  In this paper, we introduce a risk process, namely, the mixed fractional risk
process (MFRP) in which the number of claims in the associated claim process
are modelled using the mixed fractional Poisson process (MFPP). The covariance
structure of the MFRP is studied and its long-range dependence property has
been established. Also, we show that the increments of MFRP exhibit the
short-range dependence property. Another fractional risk process based on the
MFPP is introduced which we call as the MFRP-II. It differs from the MFRP in
terms of premium. Its ruin probabilities in the case of exponential and
subexponential distributions of claim sizes are obtained.
