On the discounted penalty function in the renewal risk model with general interclaim times

Transcription

1 On the disconted penalty fnction in the renewal risk model with general interclaim times Gordon E. Willmot Mnich Re Professor in Insrance Department of Statistics and Actarial Science University of Waterloo Waterloo, ON N2L 3G1 Canada Abstract The defective renewal eqation satisfied by the Gerber-Shi disconted penalty fnction in the renewal risk model with arbitrary interclaim times is analyzed. The ladder height distribtion is shown to be a mixtre of residal lifetime claim severity distribtions, which reslts in an invariance property satisfied by a large class of claim amont models. In particlar, when claims are exponentially distribted, a simple reslt follows when the penalty fnction is a fnction of the deficit only, and the Laplace transform of the (disconted) defective density of the srpls immediately prior to rin is obtained. The simplified defective renewal eqation which reslts when the penalty fnction only involves the deficit is sed to obtain moments of the disconted deficit.

2 Keywords: Sparre Andersen process, deficit at rin, srpls at rin, time of rin, defective renewal eqation, Gerber-Shi fnction, residal lifetime distribtion, compond geometric, higher-order eqilibrim distribtion, K n -class, exponential distribtion, Laplace transform Acknowledgments: Spport from a grant from the Natral Sciences and Engineering Research Concil of Canada is grateflly acknowledged, as is spport from the Mnich Reinsrance Company. 2

3 1 Introdction In this paper we will examine varios properties associated with the renewal risk (also called the Sparre Andersen) process (e.g. Rolski et al, 1999), which we now describe. The nmber of claims process N t, t is assmed to be a renewal process, with V 1 the time ntil the first claim occrs, and V i the time between the (i 1)-th and the i-th claim for i 2, 3, 4,.... It is assmed that V 1, V 2... is an independent and identically distribted (iid) seqence of positive random variables with common distribtion fnction (df) K(t) P r(v t), mean E(V ) tdk(t) <, and Laplace-Stieltjes transform (LST) k(s) E ( e sv ) e st dk(t), where V is an arbitrary V i. Individal claim amonts Y 1, Y 2,... are iid positive random variables with dfp (y) 1 P (y) P r(y y), probability density fnction (pdf) p(y) P (y), mean E(Y ) P (y)dy <, and LST p(s) E(e sy ) e sy dp (y), with Y an arbitrary Y i, the size of the i-th claim. Premims are payable continosly at rate c (1+θ)E(Y )/E(V ) per nit time where θ > is the relative secrity loading. The insrer s srpls at time t is U t + ct Nt Y i with the initial srpls. Let T inft : U t < be the time of rin, with T if U t for all t. The rin probability is ψ() P r (T < U ) E I(T < ) U where I(A) 1 if A occrs and I(A) otherwise. i1 Let U T be the srpls immediately prior to rin, and U T be the deficit at rin(if rin occrs). A sbstantial generalization of ψ() is the Gerber-Shi disconted penalty fnction (e.g. Gerber and Shi, 1998) given by m δ () E e δt w (U T, U T ) I(T < ) U, (1.1) 3

4 where w(x, y) is the penalty fnction, and the discont factor δ is explicitly denoted. There has been mch recent research in the actarial and applied probability literatre on the analysis of m δ () in the present model, primarily for particlar choices of K(t). See Li and Garrido (25) and references therein. In this paper, the renewal risk model with arbitrary K(t) is considered, and varios analytic properties of m δ () are shown to hold qite generally. In Section 2, it is demonstrated that m δ () satisfies a defective renewal eqation whose components admit a nified nderlying mathematical strctre, allowing for the retention by the ladder height distribtion of varios characteristics of the claim size distribtion, as well as a simplified defective renewal eqation when the penalty fnction depends on the deficit only. That is, with w(x, y) w 1 (y), (1.1) becomes m δ, () E e δt w 1 ( U T ) I(T < ) U. (1.2) The case with the claim size distribtion being from the fairly large K n -class (considered by Li and Garrido, 25) is also considered. Exponentially distribted claim sizes with arbitrary K(t) are considered in Section 3, where the special strctre inherent in the exponential assmption allows for soltion of the Gerber Shi fnction when w(x, y) e sx w 1 (y). A simple soltion reslts when s, and the Laplace transform of the (disconted) defective density of the srpls immediately prior to rin is obtained when w 1 (y) 1. Section 4 capitalizes on the simplification discssed above to derive and examine the soltion (which is common for all K(t) and P (y)) in the special case w 1 (y) y k, k, 1, 2,..., i.e., for the fnction r k,δ () E e δt U T k I(T < ) U. (1.3) 4

