Related papers: Stochastic equations of non-negative processes wit…
In this study, we consider a class of backward SDE driven by jump Markov process. An existence and uniqueness result to this kind of equations is obtained in a locally Lipschitz case. We essentially approximate the initial problem by…
We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion…
We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the…
Consider the linear stochastic differential equation (SDE) on $\mathbb{R}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\] where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a L\'{e}vy…
Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be obtained under the…
In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schr\"{o}dinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the…
We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for…
We analyze confining mechanisms for L\'evy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process…
It is proved that the two-sided exits of a Levy process are proper, i.e. not a.s. equal to their one-sided counterparts, if and only if said process is not a subordinator or the negative of a subordinator. Furthermore, Levy processes are…
In this paper we present an $L^p$-theory for the stochastic partial differential equations (SPDEs in abbreciation) driven by L\'e{}vy processes. Existence and uniqueness of solutions in Sobolev spaces are obtained. The coefficients of SPDEs…
We consider the dynamics of a 1D system evolving according to a deterministic drift and randomly forced by two types of jumps processes, one representing an external, uncontrolled forcing and the other one a control that instantaneously…
The escape probability is a deterministic concept that quantifies some aspects of stochastic dynamics. This issue has been investigated previously for dynamical systems driven by Gaussian Brownian motions. The present work considers escape…
In a high-frequency context, we investigate the efficient estimation of scaling and jump activity parameters for a stochastic differential equation driven by a L{\'e}vy process with both diffusion component and pure-jump component. We first…
We describe stochastic calculus in the context of processes that are driven by an adapted point process of locally finite intensity and are differentiable between jumps. This includes Markov chains as well as non-Markov processes. By…
The dynamics of the eigenvalues (semimartingales) of a L\'{e}vy process $X$ with values in Hermitian matrices is described in terms of It\^{o} stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian…
A two-type continuous-state branching process in varying environments is constructed as the pathwise unique solution of a system of stochastic equations driven by time-space noises, where the pathwise uniqueness is derived from a comparison…
We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise. In view of the L\'{e}vy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental…
Efficient estimation of a non-Gaussian stable Levy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a…
After defining non-Gaussian L\'evy processes for two-sided time, stochastic differential equations with such L\'evy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in…
We prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing…