# Fractional Risk Process in Insurance

Important models in insurance, for example the Carmér--Lundberg theory and the Sparre Andersen model, essentially rely on the Poisson process. The process is used to model arrival times of insurance claims. This paper extends the classical framework for ruin probabilities by proposing and involving the fractional Poisson process as a counting process and addresses fields of applications in insurance. The interdependence of the fractional Poisson process is an important feature of the process, which leads to initial stress of the surplus process. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process.

## Authors

• 19 publications
• 6 publications
• 3 publications
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Customers arrive at a service facility according to a Poisson process. U...
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• ### Statistical Inference in Fractional Poisson Ornstein-Uhlenbeck Process

In this article, we study the problem of parameter estimation for a disc...
12/14/2017 ∙ by Héctor Araya, et al. ∙ 0

• ### Pay-As-You-Drive Insurance Pricing Model

Every time drivers take to the road, and with each mile that they drive,...
12/17/2019 ∙ by Safoora Zarei, et al. ∙ 0

• ### Parameter estimation for fractional Poisson processes

The paper proposes a formal estimation procedure for parameters of the f...
06/07/2018 ∙ by Dexter Cahoy, et al. ∙ 0

• ### On the two-phase fractional Stefan problem

The classical Stefan problem is one of the most studied free boundary pr...
02/04/2020 ∙ by Félix del Teso, et al. ∙ 0

• ### Estimation of Poisson arrival processes under linear models

In this paper we consider the problem of estimating the parameters of a ...
03/02/2018 ∙ by Michael G. Moore, et al. ∙ 0

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## 1 Introduction

Companies, which are exposed to random orders or bookings, face the threat of ruin. But what happens if bookings fail to appear and ruin actually happens? The probability of this event cannot be ignored and is of immanent and substantial importance from economic perspective. Recall that some nations have decided to bail-out banks, so a question arising naturally is how much capital is to be expected to bail-out a company after ruin, if liquidation is not an option.

This paper introduces the average capital required to recover a company after ruin. It turns out that this quantity is naturally related to classical, convex risk measures. The corresponding arrival times of random events as bookings, orders, or stock orders on an exchange are often comfortably modelled by a Poisson process — this process is a fundamental building block to model the economy reality of random orders.

From a technical modelling perspective, the inter-arrival times of the Poisson process are exponentially distributed and independent

[17]. However, real data do not always support this property or feature (see, e.g., [13, 16, 36, 39] and references therein). To demonstrate the deviation we display arrival times of two empirical data sets consisting of oil futures high frequency trading (HFT) data and the classical Danish fire insurance data. Oil futures HFT data is considered for the period from July 9, 2007 until August 9, 2007. There is a total of 951,091 observations recorded over market opening hours. The inter-arrival times are recorded in seconds for both up-ticks and down-ticks (cf. Figures (a)a and (b)b). In addition, the Danish fire insurance data is available for a period of ten years until December 31, 1990, consisting of 2167 observations: the time unit for inter-arrival times in Figure (c)c is days. The charts in Figure 4 display the survival function of inter-arrival times of the data on a logarithmic scale and compares them with the exponential distribution. The data are notably not aligned, as is the case for the Poisson process. By inspection it is thus evident that arrival times of the data apparently do not follow a genuine Poisson process.

This paper features the fractional Poisson process. It is a parametrized stochastic counting process which includes the classical Poisson process as a special, perhaps extreme case, but with dependent arrival times. We will see that the fractional Poisson process correctly captures main characteristics of the data displayed in Figure 4.

Arrival times of claim events are of striking economic importance for insurance companies as well and our discussion puts a slight focus on this economic segment. The Cramér–Lundberg model, to give an example, genuinely builds on the Poisson process. We replace the Poisson process by its fractional extension and investigate the new economic characteristics. We observe that the fractional Poisson process exposes companies to higher initial stress. But conversely, we demonstrate that the ruin probability and the average capital required to recover a firm are identical for the classical and the fractional Poisson process. In special cases we can even provide explicit relations.

#### Outline of the paper.

The following two Sections 2 and 3 are technical, they introduce and discuss the fractional Poisson process by considering Lévy subordinators. Section 4 introduces the risk process based on the fractional Poisson process, while the subsequent Section 5 addresses insurance in a dependent framework. This section contains a discussion of average capital injection to be expected in case of ruin. We finally discuss the risk process for insurance companies and conclude in Section 6.

