V. P. Kurenok
Using the method of Krylov's estimates, we prove the existence of weak solutions of stochastic differential equations driven by purely discontinuous Levy processes satisfying an additional assumption. The diffusion coefficient is assumed to be one and the time-dependent drift is measurable and bounded.
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We consider the problem of estimation of the drift parameter of an ergodic Ornstein--Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed continuously on a long time interval $[0,T]$, $T\to\infty$. We prove that the statistical model is locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically efficient.
We stu\dd y a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy noise. We then prove a decomposition of the solution process in a stationary component and a component which vanishes asymptotically for large times in the $L^p-$sense, $p\geq1$. The law of the stationary component is identified with the unique invariant proba…
Euler-Maruyama scheme for SDE driven by Lévy process with Hölder drift
本文研究具有 Hölder 连续漂移项的 Lévy 过程驱动 SDE 的 Euler-Maruyama 数值逼近，并推导涵盖所有非退化 $α$-stable 过程在内的广泛噪声下的 $L^p$ 误差。
We are concerned about the averaging principle for the stochastic Burgers equation with slow-fast time scale. This slow-fast system is driven by Lévy processes. Under some appropriate conditions, we show that the slow component of this system strongly converges to a limit, which is characterized by the solution of stochastic Burgers equation whose coefficients are averaged with respect to the stationary measure of the fast-varying jump-diffusion. To illustrate our theoretical result, we provide some numerical simulations.
On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients
本文研究Lévy过程驱动的McKean-Vlasov随机微分方程的数值逼近，提出了一种驯服自适应Euler-Maruyama格式，并证明了其在非全局Lipschitz连续且超线性增长系数下于有限和无限时间跨度的强收敛性。
We consider an SDE in R^m of the type dX(t)=a(X(t))dt+dU(t) with a Lévy process U and study the problem for the distribution of a solution to be regular in various senses. We do not impose any specific conditions on the Lévy measure of the noise, and this is the main difference between our method and the known methods by J.Bismut or J.Picard. The main tool in our approach is the stochastic calculus of variations for a Lévy process, based on the time-stretching transformations of the trajectories. Three problems are solved in this framework. First, we prove that if the drift coefficient a is no…
The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether the Lévy process is drifting to $+\infty$, $-\infty$ or oscillates. Whenever the Lévy process drifts to $+\infty$, we prove that the $κ$-th moment of the first passage time (conditioned to be finite) exists if and only if the $(κ+1)$-th moment of the Lévy jump measure exists. This generalises a result shown earlier by Delbaen for Cramér-Lundberg risk processes \cite{Delbaen1990}. Whenever the Lévy process drifts to $-\infty$, we pr…
We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.
In this paper, based on the white noise analysis of square integrable pure-jump Levy process given by [1], we define the formal derivative of fractional Levy process defined by the square integrable pure-jump Levy process as the fractional Levy noises by considering fractional Levy process as the generalized functional of Levy process, and then we define the Skorohod integral with respect to the fractional Levy process. Moreover, we propose a class of stochastic Volterra equations driven by fractional Levy noises and investigate the existence and uniqueness of their solutions; In addition, we …
We extend the taming techniques developed in \cite{konstantinos2014,sabanis2013} to construct explicit Milstein schemes that numerically approximate Lévy driven stochastic differential equations with super-linearly growing drift coefficients. The classical rate of convergence is recovered when the first derivative of the drift coefficient satisfies a polynomial Lipschitz condition.
Parabolic Fractal Geometry of Stable Lévy Processes with Drift
该研究引入抛物Hausdorff维数，利用抛物柱体覆盖的时空非线性标度匹配自相似性，显式求出带确定性漂移的各向同性稳定Lévy过程的图像与值域的Hausdorff维数，证明漂移项的抛物维数的贡献，并推导相关公式与界。
Bôcher type theorem for elliptic equations with drift perturbed Lévy operator
本文将经典Bôcher定理推广至漂移扰动的Lévy算子，提出基于概率位势理论的新方法，适用于预解式为强Feller的算子，涵盖次临界与超临界的分数阶Laplacian。
Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime
本文研究了具有奇异漂移和分数布朗运动或莱维噪声的SDE的弱存在性，提出了在最优可积指数条件下成立的结论，并在一维情况下证明了强解的存在。
Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with Lévy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced.