5 Evidently, the rin probability is ψ() r, (), and the moments of the disconted deficit e δt U T are obtainable from (1.3). Of corse, moments of the ordinary deficit are also recovered with δ. Moments of the disconted deficit in the classical Poisson model, a special case of the present model, were discssed by Lin and Willmot (2). 2 Defective renewal eqation analysis To begin the analysis, the defective renewal eqation approach of Gerber and Shi (1998, p. 56), tilized for the classical Poisson model with K(t) 1 e λt, is applicable in the present more general model (see Li and Garrido, 25, p. 846, for example, when K(t) is from the K n -class of densities). Let h(x, y, t ) be the joint defective density of the srpls prior to rin (x), the deficit at rin (y), and the time of rin (t), given U (the word defective is sed in the context of the density to mean h(x, y, t )dxdydt ψ() < 1 ). By conditioning on the first drop in srpls below its initial level, the following integral eqation for m δ () defined by (1.1) reslts (e.g. Gerber and Shi, 1998, p. 56, or Li and Garrido, 25, p. 846); m δ () m δ ( y) e δt h(x, y, t )dtdxdy + w(x +, y ) e δt h(x, y, t )dtdxdy. (2.1) Bt the following argment of Gerber and Shi (1998, p. 53), which also applies to the renewal risk model, is relevant for the ensing analysis. The conditional pdf of the deficit 5

6 (y), given both the srpls prior to rin (x) and the time of rin (t), is given by p x (y) p(x + y), y >, (2.2) P (x) and ths h(x, y, t ) p x (y)h 1 (x, t ), (2.3) where h 1 (x, t ) is the joint defective density of the srpls prior to rin (x) and the time of rin (t) given U. The disconted density representation where e δt h(x, y, t )dt p x (y)h δ (x ), (2.4) h δ (x ) e δt h 1 (x, t )dt, (2.5) follows from (2.3), and h δ (x ) in (2.5) may be interpreted as a disconted defective marginal density of the srpls prior to rin (x), given U. Sbstittion of (2.4) into (2.1) reslts in m δ () m δ ( y) p x (y)h δ (x )dxdy + w(x +, y )p x (y)h δ (x )dxdy. (2.6) Next, let φ δ h δ (x )dx, (2.7) and f δ (y) hδ (x ) p x (y) dx, (2.8) φ δ 6

7 which when sbstitted into (2.6), reslts in the representation m δ () φ δ m δ ( y)f δ (y)dy + w(x +, y )p x (y)h δ (x )dxdy. (2.9) Now, f δ (y) is a valid pdf for y > by (2.2) and (2.7), and φ δ φ ψ() < 1 becase h (x ) is the defective marginal density of the srpls prior to rin, given U. Therefore, (2.9) is a defective renewal eqation satisfied by m δ (). If the penalty fnction only involves the deficit, so that w(x, y) w 1 (y), (2.1) then (1.1) redces to (1.2), and ths from (2.9), m δ, () satisfies m δ, () φ δ m δ, ( y)f δ (y)dy + φ δ w 1 (y )f δ (y)dy. (2.11) Frthermore, if w(x, y) w 1 (y) 1, then m δ, () G δ () 1 G δ () where G δ () E e δt I(T < ) U (2.12) satisfies G δ () φ δ G δ ( y)f δ (y)dy + φ δ F δ (), (2.13) with F δ () 1 F δ () f δ (y)dy. The soltion to (2.13) is the compond geometric tail (e.g. Willmot and Lin, 21, Section 9.1), i.e. (2.12) satisfies G δ () n1 (1 φ δ ) φ n δ F n δ (), (2.14) 7

8 where Fδ n () 1 F n δ () is the df of the n-fold convoltion of F δ () with itself, i.e. has LST e s df n δ () e s df δ () The compond geometric tail representation for G δ () was noticed by Gerber (23). n. The representation (2.8) for the ladder height pdf f δ (y) provides insight into its mathematical strctre. By (2.7), h δ (x )/φ δ is a pdf, and (2.8) implies that f δ (y) is a mixtre over x of the pdf p x (y). When δ, f (y) is the pdf of the drop in srpls below its initial level, given that a drop occrs. Becase h (x )/φ is the pdf of the srpls (or excess) above the initial level prior to this drop, f (y) in (2.8) may be obtained probabilistically by conditioning on this srpls prior to the drop. The dfp x (y) 1 P x (y) has a decreasing failre rate (DFR) if P (y) has a DFR, and becase the DFR property is preserved nder mixing (e.g. Barlow and Proschan, 1975, p. 13), it follows from (2.8) that F δ (y) has a DFR if P (y) has a DFR. This generalizes a reslt of Szekli (1986), who proved this reslt when δ. The mixtre representation (2.8) implies that the ladder height pdf f δ (y) is from the same family of distribtions as p(y) in many cases. This invariance property is mathematically similar to that enjoyed by the proper distribtion of the deficit in the classical Poisson model (e.g. Willmot, 2). For example, if p(y) is a pdf from the K n -class of distribtions with rational Laplace transforms, then p(y) may be expressed in the form (e.g. Li and Garrido, 25) p(y) k i1 j1 a ij β i (β i y) j 1 e β iy, (2.15) (j 1)! 8

9 where k i1 j1 a ij 1 becase p(y) is a pdf, bt it is not necessary that a ij for all i and j. In particlar, when r 1 then (2.15) is the pdf of a combination of exponentials. If (2.15) holds, then Ths, from (2.2), where p(x + y) k i1 m1 k i1 m1 k i1 k i1 p x (y) j1 βi m e β i(x+y) a im (x + y) m 1 (m 1)! βi m e β i(x+y) m ( ) m 1 a im x m j y j 1 (m 1)! j 1 β j i yj 1 e β iy (j 1)! j1 k i1 e β ix j1 a ij (x) e β ix i1 j1 P (x) mj mj j1 β m j i x m j a im (m j)! a im (β i x) m j (m j)! a ij (x)β i(β i y) j 1 e β iy, (j 1)! mj a im (β i x) m j (m j)!. e β ix β i (β i y) j 1 e β iy. (j 1)! Hence, sbstittion into (2.8) yields k hδ f δ (y) a ij (x)β i(β i y) j 1 e β iy (x ) dx, (j 1)! and interchanging the order of integration and smmation reslts in where f δ (y) a ij (δ) 1 φ δ k i1 j1 1 φ δ mj φ δ a ij (δ) β i(β i y) j 1 e β iy, (2.16) (j 1)! a ij (x)h δ(x )dx a im (m j)! 9 (β i x) m j h δ (x ) dx. (2.17) e β ix P (x)

10 Comparison of (2.16) with (2.15) reveals that f δ (y) is from the same family as p(y), bt with a ij replaced by a ij (δ) given by (2.17). The phase-type (2) or Coxian (2) class is obtained when (2.15) holds with k 1 and r 2 (e.g. Tijms, 1994, pp ), and for this class (2.14) is of the form G δ () c 1,δ e R 1,δ + c 2,δ e R 2,δ, (2.18) and (2.18) is also the form of the Tijms approximation to the compond geometric tail G δ () itself (Willmot and Lin, 21, Chapter 8). 3 Exponential claim amonts If P (y) 1 e βy, y, then from (2.2), p x (y) βe βy for all x, implying in trn from (2.8) that f δ (y) βe βy. We will derive an explicit soltion in this sitation for the Gerber-Shi disconted penalty fnction m δ,s () E e δt su T w 1 ( U T ) I(T < ) U, (3.1) and it is clear that (3.1) is a special case of (1.1) with w(x, y) e sx w 1 (y), and that (1.2) is a special case of (3.1) with s. To begin, let s, and the following reslt holds. Lemma 3.1 by The Gerber-Shi disconted penalty fnction m δ, () defined by (1.2) is given m δ, () Ew 1 (Y )G δ (), (3.2) where Ew 1 (Y ) w 1 (y)βe βy dy (3.3) 1