## 2 Subordinators and inverse subordinators

This section considers the inverse subordinator first and introduces the Mittag-Leffler distribution. This distribution is essential in defining the fractional Poisson process. Relating the subordinators to -stable distributions enables to simulate trajectories of the fractional Poisson process. Some results here follow Bingham [8], Veillette and Taqqu [40], [41], [32].

### 2.1 Inverse subordinator

Consider a non-decreasing Lévy process starting from which is continuous from the right with left limits (càdlàg), continuous in probability, with independent and stationary increments. Such process is known as Lévy subordinator with Laplace exponent

 ϕ(s)=μs+∫(0,∞)(1−e−sx)Π(dx), s≥0, (1)

where is the drift and the Lévy measure on satisfies

 ∞∫0min(1,x)Π(dx)<∞. (2)

This means that

 Ee−sL(t)=e−tϕ(s), s≥0. (3)

The inverse subordinator is the first-passage time of , i.e.,

 Y(t):=inf{u≥0:L(u)>t},t≥0. (4)

The process is non-decreasing and its sample paths are almost surely (a.s.) continuous if is strictly increasing. Also is, in general, non-Markovian with non-stationary and non-independent increments.

We have

 {L(ui)ui, i=1,…,n} (5)

and for any

which implies that all moments are finite, i.e.,

for any .

Let and the renewal function

 U(t):=U(1)(t)=EY(t), (6)

and let

 Hu(t):=P{Y(u)

Then for , the Laplace transform of is given by

 ~U(p)(s)=∫∞0U(p)(t)e−stdt=Γ(1+p)/[sϕp(s)],

in particular, for

 ∫∞0e−stdHu(t)=e−tϕ(s), (7)

and

 ~U(s)=∫∞0U(t)e−stdt=1sϕ(s); (8)

thus, characterizes the inverse process , since characterizes

Therefore, we get a covariance formula (cf. [40], [41])

 Cov(Y(s),Y(t))=∫min(s,t)0(U(s−τ)+U(t−τ))dU(τ)−U(s)U(t). (9)

The most important example is considered in the next section, but there are some others.

### 2.2 Inverse stable subordinator

Let be an -stable subordinator with Laplace exponent , whose density is such that has pdf

 gα(x)=g(1,x) =1π∞∑k=1(−1)k+1Γ(αk+1)k!1xαk+1sin(πkα). (10)

Then the inverse stable subordinator

 Yα(t)=inf{u≥0:Lα(u)>t}

has density

 fα(t,x)=tαx−1−1αgα(tx−1α), x≥0, t>0. (11)

Alternative form for the pdf in term of -function and infinite series are discussed in [15, 18].

Its paths are continuous and nondecreasing. For the inverse stable subordinator is the running supremum process of Brownian motion, and for this process is the local time at zero of a strictly stable Lévy process of index

Let

 Eα(z):=∞∑k=0zkΓ(αk+1), α>0, z∈C (12)

be the Mittag-Leffler function (cf. [14]) and recall the following:

1. The Laplace transform of the Mittag-Leffler function is of the form

 ∫∞0e−stEα(−tα)dt=sα−11+sα, 0<α<1, t≥0, s>0.
2. The Mittag-Leffler function is a solution of the fractional equation:

 DαtEα(atα)=aEα(atα), 0<α<1,

where the fractional Caputo-Djrbashian derivative is defined as (see [32])

 Dαtu(t)=1Γ(1−α)∫t0du(τ)dτdτ(t−τ)α, 0<α<1. (13)
###### Proposition 1

The following hold true for the processes and :

1. The Laplace transform is

 Ee−sYα(t)=∞∑n=0(−stα)nΓ(αn+1)=Eα(−stα),s>0

and it holds that

 ∫∞0e−stfα(t,x)dt=sα−1e−xsα,s≥0.
2. Both processes and are self-similar, i.e.,

 Lα(at)a1/αd=Lα(t),Yα(at)aαd=Yα(t),a>0.
3.  EYα(t)=tαΓ(1+α); E[Yα(t)]ν=Γ(ν+1)Γ(αν+1)tαν, ν>0; (14)
4.  Cov(Yα(t),Yα(s)) (15) =1Γ(1+α)Γ(α)∫min(t,s)0((t−τ)α+(s−τ)α)τα−1dτ−(st)αΓ2(1+α). =1Γ(1+α)Γ(α)[t2αB(s/t;α,1+α)+s2αB(α,1+α)]−(st)αΓ2(1+α),s

where (, resp.) is the Beta function (incomplete Beta function, resp.).