Low-Rank and Sparse Drift Estimation for High-Dimensional Lévy-Driven Ornstein--Uhlenbeck Processes
本文研究高维莱维驱动的奥本-乌尔申贝克过程，提出低秩加稀疏结构的漂移矩阵估计方法，通过核范数与ℓ1惩罚结合的凸估计器，在受限强凸性条件下推导出估计器的Frobenius风险非渐近Oracle不等式。
Chernoff-Mehler Approximation for Lévy Processes with Drift
本文提出了一种基于切尔诺夫乘积公式的近似方法，用于具有漂移的莱维过程，通过提供显式条件确保测度族收敛，并展示了其在微分方程和中心极限定理中的应用。
In this paper we study the following stochastic differential equation (SDE) in ${\mathbb R}^d$: $$ \mathrm{d} X_t= \mathrm{d} Z_t + b(t, X_t)\mathrm{d} t, \quad X_0=x, $$ where $Z$ is a Lévy process. We show that for a large class of Lévy processes ${Z}$ and Hölder continuous drift $b$, the SDE above has a unique strong solution for every starting point $x\in{\mathbb R}^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. As a consequence, we show that, when ${Z}$ is an $α$-stable-type Lévy process with $α\in (0, 2)$ and $b$ is bounded and $β$-Hölder continuous with $β\in (1- α/{…
We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cramér-Lundberg risk process.
We consider the problem of absolute continuity for the one-dimensional SDE \[X_t=x+\int_0^ta(X_s) ds+Z_t,\] where $Z$ is a real Lévy process without Brownian part and $a$ a function of class $\mathcal{C}^1$ with bounded derivative. Using an elementary stratification method, we show that if the drift $a$ is monotonous at the initial point $x$, then $X_t$ is absolutely continuous for every $t>0$ if and only if $Z$ jumps infinitely often. This means that the drift term has a regularizing effect, since $Z_t$ itself may not have a density. We also prove that when $Z_t$ is absolutely continuous, the…
We consider a branching Markov process in continuous time in which the particles evolve independently as spectrally negative Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particule. The purpose of this paper is to give asymptotic estimates for the survival function of this maximum. In particular, we show that in the critical case the asymptotics is polynomial when the underlying Lévy process oscillates or drifts towards $+\infty$, and is exponential when…
Continuous-state branching processes with Lévy-Khintchine drift-interaction: Laplace duality and Fellerian extensions
本文研究了一类由Lévy-Khintchine型漂移驱动的连续状态分支过程（CBDI），通过拉普拉斯对偶性和Feller扩展分析了其边界行为和分支与漂移-相互作用机制的交换特性。
We show that the result from Da Prato and Lunardi is valid for stochastic convolutions driven by Lévy processes.
On Two Parameter Time-Changed Poisson Random Fields with Drifts
本文研究了二元 Lévy 过程与逆子ordinator的复合，推导了其色散与自相关矩阵，并引入了带漂移的双参数时变 Poisson 随机场及其分布的分数阶微分方程。
We present the Levy's distributional property for symmetric Levy processes with generating triplet $(0, 0,ν)$ or $(σ>0, γ, ν)$ where $ν$ is a symmetric measure on $R\backslash\{0\}$. This generalizes the classical Levy's theorem about Brownian motions with drift.
