Lévy random bridges and the modelling of financial information

The information-based asset-pricing framework of Brody-Hughston-Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash-flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T. The price process of the asset is worked out, along with the prices of options.     

The singularity spectrum of Lévy processes in multifractal time

Let X=(Xt)_t?0 be a Lévy process and μ a positive Borel measure on R₊. Suppose that the integral of μ defines a continuous increasing multifractal time . Under suitable assumptions on μ, we compute the singularity spectrum of the sample paths of the process X in time μ defined as the process (XF(t))_t?0.A fundamental example consists in taking a measure μ equal to an “independent random cascade” and (independently of μ) a suitable stable Lévy process X. Then the associated process X in time μ is naturally related to the so-called fixed points of the smoothing transformation in interacting particles systems.Our results rely on recent heterogeneous ubiquity theorems.     

Lévy area for Gaussian processes: A double Wiener-Itô integral approach

Let {X₁(t)}_0≤t≤1 and {X₂(t)}_0≤t≤1 be two independent continuous centered Gaussian processes with covariance functions R₁ and R₂. We show that if the covariance functions are of finite p-variation and q-variation respectively and such that p⁻¹+q⁻¹>1, then the Lévy area can be defined as a double Wiener-Itô integral with respect to an isonormal Gaussian process induced by X₁ and X₂. Moreover, some properties of the characteristic function of that generalised Lévy area are studied.     

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.     

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.     

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.     

Optimal control and dependence modeling of insurance portfolios with Lévy dynamics

In this paper we are interested in optimizing proportional reinsurance and investment policies in a multidimensional Lévy-driven insurance model. The criterion is that of maximizing exponential utility. Solving the classical Hamilton-Jacobi-Bellman equation yields that the optimal retention level keeps a constant amount of claims regardless of time and the company’s wealth level.A special feature of our construction is to allow for dependencies of the risk reserves in different business lines. Dependence is modeled via an Archimedean Lévy copula. We derive a sufficient and necessary condition for an Archimedean Lévy generator to create a multidimensional positive Lévy copula in arbitrary dimension.Based on these results we identify structure conditions for the generator and the Lévy measure of an Archimedean Lévy copula under which an insurance company reinsures a larger fraction of claims from one business line than from another.     

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.     

Mortality modelling with Lévy processes

This paper addresses the modelling of human mortality by the aid of doubly stochastic processes with an intensity driven by a positive Lévy process. We focus on intensities having a mean reverting stochastic component. Furthermore, driving Lévy processes are pure jump processes belonging to the class of α-stable subordinators. In this setting, expressions of survival probabilities are inferred, the pricing is discussed and numerical applications to actuarial valuations are proposed.     

Parametric estimation of a bivariate stable Lévy process

We propose a parametric model for a bivariate stable Lévy process based on a Lévy copula as a dependence model. We estimate the parameters of the full bivariate model by maximum likelihood estimation. As an observation scheme we assume that we observe all jumps larger than some ε>0 and base our statistical analysis on the resulting compound Poisson process. We derive the Fisher information matrix and prove asymptotic normality of all estimates when the truncation point ε→0. A simulation study investigates the loss of efficiency because of the truncation.     

Analysis of generalized Lévy white noise functionals

Employing the Segal-Bargmann transform (S-transform for abbreviation) of regular Lévy white noise functionals, we define and study the generalized Lévy white noise functionals by means of their functional representations acting on test functionals. The main results generalize (Gaussian) white noise analysis initiated by T. Hida to non-Gaussian cases. Thanks to the closed form of the S-transform of Lévy white noise functionals obtained in our previous paper, we are able to define and study the renormalization of products of Lévy white noises, multiplication operator by Lévy white noises, and the differential operators with respect to a Lévy white noise and their adjoint operators. In the courses of our investigation we also obtain a formula for the products of multiple Lévy-Itô stochastic integrals. As applications, we discuss the existence of Hitsuda-Skorokhod integral for Lévy processes, Kubo-Takenaka formula for Lévy processes, and Itô formula for generalized Lévy white noise functionals.     

Lifting Lévy processes to hyperfinite random walks

An internal lifting for an arbitrary measurable Lévy process is constructed. This lifting reflects our intuitive notion of a process which is the infinitesimal sum of its infinitesimal increments, those in turn being independent from and closely related to each other - for short, the process can be regarded as some kind of random walk (where the step size generically will vary). The proof uses the existence of càdlàg modifications of Lévy processes and certain features of hyperfinite adapted probability spaces, commonly known as the “model theory of stochastic 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.     

The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.     

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.     

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.     

Small-time expansions for the transition distributions of Lévy processes

Let X=(X_t)_t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small-time expansions of arbitrary polynomial order in t are obtained for the tails , y>0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density p_t of X_t, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order n in t are derived for the transition densities of the process. The conditions imposed on p_t require that, away from the origin, its derivatives remain uniformly bounded as t→0. Such conditions are then shown to be satisfied for symmetric stable Lévy processes as well as some tempered stable Lévy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.