11 and G δ () ( 1 R ) δ e Rδ, (3.4) β with R δ β(1 φ δ ) a negative root of the generalized Lndberg eqation, i.e. R δ satisfies 1 R δ β k(δ + cr δ ). (3.5) Proof: Clearly, m δ, () satisfies (2.11) with f δ (y) βe βy. Noting that sing (3.3), φ δ w 1 (y )f δ (y)dy φ δ w 1 (y )βe β(y ) dy e β φ δ E w 1 (Y ) e β, (2.11) may be written as m δ, () φ δ m δ, ( y)(βe βy )dy + φ δ E w 1 (Y ) e β. (3.6) Letting m δ, (z) e z m δ, ()d, from (3.6) it follows that β m δ, (z) φ δ m δ, (z) β + z + φ δew 1 (Y ), β + z and solving for m δ, (z) yields m δ, (z) φ δew 1 (Y ) z + β(1 φ δ ). Ths, m δ, () φ δ Ew 1 (Y )e β(1 φ δ). (3.7) When w 1 (y) 1, m δ, () redces to G δ (), and (3.2) and (3.4) follow from (3.7). To prove (3.5), it follows by conditioning on the time (t) and the amont (y) of the first claim (e.g. Lin et al, 23) that G δ () e δt e β(+ct) + +ct G δ ( + ct y)βe βy dy dk(t). (3.8) 11

12 Bt G δ () satisfies (2.13) with f δ (y) βe βy, and so e β + G δ ( y)βe βy dy G δ() φ δ, (3.9) and replacing by + ct in (3.9) implies that (3.8) may be expressed as φ δ G δ () e δt G δ ( + ct)dk(t). (3.1) Becase (3.4) holds, G δ ( + ct) G δ ()e R δct, and pon division by G δ (), (3.1) becomes φ δ k(δ + cr δ ) which is (3.5) with φ δ 1 R δ /β. We remark that this reslt may also be obtained sing a martingale stopping time argment as in Gerber and Shi (1998, p. 63), who considered the classical Poisson model. Also, (3.4) is in agreement with Dickson, Hghes, and Zhang (25). In terms of φ δ 1 R δ /β, (3.5) is φ δ k δ + cβ(1 φ δ ) (3.11) i.e., the parameter φ δ satisfies (3.11). The soltion to (3.1) is given in the following theorem. Theorem 3.1 The soltion to (3.1) is m δ,s () Ew 1(Y ) kδ + c(β + s) s + β k δ + c(β + s) se (β+s) + φ δ βe β(1 φ δ). (3.12) Proof: With w(x, y) e sx w 1 (y), it follows that 12

13 w(x +, y )p x (y)h δ (x )dxdy e s(x+) w 1 (y )βe βy h δ (x )dxdy e (β+s) Ew 1 (Y ) e sx h δ (x )dx (s + φ δ β) b δ (s)e (β+s) where bδ (s) Ew 1(Y ) s + φ δ β e sx h δ (x )dx. (3.13) Ths from (2.9) and (3.13), m δ,s () φ δ m δ,s ( y)βe βy dy + (s + φ δ β) b δ (s)e (β+s). (3.14) To solve (3.14), let m δ,s (z) e z m δ,s ()d, and from (3.14), β m δ,s (z) φ δ m δ,s (z) β + z + (s + φ δβ) b δ (s). z + β + s Solving for m δ,s (z) yields m δ,s (z) (s + φ δβ) b δ (s)(z + β + s) 1 1 φ δ β(z + β) 1 bδ (s)(s + φ δ β)(z + β) (z + β + s)z + β(1 φ δ ) b s δ (s) z + β + s + φ δ β z + β(1 φ δ ) 13

14 by a partial fraction expansion. Therefore, the soltion to (3.14) is m δ,s () b δ (s) se (β+s) + βg δ () (3.15) sing (3.4) and R δ β(1 φ δ ). To determine b δ (s), it follows by conditioning on the time (t) and the amont (y) of the first claim as in Lin et al (23) that m δ,s () e δt σ δ,s ( + ct)dk(t) (3.16) where σ δ,s (t) t w(t, y t)dp (y) + m δ,s (t y)dp (y). (3.17) t With w(x, y) e sx w 1 (y) and (3.15), (3.17) becomes t σ δ,s (t) e (β+s)t E w 1 (Y ) + s b δ (s) e (β+s)t E w 1 (Y ) + βs b δ (s)e (β+s)t t e (β+s)(t y) βe βy dy + β b δ (s) t G δ (t y)βe βy dy e sy Gδ (t) dy + β b δ (s) e βt φ δ sing (3.9) with replaced by t. Ths, σ δ,s (t) e (β+s)t E w 1 (Y ) + β b δ (s) e βt e (β+s)t Gδ (t) + β b δ (s) e (β+s)t E w 1 (Y ) + β φ δ bδ (s)g δ (t) β b δ (s)e (β+s)t, φ δ e βt i.e. σ δ,s (t) ( ) E w 1 (Y ) β b δ (s) e (β+s)t + β bδ (s)g δ (t). (3.18) φ δ 14

