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.     

Probabilistic and fractal aspects of Lévy trees

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted -trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.     

The forest associated with the record process on a Lévy tree

We perform a pruning procedure on a Lévy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the Lévy tree). We prove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result of Addario-Berry, Broutin and Holmgren where only Aldous’s tree is considered. As a consequence, we obtain that the “average pruning time” of a leaf is distributed as the height of a leaf picked at random in the Lévy tree.     

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.     

De Finetti’s optimal dividends problem with an affine penalty function at ruin

In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.     

Stochastic integration for Lévy processes with values in Banach spaces

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.     

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.     

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.     

Boundary Harnack Principle for fractional powers of Laplacian on the Sierpiński carpet

We prove the Boundary Harnack Principle related to fractional powers of Laplacian for some natural regions in the two-dimensional Sierpiński carpet. This is a natural application of some more general approach based on the Ikeda-Watanabe formula, that expresses the harmonic measure in terms the Green function of a given region and the Lévy measure of the semigroup.     

Gradient estimate for Ornstein-Uhlenbeck jump processes

By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time t under the condition that the process jumps before t. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.     

BSDEs driven by time-changed Lévy noises and optimal control

We study backward stochastic differential equations (BSDEs) for time-changed Lévy noises when the time-change is independent of the Lévy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for linear BSDEs and a comparison principle. BSDEs naturally appear in control problems. Here we prove a sufficient maximum principle for a general optimal control problem of a system driven by a time-changed Lévy noise. As an illustration we solve the mean–variance portfolio selection problem.     

Hyperfinite stochastic integration for Lévy processes with finite-variation jump part

This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part.As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm.     

Poincaré inequality for linear SPDE driven by Lévy Noise

In this paper, we prove the Poincaré inequality and the integration by parts formula for the invariant measure of the linear SPDE driven by Lévy Noise. The equation was researched in Dong and Xie [5], which has proved the existence and uniqueness of the weak solution and the ergodicity of the Markov semigroup associated with the solution.     

Triangular array limits for continuous time random walks

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Lévy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Lévy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space–time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed.     

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.     

Lévy group and density measures

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group G of permutations of N. Using a new characterization of the Lévy group G we will prove that a finitely additive measure extends density if and only if it is G-invariant.     

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.     

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.     

Coupling and strong Feller for jump processes on Banach spaces

By using lower bound conditions of the Lévy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical) Lévy processes on a Banach space. Unlike in the finite-dimensional case where these properties have also been confirmed for Lévy processes without drift, in the infinite-dimensional setting the appearance of a drift term is essential to ensure the quasi-invariance of the process by shifting the initial data. Gradient estimates and exponential convergence are also investigated. The main results are illustrated by specific models on the Wiener space and separable Hilbert spaces.     

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.