In , two different assumptions on N are considered that we shall also take into account in the following:. We recall that see, e. In , the proof of the recursion for the corresponding multivariate compound distribution is based on properties of the probability generating function p. Moreover, in Section 3 we present an approximate alternative method to evaluate the compound distribution based on the Fast Fourier Transform FFT , which has the advantage of being less time consuming than the recursive method especially when one needs to evaluate the distribution's tail.
We compare the two methods on a numerical example and conclude. To simplify the writing, we introduce more notation: we denote by fs the p.
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The p. From , we have that for the general model 1 under the assumptions A1-A2 , the p. Using the properties of the p. We shall now present a shorter proof based on similar recursions already proved in . In the same time, we shall also obtain the new alternative recursive formula 4. From  see also , formulas Apart the exact methods like the recursive one presented above , some approximate techniques have also been proposed for evaluating aggregate claims distributions, with the purpose to simplify calculations and reduce the computing time for details on these methods see, e.
The Fast Fourier Transform is such a technique that strongly reduces the computing time, especially when one needs to evaluate the tail of the distribution. Moreover, this technique can be applied to models for which there is no recursion available. This is why the FFT received special attention in the actuarial literature, see, e.
In the following, we shall present the FFT algorithm corresponding to our model and compare it with the recursive method. A FFT is an algorithm that computes the discrete Fourier transform and its inverse extremely fast. For more details on Fourier transforms and their applications major application in signal and image processing see, e. To apply the FFT method, the values rj must be powers of two for all j. For our model 1 , we shall use the following algorithm proposed in  based on the FFT and its inverse IFFT , algorithm that generalizes the one considered by  in the bivariate case.
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For the last equality, we applied the formula of gN obtained in  for model 1 under the assumptions A1-A2. Step 1. Set the truncation points for the r. If necessary, the resulting vectors fj or the table fQ can be padded with zeros to force the rjs to be powers of two. Step 2. Step 3. Remark 3. The usual way to find the optimal rjs consists in gradually increasing them e.
When the claim sizes distributions are heavy tailed it is recommended to use the so-called "exponential tilting" method, which consists in applying an exponential change of measure to the claim sizes distributions that forces their tails to decrease at an exponential rate.
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Published online Nov PMID: Corresponding author. Received Apr 23; Accepted Nov 7. Abstract This paper considers the distribution of some extremum on the risk process whose income depend on the current reserve. Background Before introducing the model, we revisit some important risk models. Preliminaries In this section, we first give some notation and terminologies, and then introduce the renewal measure.
Open in a separate window. The supreme profit and the deficit In this section, some distributions on the maximum surplus and the maximal severity of ruin are given. Proof This follows from 10 and Conclusions In order to make a reasonably realistic description of the actual behavior, we investigate the risk model whose income depend on the current reserve. Authors' contributions All the authors have contributed to the manuscript equally.
Acknowledgements The authors are grateful to the anonymous referee for carefully reading, valuable comments and suggestions to improve the earlier version of the paper. Competing interests The authors declare that they have no competing interests. Contributor Information Jingmin He, Email: moc. References Asmussen S. Ruin probabilities.
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Singapore: World Scientific; Asmussen S, Albrecher H. Singapore: World Scientific Publishing; On the expectation of total discounted operating costs up to default and its applications. Adv Appl Probab. Further results for the joint distribution of the surplus immediately before and after ruin under force of interest. J Stat Theory Pract.
Martingales and insurance risk. Commun Stat Stoch Models.
The distribution of some extremum on the risk process whose income depend on the current reserve
The distributions of the time to reach a given level and the duration of negative surplus in the Erlang 2 risk model. Insur Math Econ. How many claims does it take to get ruined and recovered. Ruin estimation for a general insurance risk model. When does the surplus reach a given target.
On the time value of ruin. N Am Actuar J. Gerber—Shiu risk theory, EAA series. Berlin: Springer; Landriault D, Shi T. The book comprises selected contributions from several large research communities.