Homogeneous risk model on a bounded time interval [0, ttt]

Insurance: Mathematics and Economics 26 2000 223–238 Homogeneous risk models with equalized claim amounts q F. De Vylder a , M. Goovaerts a,b,∗ a Universiteit van Amsterdam, Amsterdam, Netherlands b Katholieke Universiteit Leuven, CRIR, Huid Eygen Heerd, Minderbroederstraat 5, 3000 Leuven, Belgium Received 1 June 1998; received in revised form 1 September 1999; accepted 24 November 1999 Abstract We consider an homogeneous risk model on a fixed bounded time interval [0, t] and we denote by N t the number of claims in that interval. The claim amounts are X 1 , X 2 , . . . , X N t . The homogeneous model is an extension of the classical actuarial risk model with N t not necessarily Poisson distributed. In the model with equalized claim amounts, each amount X k is replaced with X ∼ k = X 1 + · · · + X N t N t . Let 9t, u be the ruin probability before t in the homogenous model, corresponding to the initial risk reserve u≥0 and let 9 ∼ t, u be the corresponding ruin probability evaluated in the associ- ated model with equalized claim amounts. The essence of the classical Prabhu formula is that 9t, 0=9 ∼ t, 0. By rather systematic numerical investigations in the classical risk model, we verify that 9 ∼ t, u≤9t, u for any value of u≥0 and that 9 ∼ t, u is an excellent approximation of 9t, u. Then these conclusions must be valid in any homogeneous model and this is an interesting observation because 9 ∼ t, u can be calculated numerically, whereas no algorithms are yet avail- able for the numerical evaluation of 9t, u in general homogeneous risk models. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Risk model; Homogeneous risk model; Ruin probability; Prabhu’s formula 1. Homogeneous risk model on a bounded time interval [0, ttt] N t is the number of claims in [0, t]. The claim instants process T 1 , T 2 , . . . , T N t is any homogeneous point process on the fixed interval [0, t]. Its distribution is completely specified by the probabilities PN t = n≥0 n=0, 1, 2, . . . such that 6 n≥0 PN t = n=1. The latter probabilities may be any numbers satisfying the indicated relations see Appendix of De Vylder and Goovaerts 1999. The claim amounts are X 1 , X 2 , . . . , X N t . It is assumed that X 1 , X 2 , . . . are i.i.d. random variables with distribution function F concentrated on [0, ∞ and that these amounts are independent from the claim instants. The risk reserve process is R τ = u+cτ −S τ 0≤τ ≤t, where u≥0 is the initial risk reserve, c0 the premium income rate and S τ the total claim amount in [0, τ ], i.e. S τ = X 1 + · · · + X N t , where N τ is the number of claims in [0, τ ]. Of course, S τ = 0, if N τ = 0. q Presented at the Second International IME Congress, University of Lausanne, Switzerland, July 1998. ∗ Corresponding author. Tel.: +32-16-323-746; fax: +32-16-323-740. E-mail address: marc.goovaertsecon.kuleuven.ac.be M. Goovaerts 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 5 5 - 4 224 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238 We denote by Ut, u the probability of nonruin before t corresponding to the initial risk reserve u≥0, and by U n t, u the corresponding conditional probability of nonruin for fixed N t = n. Then U t, u = X n≥0 P N t = nU n t, u. 1 The homogeneous model with fixed N t = n is defined by the probabilities PN t = n=1 and PN t = m=0 m6=n. Then U n t, u is the nonconditional probability of ruin in the homogeneous model with fixed N t = n. In the homogeneous model with fixed N t = n, the claim instants are T 1 T 2 · · · T n and the random vector T 1 , . . . , T n has a constant density on the subset W tn = { t 1 , . . . , t n |0 t 1 t 2 · · · t n t } of R n . The Lebesgue volume of W tn is t n n. Hence the density of T 1 , . . . , T n equals nt n on W tn . Then, by the foregoing assumptions, the distribution of the random vector T 1 , . . . , T n , X 1 , . . . , X n is completely specified in the homogeneous model with fixed N t = n.

2. Nonruin probability before time ttt in the homogeneous model

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