15 Sbstittion of (3.15) and (3.18) into (3.16) reslts in bδ (s) se (β+s) + βg δ () ( E w 1 (Y ) β b δ (s) ) e δt (β+s)(+ct) dk(t) + β bδ (s) e δt G δ ( + ct)dk(t) φ δ ( ) E w 1 (Y ) β b δ (s) e (β+s) kδ + c(β + s) +β b δ (s)g δ () sing (3.1). Solving for b δ (s) yields bδ (s) E w 1(Y ) k δ + c(β + s), (3.19) s + β k δ + c(β + s) and (3.12) follows by sbstittion of (3.19) and (3.4) with R δ β(1 φ δ ) into (3.15). Clearly, (3.12) becomes (3.2) when s. Also, with w 1 (y) e zy, (3.1) becomes the trivariate Laplace transform of the defective distribtion of T, U T, and U T, i.e., E e δt su T z U T I(T < ) U e δt sx zy h(x, y, t )dxdydt (3.2) where h(x, y, t ) is the joint defective density of the srpls prior to rin (x), the deficit at rin (y), and the time of rin (t), given U, and (3.2) is obtained from (3.12) with w 1 (y) e zy. It is clear that the proper distribtion of the deficit at rin has the same exponential distribtion as the claim amonts, independently of the time of rin and the srpls prior to rin. Clearly, h (x ) h(x, y, t )dydt (3.21) 15

16 is the marginal defective density of the srpls prior to rin, given U. The following corollary gives the Laplace transform of (3.21). Corollary 3.1 The Laplace transform of the defective density of the srpls prior to rin U T, given U, is given by e sx h (x )dx kc(β + s) se (β+s) + βψ(), (3.22) s + β kc(β + s) where ψ() φ e β(1 φ ), (3.23) and φ satisfies φ k cβ(1 φ ). Proof: Clearly, (3.22) follows from (3.12) with w 1 (y) 1 and δ, together with ψ() G () given by (3.23) sing (3.4) with δ and R β(1 φ ), and (3.11) with δ. 4 Disconted moments of the deficit The focs of this section is the analysis of r k,δ () defined by (1.3), and we no longer assme as in the previos section that P (y) is an exponential df. To begin, let F,δ (y) 1 F,δ (y) F δ (y), and also µ k,δ y k df δ (y) for k, 1, 2,.... Then define the k-th order eqilibrim df F k,δ (y) 1 F k,δ (y) recrsively by F k,δ (y) as long as µ k,δ <. Then (e.g. Hesselager et al, 1998) y F k 1,δ (t)dt, k 1, 2,..., (4.1) F k 1,δ (t)dt y n df k,δ (y) µ n+k,δ ( n+k n ) µk,δ, n 1, 2, 3,..., (4.2) 16

17 and the following sefl alternative representation to (4.1) holds, namely F k,δ (y) (t y) k df δ (t) y µ k,δ. (4.3) With w 1 (y) y k, it follows from (4.3) that (2.11) becomes r k,δ () φ δ r k,δ ( y)f δ (y)dy + φ δ µ k,δ F k,δ (). (4.4) The soltion to (4.4) is given by the following lemma. Lemma 4.1 The fnction r k,δ () defined by (1.3) satisfies r k,δ () φ δµ k,δ G 1 φ δ δ ( y)df k,δ (y) + F k,δ () G δ (). (4.5) Proof: Let f k,δ (s) e sy df k,δ (y) with f δ (s) f,δ (s). Then from (2.14), g δ (s) e s dg δ () 1 φ δ 1 φ δ fδ (s), (4.6) and from (4.4), the Laplace transform r k,δ (s) e s r k,δ ()d of r k,δ () satisfies This may be expressed as φ δ µ k,δ 1 f k,δ (s) r k,δ (s) φ δµ k,δ g δ (s) 1 f k,δ (s). s 1 φ δ fδ (s) 1 φ δ s r k,δ (s) φ δµ k,δ 1 φ δ and (4.5) follows from (4.7). f k,δ (s) 1 g δ(s) s + 1 f k,δ (s) s 1 g δ(s), (4.7) s 17

