The deenition and properties of L evy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving L evy process is Brownian motion. The use of more general L evy processes permits the speciication of CARMA processes with a wide variety of marginal distributions which may be asymmetric and heavier tailed than Gaus-sian. Non-negative...

The de nition and properties of L evy driven CARMA continuous time ARMA processes are re viewed Gaussian CARMA processes are special cases in which the driving L evy process is Brownian motion The use of more general L evy processes permits the speci cation of CARMA processes with a wide variety of marginal distributions which may be asymmetric and heavier tailed than Gaus sian Non negative CAR...

We de ̄ne the class of local L¶evy processes. These are L¶evy processes time changed by an inhomogeneous local speed function. The local speed function is a deterministic function of time and the level of the process itself. We show how to reverse engineer the local speed function from traded option prices of all strikes and maturities. The local L¶evy processes generalize the class of local vol...

In this paper, we introduce diagonalization of the L evy Laplacian along its eigenfunctions. We describe new Hilbert spaces as various domains of the L evy Laplacian and construct the corresponding equi-continuous semigroups of class (C 0). Moreover, we discuss innnite dimensional stochastic processes related to these extensions and one-dimensional stable processes.

We show how classical Markov processes can be obtained from quantum L evy processes. It is shown that quantum L evy processes are quantum Markov processes, and suucient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can rest...

By a classical result of P. L evy, the Brownian motion (B t) t0 on R may be characterized as a continuous process on R such that (B t) t0 and (B 2 t ?t) t0 are martingales. Generalizations of this result are usually obtained in the setting of the so-called martingale problem. This paper contains a variant of the martingale problem for stochastic processes on compact groups with independent stat...

Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density , have been of interest for many years. In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic diierential equation models to problems in...

Three processes re°ecting persistence of volatility are formulated by evaluating three L¶evy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating the processes. The characteristic functions for the log price can be used to...