ergodicity is derived in the Wasserstein distance induced by one-dimensional increasing functions chosen according to the coefficients of the equation.
By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present the exponential ergodicity in the relative entropy and (weighted) Wasserstein distances for reflecting McKean-Vlasov SDEs under the dissipative and partially dissipative conditions respectively. The main results are illustrated by non-symmetric singular granula…
In this paper we show irreducibility and the strong Feller property for transition probabilities of stochastic differential equations with jumps and monotone coefficients. Thus, exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained.
We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to $α$-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the $L^1$-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived f…
We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results th…
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimat…
In this note, we give a necessary and sufficient condition under which the comparison theorem holds for multidimensional stochastic differential equations (SDEs) with jumps and for matrix-valued SDEs with jumps.
Let $α\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $α$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+σ(X_{t-})d L^α_t, \quad X_0=x\in\mathbb{R}^d, $$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and $σ:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally $γ$-Hölder continuous with $γ\in(0\vee(1-α)^+,1]$, $L^α_t$ is a $d$-dimensional rotationally invariant $α$-stable process. Under some dissipative and non-degenerate assumptions on $b,σ$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t…
We obtain explicit criteria for both exponential ergodicity and strong ergodicity for one-dimensional time-changed symmetric stable processes with $α\in(1,2)$. Explicit lower bounds for ergodic convergence rates are given.
We study a class of stochastic integral equations with jumps under non-Lipschitz conditions. We use the method of Euler approximations to obtain the existence of the solution and give some sufficient conditions for the strong uniqueness.
We provide a support theorem for the law of the solution to an SDE with jump noise. This theorem applies to general SDEs with jumps and is illustrated by examples of SDEs with quite degenerate jump noises where the theorem leads to an informative description of the support.
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of th…
This work develops asymptotic properties of a class of switching jump diffusion processes. The processes under consideration may be viewed as a number of jump diffusion processes modulated by a random switching mechanism. The underlying processes feature in the switching process depends on the jump diffusions. In this paper, conditions for recurrence and positive recurrence are derived. Ergodicity is examined in detail. Existence of invariant probability measures is proved.
In this work, we study ergodicity of continuous time Markov processes on state space $\mathbb{R}_{\geq 0} := [0,\infty)$ obtained as unique strong solutions to stochastic equations with jumps. Our first main result establishes exponential ergodicity in the Wasserstein distance, provided the stochastic equation satisfies a comparison principle and the drift is dissipative. In particular, it is applicable to continuous-state branching processes with immigration (shorted as CBI processes), possibly with nonlinear branching mechanisms or in Lévy random environments. Our second main result establis…
This work focuses on a class of regime-switching jump diffusion processes, in which the switching component has countably infinite many states or regimes. The existence and uniqueness of the underlying process are obtained by an interlacing procedure. Then the Feller and strong Feller properties of such processes are derived by the coupling method and an appropriate Radon-Nikodym derivative. Finally the paper studies exponential ergodicity of regime-switching jump-diffusion processes.
In this paper, we investigate stochastic continuity (with respect to the initial value), irreducibility and non confluence property of the solutions of stochastic differential equations with jumps. The conditions we posed are weaker than those relevant conditions existing in the literature. We also provide an example to support our new conditions.
We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.
We present a new approach to positive self-similar Markov processes (pssMps) by reformulating Lamperti's transformation via jump type SDEs. As applications, we give direct constructions of pssMps (re)started continuously at zero if the Lamperti transformed Levy process is spectrally negative. Our paper can be seen as a continuation of similar studies for continuous state branching processes but the approach seems to be more fruitful in the context of pssMps.
We study the jump-diffusion CIR process, which is an extension of the Cox-Ingersoll-Ross model and whose jumps are introduced by a subordinator. We provide sufficient conditions on the Lévy measure of the subordinator under which the jump-diffusion CIR process is ergodic and exponentially ergodic, respectively. Furthermore, we characterize the existence of the $κ$-moment ($κ>0$) of the jump-diffusion CIR process by an integrability condition on the Lévy measure of the subordinator.
In this paper we study the jump-diffusion CIR process (shorted as JCIR), which is an extension of the classical CIR model. The jumps of the JCIR are introduced with the help of a pure-jump Lévy process $(J_t, t \ge 0)$. Under some suitable conditions on the Lévy measure of $(J_t, t \ge 0)$, we derive a lower bound for the transition densities of the JCIR process. We also find some sufficient condition guaranteeing the existence of a Forster-Lyapunov function for the JCIR process, which allows us to prove its exponential ergodicity.
In this paper, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.
In this paper we find the transition densities of the basic affine jump-diffusion (BAJD), which is introduced by Duffie and Garleanu [D. Duffie and N. Garleanu, Risk and valuation of collateralized debt obligations, Financial Analysts Journal 57(1) (2001), pp. 41--59] as an extension of the CIR model with jumps. We prove the positive Harris recurrence and exponential ergodicity of the BAJD. Furthermore we prove that the unique invariant probability measure $π$ of the BAJD is absolutely continuous with respect to the Lebesgue measure and we also derive a closed form formula for the density func…
Ergodicity of stochastic functional differential equation with jumps and finite delay
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Levy processes. In the past decade the problem of classifying all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory. Motivated by a recent article of Chaumont, Rivero, Panti we classify via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously. Our constru…
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
Under full Hörmander's conditions, we prove the strong Feller property of the semigroup determined by an SDE driven by additive subordinate Brownian motion, where the drift is allowed to be arbitrarily growth. For this, we extend a criterion due to Malicet-Poly \cite{Ma-Po} and Bally-Caramellino \cite{Ba-Ca} about the convergence of the laws of Wiener functionals in total variations. Moreover, the example of a chain of coupled oscillators is verified.
