Limit theorems for the fractional nonhomogeneous Poisson process

Abstract

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

Fractional immigration-death processes

“…Moreover, we have shown in Lemma 3.2 that the operators G and L are continuous. Hence by [4, Corollary 2] we can conclude that p ν (t, x; y) is the unique global solution of ( 23) and (24). Now let us show the uniqueness of the solutions of the backward equation ( 16).…”

“…In the discrete case, fractional (time-changed) processes have been widely considered via different approaches. First of all, a fractional version of the Poisson process has been introduced using Mittag-Leffler distributed inter-jump times instead of exponential ones [2,23,24,29,30] (this approach has been also applied to general counting processes [12]). Such process can be also obtained using a fractional differential-difference equations approach [7,8] and by means of a time-change [31].…”

Volterra mortality model: Actuarial valuation and risk management with long-range dependence

“…We propose a stochastic mortality model incorporating LRD that retains the key advantages of the works of Biffis (2005), Delgado-Vences and Ornelas (2019), and Leonenko et al (2019). More specifically, we maintain the affine nature of Biffis (2005), reflect LRD with fBM as in Delgado-Vences and Ornelas (2019), and offers explicit expressions for some important Fourier-Laplace functional generalizing Leonenko et al (2019) for actuarial valuation. Our model is highly inspired by the affine Volterra processes (Abi Jaber et al, 2019) and hence called the Volterra mortality model.…”

Mixed linear fractional boundary value problems

“…That is, if τ βi 0 denotes the first time the process X βi ti (s) hits zero, τ βi 0 := inf{s > 0 : X βi ti (s) ≤ 0}, then the ruin probability is the quantity Ram16,KL16,CWW13,LWZ15,KLP18] for ruin probabilities of multidimensional risk models, or [AA10] for a broader treatment of ruin probabilities. Fractional version of compound Poisson processes are also of interest when looking at insurance risk processes, see [LST19]. Similar kinds of questions also appear when looking at barrier options under one-dimensional Markov models, see [MP13].…”