Conditional distributions of Mandelbrot-Van Ness fractional Lévy processes and continuous-time ARMA-GARCH-type models with long memory

Long memory effects can be found in different kind of data from finance to hydrology. Therefore, models which can reflect these properties have become more popular in recent years especially in the fields of time series analysis, econometrics and financial mathematics. Mandelbrot-Van Ness fractional Lévy processes allow for such stationary long memory effects in their increments and have been used in many settings ranging from fractionally integrated continuous-time (autoregressive) moving average processes and exponential GARCH models to general stochastic differential equations. However, their conditional distributions have not yet been considered in detail. In this paper, we provide a closed formula for their conditional characteristic functions and suggest several applications to continuous-time ARMA-GARCHtype models with long memory.