This paper studies stabilities of stochastic differential equation (SDE) driven by time-changed Lévy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas including physics, biology, engineering, finance and hydrology. Necessary conditions for solution of time-changed SDE to be stable in different senses will be established. Connection between stability of solution to time-changed SDE and that to corresponding original SDE will be disclosed. Examples related to different stabilities will be given. We study SDEs with time-changed Lév…
In this paper, we investigate stochastic partial differential equations driven by multi-parameter anisotropic fractional Levy noises, including the stochastic Poisson equation, the linear heat equation, and the quasi-linear heat equation. Well-posedness of these equations under the fractional noises will be addressed. The multi-parameter anisotropic fractional Levy noise is defined as the formal derivative of the anisotropic fractional Levy random field. In doing so, there are two folds involved. First, we consider the anisotropic fractional Levy random field as the generalized functional of t…
We investigate a class of stochastic integro differential equations driven by Levy noise.
We prove an existence and uniqueness result for generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz assumptions.
This paper studies path stabilities of the solution to stochastic differential equations (SDE) driven by time-changed Lévy noise. The conditions for the solution of time-changed SDE to be path stable and exponentially path stable are given. Moreover, we reveal the important role of the time drift in determining the path stability properties of the solution. Related examples are provided.
Drift Estimation for a Multi-Dimensional Lévy-Driven Stochastic Differential Equation Using Deep Neural Networks
本文用稀疏ReLU神经网络估计多维Lévy驱动随机微分方程的漂移项，在弱集中条件下获得与扩散模型同阶的统计界，并证明估计器在极小极大意义下最优。
In this paper we give an $L_p$-theory for stochastic parabolic equations with random fractional Laplacian operator. The driving noises are general Lévy processes.
Stochastic homogenization of diffusions in turbulence driven by non-local symmetric Lévy operators
本文研究了由非局部对称莱维算子驱动的湍流扩散的随机同质化问题，通过建立校正因子的局部$W^{1,q}$估计，揭示了在弱正则性条件下随机同质化的数学机制。
The aim of this note is to give some Burkholder-Davis-Gundy type inequalities which are valid for the Ito stochastic integral with respect to Banach valued Levy noise.
In this paper, we study the speed of extinction of continuous state branching processes in subcritical Lévy environments. More precisely, when the associated Lévy process to the environment drifts to $-\infty$ and, under a suitable exponential martingale change of measure (Esscher transform), the environment either drifts to $-\infty$ or oscillates. We extend recent results of Palau et al. (2016) and Li and Xu (2018), where the branching term is associated to a spectrally positive stable Lévy process and complement the recent article of Bansaye et al. (2021) where the critical case was studied…
We introduce a stochastic integral with respect to cylindrical Lévy processes with finite $p$-th weak moment for $p\in [1,2]$. The space of integrands consists of $p$-summing operators between Banach spaces of martingale type $p$. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process.
We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient $\varepsilon \to 0$ and $n \to \infty$ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to t…
In this article we prove the pathwise uniqueness for stochastic differential equations in $\mR^d$ with time-dependent Sobolev drifts, and driven by symmetric $α$-stable processes provided that $α\in(1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $α\in(\frac{2d}{d+1},2)$. Our proof is based on some estimates of Krylov's type for purely discontinuous semimartingales.
In the paper, we consider a type of stochastic differential equations driven by G-Lévy processes. We prove that a kind of their additive functionals has path independence and extend some known results.
By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semi-linear…
In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Levy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.
Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive Lévy process with non-zero Lévy measure. In this paper we study the asymptotic behavior of the local time process, in the spatial variable, of these processes killed at two different random times: either at the time of the first visit of the Lévy process to 0, in which case we prove results at the excursion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under gener…
In this paper, we establish a large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas in [5] plays a key role.
We consider a Stochastic Differential Equation driven by a Lévy process whose Lévy measure satisfy a tempered stable domination. We study how a perturbation of the coefficients reflects on the density of the solution. We quantify the distance between the densities in term of the proximity of the coefficients. This extend to the stable case the works of [KKM15], where the noise is Gaussian.
