We establish a novel characterisation of the law convex minorant any L\'evy process. Our self-contained elementary proof is based on analysis piecewise linear functions and requires only very basic properties processes. main result provides new simple approach to fluctuation theory processes, circumventing local time excursion theory. Easy corollaries include classical theorems, such as Rogozin...

Motivated by the need for multifractal processes with stationary in crements we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes We establish conditions for the L convergence and non degeneracy of the limit process in a general setting Proceeding then to multiply ing piecewise constant processes we proof continuity of ...

We consider a network traac model consisting of an innnite number of sources linked to a server. Sources initiate transmissions to the server at Poisson time points. The duration of each transmission has a heavy tailed distribution. We show that suitable scalings of the traac process converge to a totally skewed stable L evy motion in Skorohod space, equipped with the Skorohod M1 topology. This...

There has been growing interest in the study of L evy ights observed in the movements of biological organisms performing random walks while searching for other organisms. Here, we approach the problem of what is the best statistical strategy for optimizing the encounter rate between “searcher” and “target” organisms—either of the same or of di9erent species—in terms of a limiting generalized se...

The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-change...

We introduce a new class of higher order rewriting systems, called Interaction Systems (IS's). IS's come from Lafont's (Intuitionistic) Interaction Nets Lafont 1990] by dropping the linearity constraint. In particular, we borrow from Interaction Nets the syntactical bipartitions of operators into constructors and destructors and the principle of binary interaction. As a consequence, IS's are a ...

We model the time series of the S&P500 index by a combined process, the AR+GARCH process, where AR denotes the autoregressive process which we use to account for the short-range correlations in the index changes and GARCH denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long-range correlations in the variance. We study the AR+GARCH process ...