18 The df F k,δ (y) may easily be expressed in mixtre form sing (2.8) and Hesselager et al (1998). In this context, the k-th eqilibrim distribtion associated with the pdf p x (y) is obtainable sing Willmot and Lin (21, p. 22). Alternatively, f δ (y) has been obtained for particlar interclaim time df s K(t). Li and Garrido (25), for example, have shown that if K(t) has a K n -class distribtion, then qite generally F δ (y) is a combination of generalized eqilibrim distribtions which are discssed in detail in Lin and Willmot (1999). Then F k,δ (y) is a different combination of the corresponding k-th eqilibrim distribtions (given by Lin and Willmot, 1999), which is easily established sing the approach of Hesselager et al (1998). In the present sitation, it is not necessary to explicitly identify F k,δ (y) in order to evalate r k,δ (). In order to demonstrate this, an alternative representation will be derived. To this end, define the df of the compond geometric convoltion H k,δ () 1 H k,δ () by h k,δ (s) e s dh k,δ () g δ (s) f k,δ (s), k, 1, 2,..., (4.8) which implies that H k,δ () G δ ( y)df k,δ (y) + F k,δ (). (4.9) Then (4.5) may be expressed as r k,δ () φ δµ k,δ 1 φ δ Hk,δ () G δ (), (4.1) and the goal is to express (4.1) in a simpler form. Lemma (4.2) is instrmental in this regard. Lemma 4.2 For k, 1, 2,..., H k,δ () satisfies the recrsive relationship H k+1,δ () (k + 1)µ k,δ µ k+1,δ Hk,δ (y) G δ (y) dy. (4.11) Proof: It follows from (4.8) that the mean of H k,δ () satisfies H k,δ ()d G δ ()d + F k,δ ()d (4.12) 18

19 where from (4.6) G δ ()d φ δµ 1,δ 1 φ δ. (4.13) The eqilibrim distribtion of h k,δ () ths has LST, sing (4.6) and (4.8), given by 1 h k,δ (s) φ δ 1 f δ (s) + (1 φ δ ) 1 f k,δ (s). s H k,δ ()d s 1 φ δ fδ (s) H k,δ ()d That is, with f,δ (s) f δ (s), (4.13) implies that 1 h φ δ µ 1,δ f1,δ (s) + (1 φ δ ) F k,δ ()d f k+1,δ (s) k,δ (s) s H k,δ ()d 1 φ δ fδ (s) H k,δ ()d φ δ µ 1,δ f1,δ 1 φ δ (s) + F k,δ ()d f k+1,δ (s) g δ (s), H k,δ ()d i.e., sing (4.12) and (4.13), with 1 h k,δ (s) s H k,δ ()d g δ (s) α δ f1.δ (s) + (1 α δ ) f k+1,δ (s), (4.14) α δ G δ ()d. (4.15) H k,δ ()d Clearly, (4.14) expresses the eqilibrim distribtion as a compond geometric convoltion becase (4.12) and (4.15) imply that < α δ < 1, and the eqilibrim tail ths satisfies H k,δ (y)dy H k,δ (y)dy G δ () + αδ F 1,δ ( y) + (1 α δ )F k+1,δ ( y) dg δ (y) α δ H 1,δ () + (1 α δ )H k+1,δ (). (4.16) 19

