Constructions of coupling processes for Lévy processes

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.     

Randomly weighted self-normalized Lévy processes

Let (_Ut,_Vt)$(U_t,V_t)$ be a bivariate Lévy process, where _Vt$V_t$ is a subordinator and _Ut$U_t$ is a Lévy process formed by randomly weighting each jump of _Vt$V_t$ by an independent random variable _Xt$X_t$ having cdf F$F$. We investigate the asymptotic distribution of the self-normalized Lévy process _Ut/_Vt$U_t/V_t$ at 0 and at ∞$\infty$. We show that all subsequential limits of this ratio at 0 (∞$\infty$) are continuous for any nondegenerate F$F$ with finite expectation if and only if _Vt$V_t$ belongs to the centered Feller class at 0 (∞$\infty$). We also characterize when _Ut/_Vt$U_t/V_t$ has a non-degenerate limit distribution at 0 and ∞$\infty$.     

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.     

Stitching pairs of Lévy processes into harnesses

We consider natural exponential families of Lévy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be “stitched together” into a single harness. The stitching consists of deterministic reparametrization of the time for both processes, so that they run on adjacent time intervals, and of the choice of the appropriate law at the boundary.     

On the asymptotic behaviour of Lévy processes,Part I: Subexponential and exponential processes

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.     

On solutions to backward stochastic partial differential equations for Lévy processes

In this paper, we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations (BSPDEs, for short) driven by the Teugels martingales associated with a Lévy process satisfying some moment conditions and by an independent Brownian motion. An example is given to illustrate the theory.     

Transition density estimates for jump Lévy processes

Upper estimates of densities of convolution semigroups of probability measures are given under explicit assumptions on the corresponding Lévy measure and the Lévy-Khinchin exponent.     

A stochastic heat equation with the distributions of Lévy processes as its invariant measures

We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.     

Convergence to Lévy stable processes under some weak dependence conditions

For a strictly stationary sequence of random vectors in R^d${\mathbb{R}}^d$ we study convergence of partial sum processes to a Lévy stable process in the Skorohod space with _J1$J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.     

The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups

We establish Lamperti representations for semi-stable Markov processes in locally compact groups. We also study the particular cases of processes with values in R$\mathbb{R}$ and C$\mathbb{C}$ under the hypothesis that they do not visit 0. These Lamperti representations yield some properties of these semi-stable Markov processes.     

Median,concentration and fluctuations for Lévy processes

We estimate a median of f(_Xt)$f\left(X_t\right)$ where f$f$ is a Lipschitz function, X$X$ is a Lévy process and t$t$ is an arbitrary time. This leads to concentration inequalities for f(_Xt)$f\left(X_t\right)$. In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time.     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable 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.     

Default swap games driven by spectrally negative Lévy processes

This paper studies game-type credit default swaps that allow the protection buyer and seller to raise or reduce their respective positions once prior to default. This leads to the study of an optimal stopping game subject to early default termination. Under a structural credit risk model based on spectrally negative Lévy processes, we apply the principles of smooth and continuous fit to identify the equilibrium exercise strategies for the buyer and the seller. We then rigorously prove the existence of the Nash equilibrium and compute the contract value at equilibrium. Numerical examples are provided to illustrate the impacts of default risk and other contractual features on the players’ exercise timing at equilibrium.     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.