Limit theorems for additive functionals of continuous time random walks

For a continuous-time random walk X = {X t , t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

Stochastic simulation for time and space fractional differential equations

In this paper, we aim to study the stochastic simulation for time and space fractional differential equations. First, we prove that the stochastic solution of the time and space fractional differential equation is a stable subordinated process driven by the stable Lévy motion. Then, the absorbent term is employed for this equation; we find the corresponding parent process yields to be driven by a tempered stable process. At last, we design an algorithm to simulate the trajectory of the proposed process. The Monte Carlo methods are also employed to get the approximated solution of the fractional differential equations. The contribution of this paper is to establish the relation between the time and space fractional differential equations and stochastic process, and provide the stochastic simulation algorithm for this fractional differential equations.

On the minimal number of driving Lévy motions in a multivariate price model

In this paper we consider the factor analysis for Lévy-driven multivariate price models with stochastic volatility. Our main aim is to provide conditions on the volatility process under which we can possibly reduce the dimension of the driving Lévy motion. We find that these conditions depend on a particular form of the multivariate Lévy process. In some settings we concentrate on nondegenerate symmetric α-stable Lévy motions.