Rangrang Zhang, Guoli Zhou
In this paper, we establish a large deviation principle for a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by pure jump noise. The proof is based on the weak convergence approach.
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In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution is based on the classical Faedo-Galerkin approximation, compactness method and the Jakubowski's version of the Skorokhod representation theorem for non-metric spaces. We prove the solution is pathwise unique and further establish the existence of a strong solution in the 2D case.
Heat kernel lower bound estimates for symmetric pure jump processes via averaged jump kernels
利用平均跳跃核，在一般体积加倍度量测度空间上推导对称纯跳过程的热核下界估计，并应用于Sierpinski垫片底部布朗运动迹的转移密度函数下界。
The objective of this paper is to study the filtering problem for a system of partially observable processes $(X, Y)$, where $X$ is a non-Markovian pure-jump process representing the signal and $Y$ is a general jump-diffusion which provides observations. Our model covers the case where both processes are not necessarily quasi left-continuous, allowing them to jump at predictable stopping times. By introducing the Markovian version of the signal, we are able to compute an explicit equation for the filtering process via the innovations approach.
We present sufficient conditions, in terms of the jumping kernels, for two large classes of conservative Markov processes of pure-jump type to be purely discontinuous martingales with finite second moment. As an application, we establish the law of the iterated logarithm for sample paths of the associated processes.
In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.
We obtain a lower bound for the coarse Ricci curvature of continuous time pure jump Markov processes, with an emphasis on interacting particle systems. Applications to several models are provided, with a detailed study of the herd behavior of a simple model of interacting agents.
We aim in proving Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced in \cite{G-L} by Galves and Löcherbach to describe the behaviour of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e. that lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system.
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…
We study Talagrand concentration and Poincaré type inequalities for unbounded pure jump Markov processes. In particular we focus on processes with degenerate jumps that depend on the past of the whole system, based on the model introduced by Galves and Löcherbach in \cite{G-L}, in order to describe the activity of a biological neural network. As a result we obtain exponential rates of convergence to equilibrium.
A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory (Jacod and Shiryaev, 2003, II.1.5), the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes: the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale $X$ has a unique decomposition $$X = X_0 + X^{\mathrm{qc}}+X^{\mathrm{dp}},$$ where $X^{\mathrm{qc}}$ is quas…
In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump degenerate noise, which cover 2D stochastic Navier-Stokes equations, stochastic Burgers type equations, singular stochastic p-Laplace equations, stochastic fast diffusion equations, etc. As a further application, we establish the ergodicity of si…
Based on the weak existence and weak uniqueness, we study the pathwise uniqueness of the solutions for a class of one-dimensional stochastic differential equations driven by pure jump processes. By using Tanaka's formula and the local time technique, we show that there is no gap between the strong uniqueness and weak uniqueness when the coefficients of the Poisson random measures satisfy a suitable condition
In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the $Q$-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature, and demonstrate some possible applications.
Uniqueness for the martingale problem associated with pure jump processes of variable order
设$L$为$$L f(x)=\int[f(x+h)-f(x)-1_{(|h|\leq 1)}\nabla f(x)\cdot h]\frac{n(x,h)}{|h|^{d+α(x)}}dh.$$在$C^2$函数上定义的算子，这是一个变阶算子，对应的过程是纯跳跃型的.我们考虑与$L$相关的鞅问题。给出了解存在唯一的充分条件。对$α$ -稳定过程的转移密度估计也得到了。
Weak irreducibility of stochastic delay differential equation driven by pure jump noise
本文研究纯跳跃噪声驱动的随机延迟微分方程弱不可约性，放宽已有文献假设条件，给出简洁证明，推导弱耗散系数情形下的结果，创新在于允许退化驱动噪声，填补了该领域研究空白。
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $α$-stable noise.
