Generalized fractional Lévy processes with fractional Brownian motion limit and applications to stochastic volatility models

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernels to the more general class of regularly varying functions with the corresponding fractional integration parameter. The resulting stochastic processes are called generalized fractional Lévy processes (GFLP) which are shown to exhibit both short and long memory increments possibly with jumps. Moreover, for monotone kernels we define stochastic integrals with respect to GFLPs and investigate their second order structure and path properties. We prove a functional central limit theorem for stochastic integrals driven by a GFLP. As a specific example we present our result for Ornstein-Uhlenbeck processes driven by a time scaled GFLP. This approximation applies to a wide class of stochastic volatility models which include models where neither price nor volatility process are semimartingales.     «

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernels to the more general class of regularly varying functions with the corresponding fractional integration parameter. The resulting stochastic processes are called generalized fractional Lévy processes (GFLP) which are shown to exhibit both short and long memory increments possibly with jumps. Moreover, for monotone kernels we define stocha...     »