20 Now, (4.16) with k implies that H 1,δ () H,δ (y)dy, H,δ (y)dy and (4.9) with k together with (2.13) implies that H,δ () G δ() φ δ. (4.17) Ths, letting G 1,δ () 1 G 1,δ () be the eqilibrim df of G δ (), it follows from (4.17) that H 1,δ () H,δ (y)dy H,δ (y)dy G δ (y)dy G δ (y)dy G 1,δ (). (4.18) Noting that F k,δ ()d µ k+1,δ / (k + 1)µ k,δ from (4.2) with n 1, it follows from (4.12), (4.15), and (4.18) that (4.16) may be rewritten as H k,δ (y)dy G δ (y)dy G 1,δ() + µ k+1,δ H k+1,δ () (k + 1)µ k,δ which yields (4.11) pon solving for H k+1,δ (). We remark that (4.11) is differentiable, and ths H k+1,δ () has density h k+1,δ () H k+1,δ () given by Combining (4.1) and (4.19) yields h k+1,δ () (k + 1)µ k,δ µ k+1,δ Hk,δ () G δ (). (4.19) r k,δ () φ δ µ k+1,δ (1 φ δ )(k + 1) h k+1,δ(), (4.2) and (4.2) is ths an alternative to (4.1). 2

21 We are now in a position to restate the formla for r k,δ (). Theorem 4.1 The fnction r k,δ () defined by (1.3) may be expressed for k, 1, 2,..., as r k,δ () φ δ 1 1 φ δ φ δ (t ) k dg δ (t) k j ( ) k µ k j,δ j (t ) j dg δ (t). (4.21) Proof: Eqating (4.1) and (4.21) implies that (4.21) is eqivalent to H k,δ () G δ () + 1 φ δ µ k,δ When k, (4.22) yields, sing µ,δ 1, (t ) k dg δ (t) k j ( ) k µk j,δ j µ k,δ (t ) j dg δ (t).(4.22) which is (4.17). Also, for j, 1, 2,..., H,δ () G δ () + 1 φ δ G δ () G δ () 1 φ δ G δ () y (t y) j dg δ (t)dy 1 j + 1 (t ) j+1 dg δ (t) (4.23) by interchanging the order of integration. Ths, sing (4.23) and assming (4.22) holds, (4.11) yields H k+1,δ () (k+1)µ k,δ µ k+1,δ Hk,δ (y) G δ (y) dy (k + 1) R R (t y) k dg δ (t)dy y φ δ µ k+1,δ k+1 µ k+1,δ k j ( k j) µk j,δ (t y) j dg δ (t)dy y R (t ) k+1 dg δ (t) φ δ µ k+1,δ 1 µ k+1,δ k j ( k+1 ) µk j,δ (t ) j+1 dg j+1 δ (t) R (t ) k+1 dg δ (t) φ δ µ k+1,δ k+1 j1 ( k+1 ) µk+1 j,δ j µ k+1,δ (t ) j dg δ (t) 21

22 G δ () + 1 φ δ µ k+1,δ (t ) k+1 dg δ (t) k+1 j ( k+1 ) µk+1 j,δ j µ k+1,δ (t ) j dg δ (t) which is (4.22) with k replaced by k + 1. Hence, (4.22) holds by indction on k. Clearly, moments of the disconted deficit satisfy E (e δt U T ) k T <, U r k,kδ() r, () (4.24) where ψ() G () r, () is the rin probability. The mean of the disconted deficit is given by (4.24) with k 1, and sing (4.13), (4.21) yields i.e. E e δt U T T <, U E e δt U T T <, U (t )dg δ (t) ψ() G δ (y)dy φ δµ 1,δ G δ () 1 φ δ ψ() G 1,δ () G δ () ψ() G δ (y)dy G δ () G δ (y)dy, ψ() Gδ ( + y) G δ ()G δ (y) dy ψ(). (4.25) Moments of the ordinary deficit are obtained with δ. Ths, E U T T <, U ψ( + y) ψ() where (4.26) follows from (4.25) becase ψ() G (). ψ(y) dy, (4.26) Formla (4.21) reqires evalation of (t ) j dg δ (t) for j, 1, 2,..., k rather than F k,δ (y) and ths H k,δ (), as is the case with (4.5) and (4.1). This is convenient when G δ () has a 22

