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.     

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.     

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.     

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.     

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.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

Harnack inequalities for stochastic equations driven by Lévy noise

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 semilinear stochastic equations in Hilbert spaces.     

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 ∞$\infty$.     

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.     

Criteria for ergodicity of Lévy type operators in dimension one

Some sufficient conditions for the recurrence, the positive recurrence and the exponential ergodicity of one-dimensional Lévy type operators are presented. The conditions are classified according to different conditions on the ranges and integrability of the Lévy measure, based on the drift inequalities for the extended generator, and on a comparison with diffusion operators. A number of examples are illustrated, including the fractional Laplacian operator and the Ornstein–Uhlenbeck type operator.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

White noise analysis for Lévy processes

We construct a white noise theory for Lévy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula for Lévy processes

     

Sina?ˇ's condition for real valued Lévy processes

We prove that the upward ladder height subordinator H associated to a real valued Lévy process ξ has Laplace exponent φ that varies regularly at ∞ (respectively, at 0) if and only if the underlying Lévy process ξ satisfies Sina?ˇ's condition at 0 (respectively, at ∞). Sina?ˇ's condition for real valued Lévy processes is the continuous time analogue of Sina?ˇ's condition for random walks. We provide several criteria in terms of the characteristics of ξ to determine whether or not it satisfies Sina?ˇ's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of H, here obtained, and which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to Veraverbeke and Grübel.     

Local time–space stochastic calculus for Lévy processes

We develop a stochastic calculus on the plane with respect to the local times of a large class of Lévy processes. We can then extend to these Lévy processes an Itô formula that was established previously for Brownian motion. Our method provides also a multidimensional version of the formula. We show that this formula generates many “Itô formulas” that fit various problems. In the special case of a linear Brownian motion, we recover a recently established Itô formula that involves local times on curves. This formula is already used in financial mathematics.