Limit Theorems for the Fractional Non-homogeneous Poisson Process
The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse -stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous 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.
keywords:Fractional point processes; Limit theorem; Poisson process; additive process; Lévy processes; Time-change; Subordination.
The (one-dimensional) homogeneous Poisson process can be defined as a renewal process by specifying the distribution of the waiting times to be i.i.d. and to follow an exponential distribution. The sequence of associated arrival times
gives a renewal process and its corresponding counting process
is the Poisson process with parameter . Alternatively, can be defined as a Lévy process with stationary and Poisson distributed increments. Among other approaches, both of these representations have been used in order to introduce a fractional homogenous Poisson process (FHPP). As a renewal process, the waiting times are chosen to be i.i.d. Mittag-Leffler distributed instead of exponentially distributed, i.e.
where is the one-parameter Mittag-Leffler function defined as
The Mittag-Leffler distribution was first considered in Gnedenko and Kovalenko (1968) and Khintchine (1969). A comprehensive treatment of the FHPP as a renewal process can be found in Mainardi et al. (2004) and Politi et al. (2011).
Starting from the standard Poisson process as a point process, the FHPP can also be defined as time-changed by the inverse -stable subordinator. Meerschaert et al. (2011) showed that both the renewal and the time-change approach yield the same stochastic process (in the sense that both processes have the same finite-dimensional distribution). Laskin (2003) and Beghin and Orsingher (2009, 2010) derived the governing equations associated with the one-dimensional distribution of the FHPP.
In Leonenko et al. (2017), we introduced the fractional non-homogeneous Poisson process (FNPP) as a generalization of the FHPP. The non-homogeneous Poisson process is an additive process with deterministic, time dependent intensity function and thus generally does not allow a representation as a classical renewal process. However, following the construction in Gergely and Yezhow (1973, 1975) we can define the FNPP as a general renewal process. Following the tine-change approach, the FNPP is defined as a non-homogeneous Poisson process time-changed by the inverse -stable subordinator.
Among other results, we have discussed in our previous work that the FHPP can be seen as a Cox process. Following up on this observation, in this article, we will show that, more generally, the FNPP can be treated as a Cox process discussing the required choice of filtration. Cox processes or doubly stochastic processes (Cox (1955), Kingman (1964)) are relevant for various applications such as filtering theory (Brémaud, 1981), credit risk theory (Bielecki and Rutkowski, 2002) or actuarial risk theory (Grandell, 1991) and, in particular, ruin theory (Biard and Saussereau, 2014, 2016). Subsequently, we are able to identify the compensator of the FNPP. A similar generalization of the original Watanabe characterization (Watanabe, 1964) of the Poisson process can be found in case of the FHPP in Aletti et al. (2017).
Limit theorems for Cox processes have been studied by Grandell (1976) and Serfozo (1972a, b). Specifically for the FHPP, scaling limits have been derived in Meerschaert and Scheffler (2004) and discussed in the context of parameter estimation in Cahoy et al. (2010).
The rest of the article is structured as follows: In Section 2 we give a short overview of definitions and notation concerning the fractional Poisson process. Section 3 and 4 are devoted to the application of the Cox process theory to the fractional Poisson process which allows us to identify its compensator and thus derive limit theorems via martingale methods. A different approach to deriving asymptotics is followed in Section 5 and requires a regular variation condition imposed on the rate function of the fractional Poisson process. The fractional compound process is discussed in Section 6 where we derive both a one-dimensional limit theorem using Anscombe’s theorem and a functional limit. Finally, we give a brief discussion of simulation methods of the FHPP and verify the results of some of our results in a Monte Carlo experiment.
2 The fractional Poisson process
This section serves as a brief revision of the fractional Poisson process both in the homogeneous and the non-homogeneous case as well as a setup of notation.
Let be a standard Poisson process with parameter . Define the function
where and is locally integrable. For shorthand and we assume for . We get a non-homogeneous Poisson process , by a time-transformation of the homogeneous Poisson process with :
The -stable subordinator is a Lévy process defined via the Laplace transform
The inverse -stable subordinator (see e.g. Bingham (1971)) is defined by
We assume to be independent of . For , the fractional non-homogeneous Poisson process (FNPP) is defined as
(see Leonenko et al. (2017)). Note that the fractional homogeneous Poisson process (FHPP) is a special case of the non-homogeneous Poisson process with , where a constant. Recall that the density of can be expressed as (see e.g. Meerschaert and Straka, 2013; Leonenko and Merzbach, 2015)
where is the density of given by \linenomath
The Laplace transform of can be given in terms of the Mittag-Leffler function
and for the FNPP the one-dimensional marginal distributions are given by
Alternatively, we can construct an non-homogeneous Poisson process as follows (see Gergely and Yezhow (1973)). Let be a sequence of independent non-negative random variables with identical continuous distribution function
with . Then, let . The resulting sequence is strictly increasing, since it is obtained from the non-decreasing sequence by omitting all repeating elements. Now, we define \linenomath
where . By Theorem 1 in Gergely and Yezhow (1973), we have that is a non-homogeneous Poisson process with independent increments and
It follows via the time-change approach that the FNPP can be written as \linenomath
where we have used that if and only if is not a jump time of (see Embrechts and Hofert (2013)).