Stochastic logarithmic Schrödinger equations driven by Lévy noise
本文研究带饱和非线性乘性Lévy噪声的随机对数Schrödinger方程，通过正则化方程在Orlicz空间中构造强收敛解，证明了方程的全局适定性。
Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms for Lévy-driven stochastic differential equations
研究莱维驱动随机微分方程中欧拉-马尔日ama近似与精确解在时间-空间霍尔德范数下的误差，分析其收敛性及贡献。
We consider a stochastic model of incompressible non-Newtonian fluids of second grade on a bounded domain of $\mathbb{R}^2$ driven by Lévy noise. Applying the variational approach, global existence and uniqueness of strong probabilistic solution is established.
This paper considers stochastic population dynamics driven by Levy noise. The contributions of this paper lie in that (a) Using Khasminskii-Mao theorem, we show that the stochastic differential equation associated with the model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such model.
In this paper, we establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by Lévy noise. The weak convergence method plays an important role.
This paper investigates a damped stochastic wave equation driven by a non-Gaussian Levy noise. The weak solution is proved to exist and be unique. Moreover we show the existence of a unique invariant measure associated with the transition semigroup under mild conditions.
In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a Lévy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we extend the method based on Girsanov transformation on Wiener space and developped by Buckdahn to the canonical Lévy space.
In this work stochastic integration with respect to cylindrical Levy processes with weak second moments is introduced. It is well known that a deterministic Hilbert-Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Levy process by random Hilbert-Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic int…
Spatial asymptotic behaviors of fractional stochastic heat equations driven by additive Lévy white noise
该研究针对加性Lévy白噪声驱动的分数阶随机热方程，建立了其空间渐近行为的显式积分检验准则，证明轻尾噪声下方程在全维度具有加性物理间歇性质，证明依赖分数阶Laplacian热核估计与泊松泛函精确尾部行为。
Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable Lévy noise. The crux of our proof relies on the assumption of finite dimensional Lévy noise.
The problem of the construction of strong approximations with a given order of convergence for jump-diffusion equations is studied. General approximation schemes are constructed for Lévy type stochastic differential equation. In particular, the paper generalizes the results of Platen Kloeden and Gardo\n. The Euler and the Milstein schemes are shown for finite and infinite Lévy measure.
Blow-up of stochastic semilinear parabolic equations driven by Lévy noise
针对加性与线性乘性Lévy噪声驱动的随机半线性抛物方程，本文改进随机情形下的凹性方法，证明了有界区域上系统解的爆破现象。
Limit theorems for stochastic exponentials of matrix-valued Lévy processes
本文研究矩阵值Lévy过程随机指数的长时间行为，证明了对数化范数、元素和行列式的大数定律与中心极限定理，并给出可能的Berry-Esseen界。
In this work, we consider the stochastic Cauchy problem driven by the canonical $α$-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.
In this paper, we deal with a class of backward doubly stochastic differential equations (BDSDEs, in short) involving subdifferential operator of a convex function and driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness result by means of Yosida approximation. As an application, we give the existence of stochastic viscosity solution for a class of multivalued stochastic partial differential-integral equations (MSPIDEs, in short).
Motivated by recent studies of record statistics in relation to strongly correlated time series, we consider explicitly the drawdown time of a Levy process, which is defined as the time since it last achieved its running maximum when observed over a fixed time period [0,T]. We show that the density function of this drawdown time, in the case of a completely asymmetric jump process, may be factored as a function of $t$ multiplied by a function of T-t. This extends a known result for the case of pure Brownian motion. We state the factors explicitly for the cases of exponential down-jumps with dr…
The work is about multiscale stochastic dynamical systems driven by Lévy processes. First, we prove that these systems can approximate low-dimensional systems on random invariant manifolds. Second, we establish that nonlinear filterings of multiscale stochastic dynamical systems also approximate that of reduced low-dimensional systems. Finally, we investigate the reduction for $\e=0$ and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.
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