Time fractional stochastic differential equations driven by pure jump Lévy noise
本文提出了由纯跳Lévy噪声驱动的变阶时间分数阶微分方程以建模粒子记忆效应，利用截断论证证明了方程的适定性，并推导出解的$L^p$矩估计与Hölder正则性。
Ergodicity of Stochastic two-phase Stefan problem driven by pure jump Lévy noise
本文研究由纯跳Lévy噪声驱动的随机两相Stefan问题，证明其强解存在唯一并建立遍历性，同时刻画不变测度支撑集以给出平稳解的正则性。
Existence of density functions for SDEs driven by pure-jump processes
该研究借助跳跃过程的Bismut型Malliavin演算方法，通过证明维纳-泊松泛函运行最大值的密度函数存在性，验证了布朗运动与非截断稳定过程驱动的SDE运行最大值的密度函数存在性。
In this paper, we consider the following symmetric non-local Dirichlet forms of pure jump type on metric measure space $(M,d,μ)$: $$\mathcal{E}(f,g)=\int_{M\times M} (f(x)-f(y))(g(x)-g(y))\,J(dx,dy),$$ where $J(dx,dy)$ is a symmetric Radon measure on $M\times M\setminus {\rm diag}$ that may have different scalings for small jumps and large jumps. Under general volume doubling condition on $(M,d,μ)$ and some mild quantitative assumptions on $J(dx, dy)$ that are allowed to have light tails of polynomial decay at infinity, we establish stability results for two-sided heat kernel estimates as well…
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…
This paper derives the asymptotic behavior of realized power variation of pure-jump Itô semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled Itô semimartingale over a fixed interval.
Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise
本文针对高度退化的纯跳Lévy噪声驱动的二维Navier-Stokes方程，利用Malliavin分析与预想随机分析建立e-property和弱不可约性，从而证明遍历性并解决长期未决问题。
Stochastic curve shortening flow driven by a transport-type pure jump Lévy noise
研究由传输型纯跳莱维噪声驱动的随机曲线缩短流的强解存在性、唯一性、正则性及长时间行为，通过传输方程转换为伊藤型随机偏微分方程，结合单调方法与李雅普诺夫型条件，证明解以指数速率路径收敛于零。
Irreducibility and ergodicity of SPDEs driven by pure jump noise
本文提出一种新方法，在温和且近乎必要充分的条件下，证明乘性纯跳跃噪声驱动的SPDEs/SDEs的不可约性，并应用于消除限制条件，进而获得多类奇异随机演化方程的遍历性。
Affine pure-jump Volterra fields
本文提出仿射纯跳 Volterra 场，通过稀疏化 Poisson 测度积分定义，导出仿射变换公式，推广纯跳仿射 Volterra 过程至随机场，涵盖幅值场、Hawkes 系统等特例。
Ergodicity for stochastic 2D Boussinesq equations with a highly degenerate pure jump Levy noise
研究仅作用于温度方程的退化纯跳跃Lévy噪声驱动的随机二维Boussinesq方程的遍历性，通过Malliavin分析和随机分析证明不变测度的存在唯一性。
We study optimal investment strategies that maximize expected utility from consumption and terminal wealth in a pure-jump asset price model with Markov-modulated (regime switching) jump-size distributions. We give sufficient conditions for existence of optimal policies and find closed-form expressions for the optimal value function for agents with logarithmic and fractional power (CRRA) utility in the case of two-state Markov chains. The main tools are convex duality techniques, stochastic calculus for pure-jump processes and explicit formulae for the moments of telegraph processes with Markov…
Motivated by the multilevel Monte Carlo method introduced by Giles [5], we study the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler approximations with time steps $1/n$ and $1/nm$ of a given stochastic differential equation $X$ driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [9] …
We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.