3 The FNPP as Cox process
Cox processes go back to Cox (1955) who proposed to replace the deterministic intensity of a Poisson process by a random one. In this section, we discuss the connection between FNPP and Cox processes.
Let be a probability space and be a point process adapted to a filtration . is a Cox process if there exist a right-continuous, increasing process such that, conditional on the filtration , where
then is a Poisson process with intensity .
In particular we have by definition and
A Cox process is said to be directed by , if their relation is as in the above definition. Cox processes are also called -Cox process, doubly stochastic processes, conditional Poisson processes or -conditional Poisson process.
As both and are increasing processes, they are also of finite variation. The path induces positive measures and integration w.r.t. this measure can be understood in the sense of a Lebegue-Stieltjes integral (see p. 28 in Jacod and Shiryaev (2003)). The same holds for the paths of .
Definitions vary across the literature. The above definition can be compared to essentially equivalent definitions: in Brémaud (1981), 6.12 on p. 126 in Jacod and Shiryaev (2003), Definition 6.2.I on p. 169 in Daley and Vere-Jones (2008), where and Definition 6.6.2 on p.193 in Bielecki and Rutkowski (2002).
Cox processes find applications in credit risk modelling. In this context is referred to as hazard process (see Bielecki and Rutkowski (2002)).
In case of the FHPP, there exist characterizing theorems, for example, found in Yannaros (1994) and Grandell (1976) (Theorem 1 of Section 2.2). They use the fact that the FHPP is also a renewal process and allows for a characterization via the Laplace transform of the waiting time distributions. This has been worked out in detail in Section 2 in Leonenko et al. (2017). However, the theorem does not give any insight about the underlying filtration setting. This will become more evident from the following discussion concerning the general case of the FNPP.
In the non-homogeneous case, we cannot apply the theorems which characterize Cox renewal processes as the FNPP cannot be represented as a classicalrenewal process. Therefore, we need to resort to Definition 1 for verification. It can be shown that the FNPP is a Cox process under a suitably constructed filtration. We will follow the construction of doubly stochastic processes given in Section 6.6 in Bielecki and Rutkowski (2002). Let be the natural filtration of the FNPP
We assume the paths of the inverse -stable subordinator to be known, i.e.
We refer to this choice of initial -algebra as non-trivial initial history as opposed to the case of trivial initial history, which is .
The overall filtration is then given by
which is sometimes referred to as intrinsic history. If we choose a trivial initial history, the intrinsic history will coincide with the natural filtration of the FNPP.
Let the FNPP be adapted to the filtration as in (3.3) with non-trivial initial history . Then the FNPP is a -Cox process directed by .
This follows from Proposition 6.6.7. on p. 195 in Bielecki and Rutkowski (2002). We give a similar proof: As is -measurable we have \linenomath
where in (3.4) we used the time-change theorem (see for example Thm. 7.4.I. p. 258 in Daley and Vere-Jones (2003)) and in (3.5) the fact that the standard Poisson process has independent increments. This means, conditional on , has independent increments and
Thus, is a Cox process directed by by definition. ∎
4 The FNPP and its compensator
The idenfication of the FNPP as a Cox process in the previous section allows us to determine the compensator of the FNPP. In fact, the compensator of a Cox process coincides with its directing process. From Lemma 6.6.3. p.194 in Bielecki and Rutkowski (2002) we have the result
Let the FNPP be adapted to the filtration as in (3.3) with non-trivial initial history . Assume . Then the FNPP has -compensator , where , i.e. the stochastic process defined by is a -martingale.
4.1 A central limit theorem
Using the compensator of the FNPP, we can apply martingale methods in order to derive limit theorems for the FNPP. For the sake of completeness, we restate the definition of -stable convergence along with the theorem which will be used later.
If and X are -valued random variables on a probability space and is a sub--algebra of , then (-stably) in distribution if for all and all with ,
(see Definition A.3.2.III. in Daley and Vere-Jones (2003)).
Note that -stable converges implies weak convergence/convergence in distribution. We can derive a central limit theorem for the FNPP using Corollary 14.5.III. in Daley and Vere-Jones (2003) which we state here as a lemma for convenience.
Let be a simple point process on , -adapted and with continuous -compensator . Set
Suppose for each an -predictable process is given such that
Then the randomly normed integrals converge -stably to a standard normal variable .
Note that the above integrals are well-defined as explained in Point 2 in Remarks 1. The above theorem allows us to show the following result for the FNPP.
Let be the FNPP adapted to the filtration as defined in Section 3. Then,
First note that the compensator is continuous in . Let a constant, then \linenomath
It follows from Theorem 1 above that \linenomath
In the following, we give a more rigorous proof for the limit in Section 3.2(ii) in Leonenko et al. (2017).