5. The following asymptotic expansions hold true:

1. For fixed and large , it is not difficult to show that

 Cov(Yα(t),Yα(s))=−αsα+1tα−1Γ(α)Γ(2+α)+⋯+s2αB(1+α,α). (16)

Further,

 Var(Yα(t))=t2α(2Γ(1+2α)−1Γ2(1+α))=d(α)t2α(say).
2. For fixed , it follows

 Cor(Yα(t),Yα(s))∼B(1+α,α)d(α)sαtα,ast→∞. (17)

A finite variance stationary process

is said to have long-range-dependence (LRD) property if . Further, for a non-stationary process an equivalent definition is given as follows.

###### Definition 2 (LRD for non-stationary process)

Let be fixed and . Then the process is said to have LRD property if

 Cor(Y(s),Y(t))∼c(s)t−d,ast→∞, (18)

where the constant is depending on and .

Thus this last property (17) can be interpreted as long-range dependence of in view of (18).

###### Remark 3

There is a (complicated) form of all finite-dimensional distributions of in the form of Laplace transforms, see [8].

### 2.3 Simulation of the stable subordinator

In order to obtain trajectories for the stable subordinator it is necessary to simulate random variables with finite Laplace transform satisfying (

10). To this end it is necessary that the -stable random variable is spectrally positive, which means in the standard parametrization (also type 1 parametrization, cf. [38, Definition 1.1.6 and page 6]) of the -stable random variable

 Lα(t)∼S(α,β,γ,δ;1) (19)

 EeisLα(t)=exp(−γα|s|α(1−iβtanπα2sign(s))+iδs).α≠1.

This expression is also obtained by substituting in the Laplace transform

 Ee−sLα(t)=exp(−γαcosπα2sα−δs).

Comparing the latter with (10) reveals that the parameters of the -stable random variable , , in the parametrization (19) are

 β=1, γα=t⋅cosπα2 and δ=0.

## 3 Classical fractional Poisson processes

The first definition of the fractional Poisson process (FPP) is given by Mainardi et al. [27] (see also [28]) as a renewal process with Mittag-Leffler waiting times between the events

 Nα(t) =max{n:T1+⋯+Tn≤t}=∞∑j=1I{T1+⋯+Tj≤t} =∞∑j=1I{Uj≤Gα(t)},t≥0, (20)

where , , are iid random variables with the strictly monotone Mittag-Leffler distribution function

 Fα(t)=P(Tj≤t)=1−Eα(−λtα), t≥0, 0<α<1,j=1,2,…, (21)

and

 Gα(t)=P(T1+⋯+Tk≤t)=∫t0h(k)(x)dx.

Here we denote an indicator as and ,

, are iid uniformly distributed on

random variables. is the pdf of th convolution of the Mittag-Leffler distributions which is known as the generalized Erlang distribution and it is of the form

 h(k)(x) =αλkxkα−1(k−1)!E(k)α(−λxα) =λkxαk−1Ekα,αk+1(−λxα),α∈(0,1), x>0,

where the three-parametric Generalized Mittag-Leffler function is defined as (cf. (12) and [14])

 Eγα,β(z)=∞∑j=0(γ)jzjj!Γ(αj+β),α>0, β>0, γ>0, z∈C, (22)

where is the rising factorial (sometimes also called Pochhammer function).

Note that and

 P(Tj>t)=Ee−λS(t),t≥0.

Meershaert et al. [31] find the stochastic representation for FPP

 Nα(t)=N(Yα(t)),t≥0, α∈(0,1),

where is the classical homogeneous Poisson process with parameter which is independent of the inverse stable subordinator

One can compute the following expression for the one-dimensional distribution of FPP:

 P(Nα(t)=k) =p(α)k(t)=∫∞0e−λx(λt)kk!fα(t,x)dx =tλkαk!∫∞0e−λxxk−1−1αgα(tx−1α)dx =(λtα)kk!∞∑j=0(k+j)!j!(−λtα)jΓ(α(j+k)+1)=(λtα)kk!E(k)α(−λtα) (23) =(λtα)kEk+1α,αk+1(−λtα),k=0,1,2…,t≥0, 0<α<1,

where is the Mittag-Leffler function (12) evaluated at , and is the th derivative of evaluated at Further, is the Generalized Mittag-Leffler function (22) evaluated at

Finally, Beghin and Orsingher (cf. [5], [6]) show that the marginal distribution of FPP satisfies the system of fractional differential-difference equations

 Dαtp(α)k(t)=−λ(p(α)k(t)−p(α)k−1(t)),k=0,1,2,…

with initial conditions , , , where is the fractional Caputo-Djrbashian derivative (13).