We show the existence of a broad class of affine Markov processes in the cone of positive self-adjoint Hilbert-Schmidt operators. Such processes are well-suited as infinite dimensional stochastic volatility models. The class of processes we consider is an infinite dimensional analogue of the affine processes in the space of positive semi-definite and symmetric matrices studied in Cuchiero et al. [Ann. Appl. Probab. 21 (2011) 397-463]. As in the finite dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well jumps governed by a jump measure …
Central Limit Theorem for Stochastic Nonlinear Heat Equation with Pure-Jump Lévy White Noise
本文针对一维Lévy白噪声驱动的随机非线性热方程，在特定矩条件下证明了空间平均的定量和泛函中心极限定理，将高斯波动理论推广至非线性情形。
In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions $ρ$ to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe $ρ(x,t)$ as a stochastic process in $x$ with $t$ fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of $ρ(x,t)$ over $0 \leq x \leq L$ for $L > 0$, with a suitable random boundary condition at $x = L$.
In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.
We study the stochastic Camassa-Holm equation with pure jump noise. We prove that if the initial condition of the solution is a solitary wave solution of the unperturbed equation, the solution decomposes into the sum of a randomly modulated solitary wave and a small remainder. Moreover, we derive the equations for the modulation parameters and show that the remainder converges to the solution of a stochastic linear equation as amplitude of the jump noise tends to zero.
We present simple new examples of pure-jump strict local martingales. The examples are constructed as exponentials of self-exciting affine Markov processes. We characterize the strict local martingale property of these processes by an integral criterion and by non-uniqueness of an associated ordinary differential equation. Finally we show an alternative construction for our examples by an absolutely continuous measure change in the spirit of (Delbaen and Schachermayer, PTRF 1995).
We consider a class of pure jump Markov processes in $\rr^d$ whose jump kernels are comparable to those of symmetric stable processes. We prove a support theorem, a lower bound on the occupation times of sets, and show that we can approximate resolvents using smooth functions.
Well-posedness and asymptotic behavior of the stochastic convective Brinkman-Forchheimer equations perturbed by pure jump noise
本文研究乘性纯跳跃噪声驱动的随机对流 Brinkman-Forchheimer 方程，建立了强解的适定性、指数稳定性、镇定结果及唯一遍历不变测度。
Symmetric jump processes: localization, heat kernels, and convergence
我们考虑纯跳型对称过程。我们证明了当地的估计退出球的概率，Hölder连续性的调和函数和热核，和收敛的一系列这样的过程。
Inverting the Markovian projection for pure jump processes
本文研究了纯跳过程的马尔可夫投影逆向问题，旨在构建用于信用风险的局部随机强度模型。
Fluid Limits of Pure Jump Markov Processes: a Practical Guide
在仔细指定的假设下，重新标度的马尔可夫链以概率一致地收敛到常微分方程的解。介绍比那些在外面的文献是简单得多。其结果可用于建立大型随机或伪随机系统的简约模型。
Volterra clocks and their pure-jump limits: hitting times of curved boundaries
本文引入一类连续Volterra过程（Volterra时钟），研究其记忆核坍缩为Dirac质量时的奇异极限，通过拓扑框架证明极限为布朗运动到曲线边界的首达时间所定义的纯跳过程，统一了多种感兴趣机制。
We explore martingale and convex duality techniques to study optimal investment strategies that maximize expected risk-averse utility from consumption and terminal wealth. We consider a market model with jumps driven by (multivariate) marked point processes and so-called non-linear wealth dynamics which allows to take account of relaxed assumptions such as differential borrowing and lending interest rates or short positions with cash collateral and negative rebate rates. We give suffcient conditions for existence of optimal policies for agents with logarithmic and CRRA power utility. We find c…
Littlewood--Paley estimates for pure-jump Dirichlet forms
本文利用Hardy--Stein恒等式的最新推广和Revuz对应理论，将Littlewood--Paley估计推广到一般纯跳Dirichlet型，放宽了假设条件，并通过反例纠正了先前工作中的错误。
Efficient estimation of jump parameters for stochastic differential equations driven by L{é}vy processes
针对带扩散与纯跳跃分量的Lévy过程驱动的随机微分方程，研究高频场景下跳跃相关参数的有效估计，证明参数联合估计的LAN性质，并构造出达到最优收敛速率的拟似然估计量。
In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Lévy process with finite Lévy measure, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.