We can check that the Laplace transform of the density of the inverse -stable subordinator converges to the Laplace transform of the delta distribution:
We may take the limit as the power series representation of the (entire) Mittag-Leffler function is absolutely convergent. Thus (4.2) implies
As convergence in distribution to a constant automatically improves to convergence in probability, we have
By the continuous mapping theorem, it follows that
which concludes the proof. ∎
5 Regular variation and scaling limits
In this section we will work with the trivial initial filtration setting (), i.e. is assumed to be the natural filtration of the FNPP. In this setting, the FNPP can generally not be seen as a Cox process and although the compensator of the FNPP does exist, it is difficult to give a closed form expression for it.
Instead, we follow the approach of results given in Grandell (1976), Serfozo (1972a), Serfozo (1972b), which require conditions on the function . Recall that a function is regularly varying with index if
Under the mild condition of measurability, one can show that the above limit is quite general in the sense that if the quotient of the right hand side of (5.1) converges to a function , has to be of the form (see Thm. 1.4.1 in Bingham et al. (1989)).
We check whether typical rate functions (taken from Remark 2 in Leonenko et al. (2017)) fulfill the regular variation condition.
Weibull’s rate function
is regulary varying with index . This can be seen as follows
Makeham’s rate function
is not regulary varying, since \linenomath
does not fulfill (5.1).
In the following, the condition that is regularly varying is useful for proving limit results. We will first show a one-dimensional limit theorem before moving on to the functional analogue.
5.1 A one-dimensional limit theorem
Let the FNPP be defined as in Equation (2.1). Suppose the function is regularly varying with index . Then the following limit holds for the FNPP:
We will first show that the characteristic function of the random variable on the left hand side of (5.2) converges to the characteristic function of the right hand side.
By self-similarity of we have
Therefore, it follows for the characteristic function of that \linenomath
where we used a conditioning argument in (5.3), is the density function of the distribution of . In the last step in (5.4) we may insert the characteristic function of a Poisson distributed random variable with parameter evaluated at the point .
In order to pass to the limit, we need to justify that we may exchange integration and limit. It can be observed that the integrand is dominated by an integrable function independent of . By Jensen’s inequality \linenomath
This allows us to use the dominated convergence theorem to get \linenomath
We are left with calculating the limit in the square bracket in (5.5). To this end, consider a power series expansion of to observe that \linenomath
where we have used that is regularly varying with index in the last step. Inserting this result into (5.5) yields \linenomath
Applying Lévy’s continuity theorem concludes the proof. ∎
The above result can be shown alternatively using Theorem 3.4 in Serfozo (1972a) or Theorem 1 on pp. 69-70 in Grandell (1976). The limit distribution of is the sum of the limit distribution of the inner process and a normal distribution (the limit of the outer process, the Poisson process). The variance of the normal distribution is determined by the norming constants in the inner process limit. In our case the variance is and we are left with as limit of the overall process.
As a special case of the theorem we get for , for constant
which means is regularly varying with index . It follows that
This is in accordance to the scaling limit given in Cahoy et al. (2010) who showed
5.2 A functional limit theorem
The one-dimensional result in Theorem 5 can be extended to a functional limit theorem. In the following we consider the Skorohod space endowed with a suitable topology (we will focus on the and topology). For more details see Meerschaert and Sikorskii (2012).
Let the FNPP be defined as in Equation (2.1). Suppose the function is regularly varying with index . Then the following limit holds for the FNPP:
As the limit process has continuous paths the mode of convergence improves to local uniform convergence. Also in this theorem, we will denote the homogeneous Poisson process with intensity parameter with .
In order to proof the Theorem we need Theorem 2 on p. 81 in Grandell (1976), which we will state here for convenience.
Let be a stochastic process in with and let be the corresponding doubly stochastic process. Let with and a positive regularly varying function with index such that \linenomath
where is a stochastic process in . Then
where and and are independent. is the standard Brownian motion in .
Proof of Thm. 6.
We apply Theorem 7 and choose and . Then it follows that and it can be checked that is regularly varying with index :
by the regular variation property in (5.1).
We are left to show that
This can be done by following the usual technique of first proving convergence of the finite-dimensional marginals and then tightness of the sequence in the Skorohod space .
Concerning the convergence of the finite-dimensional marginals we show convergence of their respective characteristic functions. Let be fixed at first, and denote the scalar product in . Then, we can write the characteristic function of the joint distribution of the vector
where and is the density of the joint distribution of . We can find a dominating function by the following estimate:
The upper bound is an integrable function which is independent of . By dominated convergence we may interchange limit and integration: \linenomath
where in the last step we used the continuity of the exponential function and the scalar product to calculate the limit. By Lévy’s continuity theorem we may conclude that for
In order to show tightness, first observe that for fixed both the stochastic process on the left hand side and the limit candidate have increasing paths. Moreover, the limit candidate has continuous paths. Therefore we are able to invoke Thm. VI.3.37(a) in Jacod and Shiryaev (2003) to ensure tightness of the sequence and thus the assertion follows. ∎
By applying the transformation theorem for probability densities to (2.2), we can write for the density of the one-dimensional marginal of the limit process as \linenomath
Note that this is not the density of .
A further limit result can be obtained for the FHPP via a continuous mapping argument.
Let be a homogeneous Poisson process and be the inverse -stable subordinator. Then
where is a standard Brownian motion.
The classical result