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. 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 , Veillette and Taqqu , , .
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
where is the drift and the Lévy measure on satisfies
This means that
The inverse subordinator is the first-passage time of , i.e.,
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.
and for any
which implies that all moments are finite, i.e.,for any .
Let and the renewal function
Then for , the Laplace transform of is given by
in particular, for
thus, characterizes the inverse process , since characterizes
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
Then the inverse stable subordinator
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
be the Mittag-Leffler function (cf. ) and recall the following:
The Laplace transform of the Mittag-Leffler function is of the form
The Mittag-Leffler function is a solution of the fractional equation:
where the fractional Caputo-Djrbashian derivative is defined as (see )
The following hold true for the processes and :
The Laplace transform is
and it holds that
Both processes and are self-similar, i.e.,
where (, resp.) is the Beta function (incomplete Beta function, resp.).
The following asymptotic expansions hold true:
For fixed and large , it is not difficult to show that
For fixed , it follows
A finite variance stationary processis 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
where the constant is depending on and .
There is a (complicated) form of all finite-dimensional distributions of in the form of Laplace transforms, see .
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
This expression is also obtained by substituting in the Laplace transform
3 Classical fractional Poisson processes
where , , are iid random variables with the strictly monotone Mittag-Leffler distribution function
Here we denote an indicator as and ,
, are iid uniformly distributed onrandom 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
where is the rising factorial (sometimes also called Pochhammer function).
Note that and
Meershaert et al.  find the stochastic representation for FPP
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:
with initial conditions , , , where is the fractional Caputo-Djrbashian derivative (13).
Remark 4 (Expectation and variance)
where is the classical homogeneous Poisson process with parameter , which is independent of the inverse stable subordinator Note that
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
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 .
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
Remark 6 (Consistency with equivalence principle)
A main motivation for considering the risk process (27) comes from the fact that the stochastic processes
and its non-homogeneous analogon
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.
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. . We have
since the compensated FPP is a martingale. Thus
This completes the proof.
Remark 8 (Marginal moments of the risk process)
4.1 A variant of the surplus process
or, more generally,
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 , 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
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
In this section, we discuss the long range dependency property (LRD property, see Definition 2) of the risk process .
The covariance structure of risk process is given by
Further, the variance of is
Proof. For , we have .