###### Remark 4 (Expectation and variance)

Note that

 ENα(t)=∫∞0[EN(u)]fα(t,u)du=λtαΓ(1+α), (24)

where is given by (11); Leonenko et al. [23]) show that

 Cov(Nα(t),Nα(s)) =∫∞0∫∞0[VarN(1)]min(u,v)Ht,s(du,dv) (25) +(EN(1))2Cov(Yα(t),Yα(s)) =λ(min(t,s))αΓ(1+α)+λ2Cov(Yα(t),Yα(s)),

where is given in (15), and In particular

 VarNα(t) =λ2t2α(2Γ(1+2α)−1Γ2(1+α))+λtαΓ(1+α) =λ2t2αΓ2(1+α)(αΓ(α)Γ(2α)−1)+λtαΓ(1+α),t≥0.

For the definition of Hurst index for renewal processes is discussed in [12]. For the FPP it is equal

 H=inf{β:limsupt→∞VarNα(t)tβ<∞},

thus

Finally, Leonenko et al. [26, 25] introduced a fractional non-homogeneous Poisson process (FNPP) with an intecity function as

 NNα(t) :=N(Λ(Yα(t))),t≥0, α∈(0,1) and Λ(t) :=∫t0λ(s)ds, (26)

where is the classical homogeneous Poisson process with parameter , which is independent of the inverse stable subordinator Note that

 P(NNα(t)=k)=∫∞0e−Λ(u)Λ(u)kk!fα(t,u)du,k=0,1,2…,

where is given by (11). Alternatively (cf. [25]),

 NNα(t)=∞∑n=0nI{Lα(ζn)≤t

where is a sequence of non-negative iid random variables such that and , where with The resulting sequence is strictly increasing, since it is obtained from the non-decreasing sequence by omitting all repeating elements.

## 4 Fractional risk processes

The risk process is of fundamental importance in insurance. Based on the inverse subordinator process (cf. (4)) we extend the classical risk process (also known as surplus process) and consider

 Rα(t):=u+μλ(1+ρ)Yα(t)−Nα(t)∑i=1Xi,t≥0. (27)

Here, is the initial capital relative to the number of claims per time unit (the Poisson parameter ) and the iid variables , with mean model claim sizes. , is an independent of , , fractional Poisson process. The parameter is the safety loading; we demonstrate in Proposition 7 below that the risk process (27) satisfies the net profit condition iff (cf. Mikosch [33, Section 4.1]), which is economically that the company will not necessarily go bankrupt iff .

###### Remark 5

Note that for so that the model (27) extends the classical ruin process considered in risk theory.

It is essential in (27) to observe that the counting process and the payment process follow the same time scale. These coordinated, or harmonized clocks are essential, as otherwise the model would over-predict too many claims (too many premiums, respectively). As well, different clocks for these two processes violated the profit condition (cf. Proposition 7 below).

The process for the non-homogeneous analogue of (27) is

 Rα(t)=u+μ(1+ρ)Λ(Yα(t))−NNα(t)∑i=1Xi, t≥0.
###### Remark 6 (Consistency with equivalence principle)

A main motivation for considering the risk process (27) comes from the fact that the stochastic processes

 Nα(t)∑i=1Xi−μλYα(t),t≥0

and its non-homogeneous analogon

 NNα(t)∑i=1Xi−μΛ(Yα(t)),t≥0

are martingales with respect to natural filtrations (cf. Proposition 7 below).

It follows from this observation that the time change imposed by does not affect or violate the net or equivalence principle. Further, the part of the risk process corresponding to the premium (i.e., without , or by setting ) is the fair premium of the remaining claims process even under the fractional Poisson process.

###### Proposition 7

The fractional risk process introduced in (27) is a submartingale (martingale, supermartingale, resp.) for (, , resp.) with respect to the natural filtration.

Proof. Note that the compensated FPP is a martingale with respect to the filtration , cf. [2]. We have

