Mixed fractional risk process

Abstract

In this paper, we introduce a compound version of the mixed fractional Poisson process (MFPP). We obtain its mean, variance and the system of fractional differential equations that governs its state probabilities. By using the compound MFPP as claim process, we introduce a risk process, namely, the mixed fractional risk process (MFRP). It is shown that the MFRP is a martingale with respect to a suitable filtration. Its variance and covariance are obtained using which its long-range dependence property is established. Also, we show that its increments exhibit the short-range dependence property. Later, we consider a different fractional risk process based on the compound MFPP. We call it the mixed fractional risk process-II (MFRP-II) and it differs from the MFRP in terms of premium. For exponentially distributed claim sizes the probability density function of the ruin time of MFRP-II is obtained. An asymptotic behaviour of its finite time ruin probability is derived when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large.