In the present paper we study a stochastic evolution equation for shell (SABRA \& GOY) models with pure jump \levy noise $L=\sum_{k=1}^\infty l_k(t)e_k$ on a Hilbert space $\h$. Here $\{l_k, k\in \mathbb{N}\}$ is a family of independent and identically distributed (i.i.d.) real-valued pure jump \levy processes and $\{e_k, k\in \mathbb{N}\}$ is an orthonormal basis of $\h$. We mainly prove that the stochastic system has a unique invariant measure. For this aim we show that if the \levy measure of each component $l_k(t)$ of $L$ satisfies a certain order and a non-{degeneracy} condition and is ab…
We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an $L^2-$valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differe…
By a coupling method, we prove that a family of stochastic partial differential equations (SPDEs) driven by highly degenerate pure jump Lévy noises are exponential mixing. These pure jump Lévy noises include $α$-stable process with $α\in (0,2)$
We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the Lévy system formula and a near-diagonal heat kernel lower bound.
The objective of this paper is to give conditions ensuring that the backward partial integro differential equation associated with a multidimensional jump-diffusion with a pure jump component has a unique classical solution; that is the solution is continuous, twice differentiable in the diffusion component and differentiable in time. Our proof uses a probabilistic arguments and extends the results of Pham (1998) to processes with a pure jump component where the jump intensity is modulated by a diffusion process. This result is particularly useful in some applications to pricing and hedging of…
By using the $Φ$-entropy inequality derived in \cite{Wu, Ch} for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump Lévy processes. The semigroup $Φ$-entropy inequality for SDEs driven by Poisson point processes as well as a sharp result on the existence of invariant probability measures are also presented.
Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes
本文在变分框架下利用 Kurtz 结果，将 Yamada-Watanabe 理论推广到由 Wiener 过程与 Poisson 随机测度共同驱动的非线性 SPDE，证明鞅解存在加轨道唯一性可推出唯一强解存在。
Ergodicity for the fractional Magneto-Hydrodynamic equations driven by a degenerate pure jump noise
本文研究了由高度退化纯跳跃 Lévy 噪声驱动的随机 2D 分数阶磁流体动力学方程的遍历性，针对噪声仅作用于四个方向的高退化情形，结合 Malliavin 变分法证明半群 e-property 与弱不可约性，确立了不变测度的唯一性。
A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein--Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applicati…
Estimation of a pure-jump stable Cox-Ingersoll-Ross process
本文研究了纯跳跃稳定CIR过程的参数估计，提出基于似然近似的相合估计量，证明其渐近性质，并通过一步法改进初步估计量。
We construct a martingale solution of the stochastic nonlinear Schrödinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood-Paley decomposition complement the classical orthogonal projections to gu…
A pathwise large deviation result is proved for the pure jump models of $k$-nary interacting particle system introduced by Kolokoltsov that generalize classical Boltzmann's collision model, Smoluchovski's coagulation model and many others. The upper bound is obtained by following the standard methods of using a process "perturbed" by a regular function. To show the lower bound, we propose a family of orthogonal martingale measures and prove a coupling for the general perturbations. The rate function is studied based on the idea of Léonard with a simplification by considering the conjugation of…
We consider the problem of utility maximization with exponential preferences in a market where the traded stock/risky asset price is modelled as a Lévy-driven pure jump process (i.e. the driving Lévy process has no Brownian component). In this setting, we study the terminal utility optimization problem in the presence of a European contingent claim. We consider in detail the BSDE (backward stochastic differential equation) characterising the value function when using an exponential utility function. First we analyse the well-definedness of the generator. This leads to some conditions on the ma…
The stochastic MHD equations driven by pure jump noise in $L^p$ spaces
研究由纯跳跃噪声驱动的随机不可压缩磁流体动力学方程，在 $L^p$ 空间中建立了温和解的局部存在唯一性，并在二维情形下证明了全局存在性。
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