 E[ Rα(t)−Rα(s)∣Fs] =E⎡⎣λμ(1+ρ)Yα(t)−Nα(t)∑i=1Xi−⎛⎝λμ(1+ρ)Yα(s)−Nα(s)∑i=1Xi⎞⎠∣∣∣Fs⎤⎦ =E⎡⎣⎛⎝λμ(1+ρ)(Yα(t)−Yα(s))−Nα(t)∑i=Nα(s)+1Xi⎞⎠∣∣∣Fs⎤⎦ =E[(λμ(1+ρ)(Yα(t)−Yα(s)))∣∣∣Fs]−E⎡⎣E⎡⎣Nα(t)∑i=Nα(s)+1Xi∣∣∣Ft⎤⎦∣∣∣Fs⎤⎦ =E[(λμ(1+ρ)(Yα(t)−Yα(s)))∣∣∣Fs]−E[(Nα(t)−Nα(s))μ∣∣∣Fs] =−μE[(Nα(t)−λYα(t))−(Nα(s)−λYα(s))∣∣∣Fs]+λμρE[Yα(t)−Yα(s)∣∣∣Fs] =λμρE[Yα(t)−Yα(s)∣∣∣Fs],

since the compensated FPP is a martingale. Thus

 E[Rα(t)−Rα(s)∣Fs]=⎧⎨⎩>0ifρ>0,=0ifρ=0,<0ifρ<0.

This completes the proof.

###### Remark 8 (Marginal moments of the risk process)

The expectation and the covariance function of the process loss in (27) can be given as

 ENα(t)∑i=1Xi=λtαΓ(1+α)EXi (28)

and, by employing [23],

 Cov⎛⎝Nα(t)∑i=1Xi,Nα(s)∑i=1Xi⎞⎠= λtEX2iλ(min(t,s))αΓ(1+α) +[λEX2i+λ2(EXi)2]Cov(Yα(t),Yα(s)),

where is given in (15).

### 4.1 A variant of the surplus process

A seemingly simplified version of the surplus process (27) is obtained by replacing the processes or by their expectations (explicitly given in (14)) and to consider

 ~Rα(t)=u+μλ(1+ρ)tαΓ(1+α)−Nα(t)∑i=1Xi,t≥0, (29)

or, more generally,

 ~Rα(t)=u+μEΛ(Yα(t))−NNα(t)∑i=1Xi,t≥0

for the non-homogeneous process. As above, is the initial capital, is the constant premium rate and the sequence of iid random variables is independent of FFP . The net profit condition, as formulated by Mikosch [33], involves the expectation only. For this reason, the adapted surplus process (29) satisfies Mikosch’s net profit condition as well.

For we have that so that the simplified process (29) coincides with the classical surplus process considered in insurance (cf. Remark 5). However, the simplified risk process , in general, is not a martingale unless and , while is martingale for any and by Proposition 7.

###### Remark 9 (Time shift)

The formula (29) reveals an important property of the fractional Poisson process and the risk process . Indeed, for small times , there are more claims to be expected under the FPP regime, as

 tαΓ(1+α)>tfor t small.

The inequality reverses for later times. This means that the premium income rate decays later. The time change imposed by FPP postpones claims to later times — another feature of the fractional Poisson process and a consequence of the martingale property.

However, the premium income is in a real world situation. For this we conclude that the FPP can serve as a stress test for insurance companies within a small, upcoming time horizon.

### 4.2 Long range dependence of risk process Rα(t)

In this section, we discuss the long range dependency property (LRD property, see Definition 2) of the risk process .

###### Proposition 10

The covariance structure of risk process is given by

 Cov(Rα(t),Rα(s))=μ2λ2ρ2Cov(Yα(t),Yα(s))+λ(EX2i)sαΓ(1+α),s≤t. (30)

Further, the variance of is

 Var(Rα(t))=μ2λ2ρ2Var(Yα(t))+λ(EX2i)tαΓ(1+α).

Proof. For , we have .

 Cov ⎛⎝Nα(t)∑i=1Xi,Nα(s)∑j=1Xj⎞⎠ = E(∞∑i=1∞∑j=1XiXjI{Nα(s)≥j,Nα(t)≥i})−(EXi)2E(Nα(t))E(Nα(s)) = E(∞∑j=1X2jI{Nα(s)≥j})+E⎛⎝∑∑i≠jXiXjI{Nα(s)≥j,Nα(t)≥i}⎞⎠ −(EXi)2E(Nα(t))E(Nα(s)) = E(X2j)∞∑j=1P(Nα(s)≥j) +(EXi)2(∞∑i=1∞∑j=1P(Nα(s)≥j,Nα(t)≥i)−∞∑j=1P(Nα(s)≥j)) −(EXi)2E(Nα(t))E(Nα(s)) = E(X2i)ENα(s)+(EXi)2E(Nα(s)Nα(t)) −(EXi)2ENα(s)−(EXi)2E(Nα(t))E(Nα(s)) = E(X2i)ENα(s)+(EXi)2Cov(Nα(t),Nα(s))−(EXi)2ENα(s) = Var(Xi)ENα(s)+(EXi)2Cov(Nα(t),Nα