23 simple analytic form, as for example if (2.18) holds. Integration by parts yields (t ) j dg δ (t) j (t ) j 1 G δ (t)dt; j 1, 2, 3,..., (4.27) and ths if (2.18) holds, (4.27) yields (t ) j dg δ (t) j j j! 2 k1 c k,δ (t ) j 1 e R k,δt dt 2 c k,δ e R k,δ k1 t j 1 e R k,δt dt 2 c k,δ (R k,δ ) j e Rk,δ, k1 which also holds when j, and (4.21) follows easily. An important special case of the K n -class of distribtions is now considered. Example 4.1 Mixed Erlang claim amonts Sppose that p(y) j1 β(βy) j 1 e βy q j, (4.28) (j 1)! where r j1 q j 1 and q j. This is a special case of (2.15) with k 1, and any continos distribtion on (, ) may be approximated arbitrarily accrately (e.g. Tijms, 1994, p. 163) by a pdf of the form (4.28). One has from (4.28) that (e.g. Willmot and Lin, 21, p. 12), where Q j r kj+1 r 1 P (x) e βx j (βx) j Q j, j! q k for j, 1, 2,..., r 1. Then from (2.16), f δ (y) j1 q j (δ) β(βy)j 1 e βy (j 1)! 23 (4.29)

24 where from (2.17), q j (δ) 1 φ δ mj q m (m j)! (βx) m j h δ (x ) r 1 m Q m (βx) m m! dx. (4.3) Becase < q j (δ) < 1 in (4.3), (4.29) is of the same form as (4.28). Also, G δ () is a compond geometric tail from (2.14), and it follows from Example of Willmot and Lin (21, pp ) that where C m (δ) km+1 generating fnction G δ () e β m C m (δ) (β)m, (4.31) m! c k (δ), and c k (δ); k, 1, 2,... has compond geometric probability c k (δ)z k k 1 φ δ. 1 φ δ q j (δ)z j j1 Ths to evalate r k,δ (), it follows that (t ) j dg δ (t) c k (δ) k1 k1 c k (δ)β k (k 1)! e β k1 (t ) j β(βt)k 1 e βt dt (k 1)! c k (δ)β k (k 1)! e β c k (δ) k1 t j (t + ) k 1 e β(t+) dt k 1 m k 1 m (j!)β j e β c k (δ) k1 ( k 1 m ) m t j+k 1 m e βt dt (j + k 1 m)! m!(k 1 m)! m β m j, k 1 m ( ) j + k 1 m (β) m. j m! Interchanging the order of smmation yields (t ) j dg δ (t) (j!)β j e β m km+1 24 ( ) j + k 1 m (β) m c k (δ), (4.32) j m!

25 which redces to (4.31) when j. Sbstittion of (4.32) into (4.21) then yields r k,δ () in a straightforward manner, and the details are omitted. 5 References Barlow, R., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Dickson, D.C.M., Hghes, B., and Zhang, L. (25). The density of the time to rin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scandinavian Actarial Jornal, Gerber, H. (23). Personal Commnication. Gerber, H.U., and Shi, E.S.W. (1998). On the Time Vale of Rin. North American Actarial Jornal, 48-78, Discssions Hesselager, O., Wang, S., and Willmot, G.E. (1998). Exponential and scale mixtres and eqilibrim distribtions. Scandinavian Actarial Jornal, Li, S., and Garrido, J. (25). On a general class of renewal risk process: analysis of the Gerber-Shi fnction. Advances in Applied Probability, 37, Lin, X., and Willmot, G.E. (1999). Analysis of a defective renewal eqation arising in rin theory. Insrance: Mathematics & Economics, 25, Lin, X., and Willmot, G.E. (2). The moments of the time of rin, the srpls before rin, and the deficit at rin. Insrance: Mathematics & Economics, 27, Lin, X., and Willmot, G.E., and Drekic, S. (23). The classical risk model with a constant dividend barrier: analysis of the Gerber-Shi disconted penalty fnction. Insrance: Mathematics & Economics, 33,

26 Rolski, T., Schmidli, H., Schmidt, V., and Tegels, J. (1999). Stochastic Processes for Insrance and Finance. John Wiley, Chichester. Szekli, R. (1986). On the concavity of the waiting time distribtion in some GI/G/1 qees. Jornal of Applied Probability, 223, Tijms, H. (1994). Stochastic Models - An Algorithmic Approach. John Wiley, Chichester. Willmot, G.E. (2). On evalation of the conditional distribtion of the deficit at the time of rin. Scandinavian Actarial Jornal, Willmot, G.E., and Lin, X.S. (21). Lndberg Approximations for Compond Distribtions with Insrance Applications. Springer-Verlag, New York. 26