Canonical Lévy process and Malliavin calculus

A suitable canonical Lévy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Lévy processes proved by Itô using multiple two-parameter integrals. In this setup, the two-parameter derivative _Dt,x$D_{t,x}$ is studied, depending on whether x=0$x=0$ or x≠0$x\ne 0$; in the first case, we prove a chain rule; in the second case, a formula by trajectories.     

Finite approximation schemes for Lévy processes,and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T]$[0,T]$, and having a finite number of states, for a pure jump Lévy process _Lt$L_t$. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.     

Simulation of the continuous time random walk of the space-fractional diffusion equations

In this article, we discuss the solution of the space-fractional diffusion equation with and without central linear drift in the Fourier domain and show the strong connection between it and the α$\alpha$-stable Lévy distribution, 0<α<2$0<\alpha <2$. We use some relevant transformations of the independent variables x$x$ and t$t$, to find the solution of the space-fractional diffusion equation with central linear drift which is a special form of the space-fractional Fokker–Planck equation which is useful in studying the dynamic behaviour of stochastic differential equations driven by the non-Gaussian (Lévy) noises. We simulate the continuous time random walk of these models by using the Monte Carlo method.     

Multivariate CARMA processes

A multivariate Lévy-driven continuous time autoregressive moving average (CARMA) model of order (p,q$p,q$), q<p$q<p$, is introduced. It extends the well-known univariate CARMA and multivariate discrete time ARMA models. We give an explicit construction using a state space representation and a spectral representation of the driving Lévy process. Furthermore, various probabilistic properties of the state space model and the multivariate CARMA process itself are discussed in detail.     

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

Functional limit theorems for renewal shot noise processes with increasing response functions

We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space D[0,∞)$D[0,\infty )$ under the _J1$J_1$ or _M1$M_1$ topology. The limiting processes are either spectrally nonpositive stable Lévy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of regular variation is positive). The proof exploits fine properties of renewal processes, distributional properties of stable Lévy processes and the continuous mapping theorem.     

Time regularity of solutions to linear equations with Lévy noise in infinite dimensions

The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space H$H$, driven by a Lévy process taking values in a Hilbert space U?H$U?H$ is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite systems of linear equations driven by independent Lévy processes.     

Regularity of semigroups generated by Lévy type operators via coupling

By adopting the coupling method, we obtain new verifiable sufficient conditions about the _Cb(R^d)$C_b\left({\mathbb{R}}^d\right)$-Feller continuity, the Lipschitz continuity and the strong Feller continuity of the semigroups associated with Lévy type operators. These results easily apply to jump–diffusion processes and stochastic differential equations driven by Lévy processes. Our results also yield the criterion for the e$e$-property (namely the characterization about the equi-continuity of semigroups acting on bounded Lipschitz functions) of Lévy type operators, and show that both genuine Lévy processes and the Ornstein–Uhlenbeck type processes are e$e$-processes.     

On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps

We study the Wiener–Hopf factorization for Lévy processes with bounded positive jumps and arbitrary negative jumps. We prove that the positive Wiener–Hopf factor can be expressed as an infinite product involving solutions to the equation ψ(z)=q$\psi \left(z\right)=q$, where ψ$\psi$ is the Laplace exponent. Under additional regularity assumptions on the Lévy measure we obtain an asymptotic expression for these solutions. When the process is spectrally negative with bounded jumps, we derive a series representation for the scale function. In order to illustrate possible applications, we discuss the implementation of numerical algorithms and present the results of several numerical experiments.     

The maximum of a Lévy process reflected at a general barrier

We investigate the reflection of a Lévy process at a deterministic, time-dependent barrier and in particular properties of the global maximum of the reflected Lévy process. Under the assumption of a finite Laplace exponent, ψ(θ)$\psi \left(\theta \right)$, and the existence of a solution θ^∗>0${\theta }^{\ast }>0$ to ψ(θ)=0$\psi \left(\theta \right)=0$ we derive conditions in terms of the barrier for almost sure finiteness of the maximum. If the maximum is finite almost surely, we show that the tail of its distribution decays like Kexp(−θ^∗x)$Kexp(-{\theta }^{\ast }x)$. The constant K$K$ can be completely characterized, and we present several possible representations. Some special cases where the constant can be computed explicitly are treated in greater detail, for instance Brownian motion with a linear or a piecewise linear barrier. In the context of queuing and storage models the barrier has an interpretation as a time-dependent maximal capacity. In risk theory the barrier can be interpreted as a time-dependent strategy for (continuous) dividend pay out.     

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C^1,1$C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C^1,1$C^{1,1}$ open set D$D$ and identify the Martin boundary of D$D$ with respect to the subordinate Brownian motion with the Euclidean boundary.     

On non-parametric estimation of the Lévy kernel of Markov processes

We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations _X0,_XΔ,…,_XnΔ$X_0,X_{\Delta },\dots ,X_{n\Delta }$, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞$n\Delta \to \infty$ and Δ→0$\Delta \to 0$) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided.