Convolution of equilibrium distributions and compound geometric tails

24 X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44

3. Convolution of equilibrium distributions and compound geometric tails

In this section we introduce two auxiliary functions which involve the convolution of the compound geometric tail ¯ Ku and the nth equilibrium tail ¯ G n x of the “claim size” distribution Gx = G x. Both functions are necessary in computing the moments of the discounted deficit and the moments at the time of ruin as well as other important quantities in classical ruin theory, as will become clear in Sections 4 and 6. Define g −1 ρ = p 1 , µ ρ = 1, and g n ρ to be the mean of the nth equilibrium DF G n x of Gx. It follows from 2.3 that g n ρ = µ n+1 ρ n + 1µ n ρ , n = 0, 1, 2, . . . 3.1 Also define α n u, ρ = g n ρ Z u ¯ Ku − x dG n+1 x + ¯ G n+1 u , n = −1, 0, 1, 2, . . . 3.2 Evidently, α −1 u, ρ = p 1 1 + β ¯ Ku from 1.12, and α n u, ρ = Z u ¯ Ku − x ¯ G n x dx + Z ∞ u ¯ G n x dx, n = 0, 1, 2, . . . 3.3 We will derive an alternative representation for α n u, ρ, beginning with the following recursive relation. Lemma 3.1. α n+1 u, ρ = 1 g n ρ Z ∞ u α n t, ρ dt − Z ∞ u ¯ Kt dt, n = −1, 0, 1, 2, . . . 3.4 Proof. For n = −1, 0, 1, 2, . . . , one has from 3.2 that 1 g n ρ Z ∞ u α n t, ρ dt = Z ∞ u Z t ¯ Kt − x dG n+1 x dt + Z ∞ u ¯ G n+1 t dt. By interchanging the order of integration, we have Z ∞ u Z t ¯ Kt − x dG n+1 x dt = Z u Z ∞ u ¯ Kt − x dt dG n+1 x + Z ∞ u Z ∞ x ¯ Kt − x dt dG n+1 x = Z u Z ∞ u−x ¯ Kt dt dG n+1 x + ¯ G n+1 u Z ∞ ¯ Kt dt. Integration by parts yields Z u Z ∞ u−x ¯ Kt dt dG n+1 x = − ¯ G n+1 x Z ∞ u−x ¯ Kt dt u x=0 + Z u ¯ Ku − x ¯ G n+1 x dx = − ¯ G n+1 u Z ∞ ¯ Kt dt + Z ∞ u ¯ Kt dt + Z u ¯ Ku − x ¯ G n+1 x dx. Thus 1 g n ρ Z ∞ u α n t, ρ dt = Z ∞ u ¯ Kt dt + Z u ¯ Ku − x ¯ G n+1 x dx + Z ∞ u ¯ G n+1 t dt = Z ∞ u ¯ Kt dt + α n+1 u, ρ using 3.3. X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44 25 We have the following alternative representation for α n u, ρ from the lemma. Theorem 3.1. α u, ρ = β Z ∞ u ¯ Kx dx 3.5 and for n = 1, 2, 3, . . . , α n u, ρ = β µ n ρ Z ∞ u x − u n ¯ Kx dx − n−1 X j =0 n j µ n−j ρ µ n ρ Z ∞ u x − u j ¯ Kx dx. 3.6 Proof. Since α −1 u, ρ = p 1 1 + β ¯ Ku, 3.4 yields with n = −1, α u, ρ = p 1 1 + β p 1 Z ∞ u ¯ Kt dt − Z ∞ u ¯ Kt dt = β Z ∞ u ¯ Kt dt, proving 3.5. For j ≥ 0, one has by interchanging the order of integration Z ∞ u Z ∞ t y − t j ¯ Ky dy dt = 1 j + 1 Z ∞ u y − u j +1 ¯ Ky dy. 3.7 With the help of 3.7 and Lemma 3.1, it is straightforward to verify that 3.6 holds by induction on n. In the special case when δ = 0, it is convenient to define τ n u = Z u ψu − x ¯ P n x dx + Z ∞ u ¯ P n x dx, n = 1, 2, 3, . . . , 3.8 and so τ n u = p n+1 n + 1p n Z u ψu − x dP n+1 x + ¯ P n+1 u , n = 1, 2, 3, . . . 3.9 We have the following corollary. Corollary 3.1. τ 1 u = θ Z ∞ u ψx dx 3.10 and for n = 2, 3, 4, . . . , τ n u = np 1 θ p n Z ∞ u x − u n−1 ψx dx − n−2 X j =0 n j p n−j p n Z ∞ u x − u j ψx dx. 3.11 Proof. When δ = 0, ρ = 0, Gx = P 1 x, and 2.6 holds. Thus ¯ P n x = ¯ G n−1 x. Since ¯ Ku = ψu in this case, it follows from 3.3 and 3.8 that τ n u = α n−1 u, 0, n = 1, 2, 3, . . . 3.12 Since β = θ when ρ = 0, 3.5 becomes 3.10. Then 3.11 follows directly from 2.6 and 3.6. 26 X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44 It is evident in the proofs of the above theorems that the existence of certain moments of the claim size distribution P x is required. For instance, if p n+2 is finite, then α n u, ρ and τ n+1 u are both finite. In order to focus on analytical representation of functions of interest, we always assume that required moments are finite throughout the rest of the paper. In the following examples we demonstrate how to evaluate α n u, ρ and τ n u when P x is a combination of exponentials or a mixture of Erlangs. The case when P x is a combination of exponentials is considered in Gerber et al. 1987 and Dufresne and Gerber 1988. From a formal standpoint, a combination of two exponentials is very important since it may be viewed as providing a Tijms approximation to the relevant quantity see Willmot 1997 and references therein. In particular, Tijms approximations for α n u, ρ and τ n u allow for corresponding approximations for moments at the time of ruin and the deficit of ruin, as is discussed in later sections. In what follows, a combination of two exponentials may be viewed in this context. Example 3.1 Combinations of exponentials. The case where P x is a combination of exponentials is first considered in Gerber et al. 1987 and Dufresne and Gerber 1988 in which a closed form expression for the probability of ruin and the distribution of the deficit at the time of ruin are derived. We begin with an exponential claim size DF ¯ P x = e −µx . In this case, for all n = 0, 1, . . . , ¯ P n x = e −µx . It follows from 2.8 that ¯ G n x = e −µx . Thus, µ n ρ = µ n 0 = nµ n from 2.7. Moreover, ¯ Ku = 1 1 + β e −Ru , 3.13 where 1 1 + β = µ ρ + µ 1 1 + θ , R = βµ 1 + β = θ µ 1 + θ + ρ ρ + µ µ 1 + θ see Example 4.1 of Lin and Willmot 1999. A direct computation from 3.2 yields α n u, ρ = 1 µ e −Ru . 3.14 We now assume that the function Ku is of the form ¯ Ku = C 1 e −R 1 u + C 2 e −R 2 u . This is the case when P x is a combination of two exponentials or a Tijms approximation is used. A detailed discussion of this is also given in Example 4.1 of Lin and Willmot 1999. Since Z ∞ u x − u j ¯ Kx dx = C 1 e −R 1 u Z ∞ x j e −R 1 x dx + C 2 e −R 2 u Z ∞ x j e −R 2 x dx = C 1 j R −j −1 1 e −R 1 u + C 2 j R −j −1 2 e −R 2 u , Theorem 3.1 yields α n u, ρ = C ∗ 1,n e −R 1 u + C ∗ 2,n e −R 2 u , 3.15 X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44 27 where C ∗ 1,n = C 1   βnR −n−1 1 µ n ρ − n−1 X j =0 n j µ n−j ρj R −j −1 1 µ n ρ   , C ∗ 2,n = C 2   βnR −n−1 2 µ n ρ − n−1 X j =0 n j µ n−j ρj R −j −1 2 µ n ρ   . With ρ = 0, we obtain from 2.6 that τ n u = C ∗∗ 1,n e −R 1 u + C ∗∗ 2,n e −R 2 u , 3.16 C ∗∗ 1,n = nC 1 R −n 1 p n   p 1 θ − n−1 X j =1 p j +1 R j 1 j + 1   , C ∗∗ 2,n = nC 2 R −n 2 p n   p 1 θ − n−1 X j =1 p j +1 R j 2 j + 1   . For a general linear combination of exponentials, the derivation is similar and is omitted here. Mixtures of Erlangs are an important distributional class in modeling insurance losses for the reason that any continuous distribution on 0, ∞ may be approximated arbitrarily accurately by a distribution of this type see Tijms 1994, pp. 162–164. Example 3.2 Mixtures of Erlangs. The density function of a mixture of Erlangs is given by P ′ x = r X k=1 q k µµx k−1 e −µx k − 1 , x ≥ 0, where {q 1 , q 2 , . . . , q r } is a probability distribution. Example 4.2 of Lin and Willmot 1999 shows that ¯ Ku = e −µu ∞ X j =0 ¯ C j µu j j , u ≥ 0, where ¯ C j = 1 1 + β j X k=1 q ∗ k ¯ C j −k + 1 1 + β ∞ X k=j +1 q ∗ k , j = 1, 2, 3, . . . . Here it is assumed that ¯ C = 1 + β −1 , and q ∗ k = P r j =k q j µ µ+ρ j −k P r j =1 q j P j −1 i=0 µ µ+ρ i , for k = 1, 2, . . . , r, and q ∗ k = 0, for k r. 28 X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44 Now, for any i = 0, 1, . . . , and j = 0, 1, . . . , Z ∞ u x − u j µx i i e −µx dx = µ i i e −µu Z ∞ x j x + u i e −µx dx = µ i i e −µu i X k=0 i k u k Z ∞ x j +i−k e −µx dx = µ −j −1 e −µu i X k=0 j + i − k i − k µu k k . Thus, α n u, ρ = e −µu ∞ X k=0 ¯ C ∗ k,n µu k k , 3.17 where ¯ C ∗ k,n = ∞ X i=0 ¯ C k+i   βµ −n−1 µ n ρ n + i i − n−1 X j =0 n j µ n−j ρµ −j −1 µ n ρ j + i i   . In the rest of this paper, we turn to evaluation of φu for a particular weight function wx, y.

4. The deficit at the time of ruin