On Multidimensional Diffusion Processes with Jumps (original) (raw)

Density Estimation of L^|^eacute;vy Measures for Discretely Observed Diffusion Processes with Jumps

JOURNAL OF THE JAPAN STATISTICAL SOCIETY, 2006

We study a nonparametric estimation of Lévy measures for multidimensional jump-diffusion models from some discrete observations. We suppose that the jump term is driven by a Lévy process with finite Lévy measure, that is, a compound Poisson process. We construct a kernel-estimator of the Lévy density under a sampling scheme where the terminal time tends to infinity and at the same time the distance between the observations tends to zero fast enough, and show the L 2consistency and the optimal rate in the MSE sense. First, we consider the case where the observations are given continuously and then compare it to the discretely observed case.

Weak Dirichlet processes with jumps

Stochastic Processes and their Applications, 2017

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N, A] = 0, for any continuous local martingale N. Given a function u : [0, T ] × R → R, which is of class C 0,1 (or sometimes less), we provide a chain rule type expansion for u(t, X t) which stands in applications for a chain Itô type rule.

Regularity and stability for the semigroup of jump diffusions with state-dependent intensity

The Annals of Applied Probability, 2018

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc.). First, we give sufficient conditions guaranteeing that the semigroup associated with such an equation preserves regularity by mapping the space of k-times differentiable bounded functions into itself. Furthermore, we give an upper estimate of the operator norm. This is the key-ingredient in a quantitative Trotter-Kato-type stability result: it allows us to give an upper estimate of the distance between two semigroups associated with different sets of coefficients in terms of the difference between the corresponding infinitesimal operators. As an application, we present a method allowing to replace "small jumps" by a Brownian motion or by a drift component. The example of the 2D Boltzmann equation is also treated in all detail.

The Doeblin condition for a class of diffusions with jumps

We prove non-explosiveness and a lower bound of the spectral gap via the strong Doeblin condition for a large class of stochastic processes evolving in the inte-rior of a region D ⊆ R d with boundary ∂D according to an underlying Markov process with transition probabilities p(t, x, dy), undergoing jumps to a random point x in D with distribution ν ξ (dx) as soon as they reach a boundary point ξ. Besides usual regularity conditions on p(t, x, dy), we require a tightness condition on the family of measures ν ξ , preventing mass from escaping to the boundary. The setup can be applied to a multitude of models considered recently, including a particle system with the Bak-Sneppen dynamics from evolutionary biology.

Topics on diffusion semigroups on a path space with Gibbs measures, (ギブス測度に関する経路空間上の拡散半群の話題）

Proceedings of RIMS Workshop on Stochastic Analysis and Applications, RIMS Kokyuroku Bessatsu, 2008

In this paper, we give a summary of recent results on symmetric diffusion semigroups associated with classical Dirichlet forms on an infinite volume path space C(\mat hr m{ R} , \mat hr m{ R}{ d}) with Gibbs measures. First, we discuss essential self-adjointness of diffusion operators (Dirichlet operators) associated with the Dirichlet forms. We also show the connection between the corresponding diffusion semigroup and the solution of a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on R. Next, we present some functional inequalities for the diffusion semigroup. As applications of these inequalities, we have the existence of a gap at the lower end of spectrum of the Dirichlet operator and the boundedness of the Riesz transforms.

Heat Kernels of Non-symmetric Jump Processes: Beyond the Stable Case

Potential Analysis

Let J be the Lévy density of a symmetric Lévy process in R d with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator L κ f (x) := lim ε↓0 {z∈R d :|z|>ε} (f (x + z) − f (x))κ(x, z)J(z) dz , where κ(x, z) is a Borel function on R d × R d satisfying 0 < κ 0 ≤ κ(x, z) ≤ κ 1 , κ(x, z) = κ(x, −z) and |κ(x, z) − κ(y, z)| ≤ κ 2 |x − y| β for some β ∈ (0, 1]. We construct the heat kernel p κ (t, x, y) of L κ , establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel p κ .

Conservation property of symmetric jump processes

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2011

Motivated by the recent development in the theory of jump processes, we investigate its conservation property. We will show that a jump process is conservative under certain conditions for the volume-growth of the underlying space and the jump rate of the process. We will also present examples of jump processes which satisfy these conditions. Résumé. Motivés par les récents développements dans la théorie des processus de sauts, nous étudions leur propriété de conservation. Nous montrons qu'un processus de saut est conservatif sous certaines conditions sur la croissance du volume de l'espace sous-tendant et sur le taux de saut du processus. Nous donnons des examples de processus satisfaisant ces conditions.

Integro-differential operators associated with diffusion processes with jumps

Applied Mathematics & Optimization, 1982

We show existence and W12~ p (3 Wl,°%regularity results for the integro-differential equation, associated with a diffusion process with jumps on a bounded domain. The second order elliptic partial differential operator and the integral operator involved here are both maximum principle type operators, which enables us to make W ~'°° a priori estimates.

Ergodicity for multidimensional jump diffusions with position dependent jump rate

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2017

We consider a jump type diffusion X = (X t) t with infinitesimal generator given by Lψ(x) = 1 2 1≤i,j≤d a ij (x) ∂ 2 ψ(x) ∂x i ∂x j + g(x)∇ψ(x) + R d (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) where µ is of infinite total mass. We prove Harris recurrence of X using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals.