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Risks 2013, 1(3), 192-212; doi:10.3390/risks1030192
Abstract: This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process.
Let us consider a continuous-time risk model, whose reserves at time t are of the form:
In the classical Cramér–Lundberg model, is modeled by a compound Poisson process. A large literature is devoted to the derivation of the ruin probabilities and other related ruin quantities, for this model and various extensions or modifications. Much can be found, e.g., in the comprehensive books [1,2,3].
The present paper is concerned with the case where is a Lévy subordinator without the drift, implying that is a spectrally negative Lévy process. This model and more general Lévy risk processes have been the object of much research work in recent years. The reader is referred, e.g., to the books [4,5] and the papers [6,7,8,9,10,11,12,13].
Our purpose is to present a direct approach for determining the finite time (and ultimate) ruin probabilities (Section 2), the distribution of the ruin severity and the reserves prior to ruin (Section 3) and the Laplace transform of the ruin time (Section 4). This method relies on simple probabilistic arguments. In particular, we will operate a time reversal that allows us to argue with a dual risk model for which ruin problems are more easily studied. The power of duality is well recognized in ruin theory (see, e.g., [14,15,16]), as well as for stochastic processes with independent stationary increments (see, e.g., the books [17,18,19]).
In the sequel, it is assumed that . Special attention is paid to the standard case where , that is when the net profit condition holds, so that ultimate ruin is not a.s. Nevertheless, we also examine the case where , which could arise and be temporarily allowed for certain branches in a large insurance company. The case , quite different, will not be considered in Section 3 and Section 4; its practical relevance, however, is minor.
Most results obtained when generalize those known for the compound Poisson claim process (see, e.g., [16,20]). This is in agreement with an observation of . A main interest of our work comes from the direct and systematic study of the model made using simple probabilistic methods. Note that a similar approach could be applied to certain queueing and storage models (as, e.g., in ). Of course, other methods can be followed to analyze ruin problems with a subordinator. For instance, one can approximate the subordinator by a sequence of compound Poisson processes (a strategy adopted, e.g., in [8,11]). A powerful alternative is by using the fluctuation theory for Lévy processes (see, e.g., ).
In forthcoming work, we will show, for the classical compound Poisson risk model, that different results can be obtained by exploiting the analycity in time of the distribution of . For the ruin probabilities, the formulas are those obtained by [23,24,25]; see, also, .
Throughout the paper, we denote for any Borel set, A. In particular, and , . For clarity, the main results will be stated at the beginning of each Section, their proofs being presented afterwards. Some more technical results are also given in an Appendix.
2. Non-Ruin Probabilities
The claim process is a Lévy subordinator, , without the drift and with (see, e.g., the books [19,27,28]). By the Lévy–Khintchine formula, the Laplace transform of its probability distribution, , is given by:
A basic subordinator is the classical compound Poisson process with parameter λ and positive i.i.d. claim amounts (distributed as X, say); here, and . For the other subordinators, , i.e., the process has infinitely many small jumps. This is the case with the gamma process, the α-stable subordinator and the (generalized) inverse Gaussian process (see the Appendix).
As a preliminary, consider the case where there are no initial reserves (). Non-ruin probabilities then have a (known) explicit expression.
Formula (4) was first derived by  in a special case. Later, it was obtained by  (Theorem 2) for a process with non-negative interchangeable increments. It was shown to hold, too, for a general spectrally negative Lévy process (see, e.g.,  (Corollary 7.3), ). For this reason, a proof of Equation (4) will not be included here.
Let be the expected claim amount per time unit, i.e.:
Corollary 3. If :
For the compound Poisson model, these formulas can be found in most books on ruin theory (e.g., ). As shown by  (Theorems 3 and 4), Equation (5) holds, too, for a process with non-negative interchangeable increments and Equation (6) for a Lévy subordinator process. We will rederive Formulas (5) and (6) by arguing through the dual risk model.
It is worth mentioning that an alternative expression for is provided by a Pollaczek–Khintchine-type formula. This result is omitted here for brevity reasons. We refer, e.g., to  (Theorem 5) for the case of a Lévy subordinator and to [6,32] for a perturbed subordinator model.
Let be the density function of , if it exists.
2.1.1. Special Cases
(i) Consider the Cramér–Lundberg model, i.e., is a compound Poisson process with parameter λ and i.i.d. claim amounts . The case where the ’s are positive arithmetic random variables was studied in  when the cumulated premium income is a linear function, , as here, but also for any non-decreasing deterministic function. Let us rather examine the case where the ’s are valued in with density . Therefore, has an atom at state 0, and otherwise, it is continuous with density:
Then, Equation (5) yields Seal’s relation:
(iii) If the Lévy subordinator is an inverse Gaussian process with parameter , then Equation (6) gives, when :
Non-ruin until time t means, of course, that over the period , the process, , remains below the upper boundary, F, of equation . It is easily seen that non-crossing through a lower boundary is an easier problem, because crossing here means necessarily meeting. Therefore, we choose to tackle the non-ruin problem by first studying the case of a lower boundary.
2.2.1. Step 1: First-Meeting in a Lower Boundary.
Consider a point, , with and a trajectory joining the origin to . To construct the dual model, we make a rotation of with center and then reverse the two axes. In these new axes, the corresponding trajectory, , joins to , and the straight line becomes a lower boundary, , for this trajectory. Clearly, , and is of equation . Moreover, the process, , over is defined exactly as (e.g., , p. 43).