We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed including diffusion processes on Riemannian manifolds and Ornstein-Uhlenbeck processes driven by symmetric $α$-stable processes. For SDE driven by $α$-stable process ($α\in (0,2]$) with polynomial drift, the strong ergodicity or not is independent on $α$.
This paper considers multidimensional jump type stochastic differential equations with super linear growth and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient non-Lipschitz conditions for pathwise uniqueness. The non confluence property for solutions is investigated. Feller and strong Feller properties under 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 di…
The paper is dedicated to studying the problem of existence and uniqueness of solutions as well as existence of and exponential convergence to invariant measures for McKean-Vlasov stochastic differential equations with Markovian switching. Since the coefficients are only locally Lipschitz, we need to truncate them both in space and distribution variables simultaneously to get the global existence of solutions under the Lyapunov condition. Furthermore, if the Lyapunov condition is strengthened, we establish the exponential convergence of solutions' distributions to the unique invariant measure …
We extend some methods developed by Albeverio, Brzeźniak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.
We consider a class of Lévy-driven stochastic differential equations (SDEs) with McKean-Vlasov (MK-V) interaction in the drift coefficient. It is assumed that the coefficient is bounded, affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration scheme that app…
In this paper, we study the long-time behavior of a fluid particle immersed in a turbulent fluid driven by a diffusion with jumps, that is, a Feller process associated with a non-local operator. We derive the law of large numbers and central limit theorem for the evolution process of the tracked fluid particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic or (iii) is a class of Lévy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.
Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity
In this paper, we introduce a new class of processes which are diffusions with jumps driven by a multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. In the case of exponential memory kernels for the underlying Hawkes process we establish conditions for the positive Harris recurrence of the couple (X, Y), where X denotes the diffusion process and Y the piecewise deterministic Markov process (PDMP) defining the stochastic intensity of the driving Hawkes. As a direct consequence of the Harris recurrence, we obtain the ergodic theorem for X. Furthermore, we provid…
We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state $0$, say, subject to a perturbation by a stochastic integral which is driven by an $\varepsilon$-small and $(1/\varepsilon)$-accelerated Lévy process with exponentially light jumps. For this purpose we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel we solve the associated asymptotic first escape problem from the bounded ne…
In this paper, we investigate new sufficient conditions to ensure the existence of a unique global strong solution of stochastic differential equations with jumps. By using Euler approximation and by utilising a new test function $φ_δ$ (see the following definition (\ref{pntas1})), we prove that there is a unique global strong solution for the initial value problem of the equation. The condition we posed is even weaker than the local Lipschitzian continuity of the coefficients. We also present an example to show that our conditions are indeed weaker than those relevant conditions existing in t…
We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold. First, when the driving vector fields satisfy Hörmander's celebrated "Lie bracket condition," we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential eq…
We present a new algorithms to discretize a decoupled forward backward stochastic differential equations driven by pure jump Lévy process (FBSDEL in short). The method is built in two steps. Firstly, we approximate the FBSDEL by a forward backward stochastic differential equations driven by a Brownian motion and Poisson process (FBSDEBP in short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps $\eps$ goes to 0. In the second step, we obtain the $L^p$ Hölder continuity of the solution of FBSDEBP and we c…
Ergodicity for stochastic 2D Boussinesq equations with a highly degenerate pure jump Levy noise
College of Science, National University of Defense Technology（国防科技大学理学院）；MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burd…
In this paper we study the transition density and exponential ergodicity in total variation for an affine process on the canonical state space $\mathbb{R}_{\geq0}^{m}\times\mathbb{R}^{n}$. Under a Hörmander-type condition for diffusion components as well as a boundary non-attainment assumption, we derive the existence and regularity of the transition density for the affine process and then prove the strong Feller property. Moreover, we also show that under these and the additional subcritical conditions the corresponding affine process on the canonical state space is exponentially ergodic in t…
In this paper, Wang's log-Harnack inequality and exponential ergodicity are derived for two types of distribution dependent SDEs: one is the CKLS model, where the diffusion coefficient is a power function of order $θ$ with $θ\in[\frac{1}{2},1)$; the other one is Vasicek model, where the diffusion coefficient only depends on distribution. Both models in the distribution independent case can be used to characterize the interest rate in mathematical finance.
Considering irreducibility is fundamental for studying the ergodicity of stochastic dynamical systems. In this paper, we establish the irreducibility of stochastic complex Ginzburg-Laudau equations driven by pure jump noise. Our results are dimension free and the conditions placed on the driving noises are very mild. A crucial role is played by criteria developed by the authors of this paper and T. Zhang for the irreducibility of stochastic equations driven by pure jump noise. As an application, we obtain the ergodicity of stochastic complex Ginzburg-Laudau equations. We remark that our ergodi…
By Girsanov's thoerem and using the existing log-Harnack inequality for distribution independent SDEs, the log-Harnack inequality is derived for path-distribution dependent stochastic Hamiltonian systems. As an application, the exponential ergodicity in relative entropy is obtained by combining with transportation cost inequality. In addition, the quantitative propagation of chaos in the sense of Wasserstein distance, which together with the coupling by change of measure implies the quantitative propagation of chaos in total variation norm as well as relative entropy are obtained.