Let us now pass to the whole positive quadrant in which is again a Lévy subordinator. Two lower boundaries are considered: defined by for followed by a vertical line of abscissa t and defined as before, but on . The first-meeting times of with these boundaries are denoted and , respectively.
We want to determine , the probability that meets for the first time at the point , i.e., . For that, we will have to compute , the probability that meets for the first time at the level, z, i.e., .
The lemma below shows that the probability, ν, can be easily calculated.
Lemma 5. For ,
Proof. Consider a first-meeting of with at some point M of height z and, thus, at time ; so, . Returning to the original axes, but with origin M, we observe that corresponds to the probability . Formula (9) then follows directly from Equation (4). ☐
Thanks to ν, the probability can be determined from the result (10) below. Subsequently, a product measure between two measures and ν will be denoted by (this should not be confused with a convolution product, which has already been used and denoted by ).
Lemma 6. For ,
Proof. The argument is standard (see e.g., ). Any trajectory of can reach the point either without meeting the boundary , or after a first meeting with at some point of height z, thus at time , followed by an increment of during the period . Therefore,
Note that Equation (10) can be extended to any increasing lower boundary, (i.e., linearity has no simplifying role at this step). The next lemma is straightforward.
Lemma 7. For ,
Proof. Evidently, for all , which implies that . Given any probability measure ρ satisfying (for instance, ), the l.h.s. of Equation (11) corresponds to the quotient of the Radon–Nikodym derivatives with respect to ρ, which is equivalent to . ☐
2.2.2. Step 2: Back to the Ruin Problem
We are now ready to derive the announced formulas for the ruin time distribution.
Proof. Operating the rotation of axes described above, we get:
Proof of Corollary 4. Formula (5) can be rewritten as:
When , Equation (6) gives:
Proof. By the independence and stationarity of the increments in , we have:
Proof of Proposition 4. It suffices to adapt the proof of Proposition 2, starting from Equation (16). ☐
3. Reserves at and Prior to Ruin
In this Section, we focus on the joint distribution of the reserves at and just prior to ruin (when it occurs) and some of its implications.
Before this, we introduce a useful parameter inside the subordinator model. For any , consider the equation in θ:
Let be the reserves just prior to ruin and the severity of ruin. Note that is the claim amount that causes the ruin. Our main goal is to determine the probability:
The following function will have a key role. For any real v:
Note that the domain of integration in Equation (21) is if , while it is if . In Property 18, we will show that if , the function, , is finite a.e. On the contrary, when , may be infinite (see the remark in the Appendix).
Proposition 10. If :
Corollary 11. If :
Corollary 12. If , for :
Corollary 13. If , for and , and provided these moments exist:
The result below is proved by following the same argument as for Lemma 9.
Proof. We have, for :
For Equation (33), we get:
For Equation (34), we have:
Finally, suppose that . In the r.h.s. of Equation (22), the second factor, , becomes:
Proof of Corollary 12. By Equation (22), we can write:
Proof of Corollary 13. By Equation (22):
For Equation (30), a similar argument leads to:
4. Ruin Time
In this Section, our main goal is to determine the expectation:
To begin with, we introduce a function that generalizes defined in Equation (21): for any and real v, let:
Proposition 15. If :
Corollary 16. If :
Formula (47) and its corollaries cover several results known for the compound Poisson case when (see, e.g., [16,20]). Clearly, a similar approach would allow us to determine the expected discounted penalty function introduced in . For a study of this function in a Lévy framework, see, e.g., [6,7,8,10].
Proof of Proposition 15. Starting with Equation (31), we have:
When in Equation (47), the second factor, , reduces to:
For Equation (49), putting in Equation (47) gives:
Formula (50) is obtained by a different method. Clearly:
We thank the two referees for useful remarks and suggestions. This work was conducted while C. L. was visiting the Institut de Science Financière et d’Assurances, Université de Lyon. C. L. received support from the Belgian FNRS and the ARC project IAPAS of the Fédération Wallonie-Bruxelles.
Conflicts of Interest
The authors declare no conflict of interest.
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A. Special Cases
It may be worth recalling a few (standard) examples of a Lévy subordinator that could be used to represent the claim process, (see, e.g., ).
A.1. Compound Poisson Process
For such a process with Poisson parameter λ and claim amounts distributed as X with distribution function F, then:
A.2. Gamma Process
For such a process with parameters , then:
A.3. α-Stable Subordinator
For such a process with parameter , then:
A.4. Inverse Gaussian Process
For such a process with parameter , then:
B. Useful Properties
The following property is used in the proof of Lemmas 9 and 14. It is surely known, but we have no precise reference to give. A proof is presented below for completeness. Note that the result is easy to see in the compound Poisson case.
Proof. Define as a compound Poisson measure with parameter and jump distribution (so, ). For , let and denote the associated Laplace transforms. Observing that:
Let in Equation (A2). Using , we get:
Now, let . For any , we may consider the function:
The next property is used when proving Corollary 3 and Proposition 10.
Property 18. If , then for any :
Now, consider . It is sufficient to establish that the Laplace transform of χ, i.e.:
Remark. Property 18 is not true when . To show this, consider, for instance, the compound Poisson model with parameter λ and i.i.d. claim amounts of the exponential law with parameter . Of course, . By definition:
Property 19. For ,
Thus, the l.h.s. of Equation (A7) can be written as:
Now, for defined through Equation (19), we know that the process, , is a martingale. Applying the optional stopping theorem at time then gives:
By differentiating with respect to s, we have:
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