On the number of claims until ruin in a two-barrier renewal risk model with Erlang mixtures

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1 On the number of claims until ruin in a two-barrier renewal risk model with Erlang mixtures October 7, 2015 Boutsikas M.V. 1, Rakitzis A.C. 2 and Antzoulakos D.L. 3 1 Dept. of Statistics & Insurance Science, Univ. of Piraeus, Greece (mbouts@unipi.gr). 2 Dept. of Mathematics, Univ. of the Aegean, Greece (arakitz@aegean.gr ). 3 Dept. of Statistics & Insurance Science, Univ. of Piraeus, Greece (dantz@unipi.gr). Abstract In this paper, we consider the renewal risk model and we are interested in the distribution of the number ν of claims until the first time that insurer s surplus process falls below zero (ruin) or exceeds a predefined upper barrier b > u (safety level), immediately after the payment of a claim. By using exponentially tilted measures we derive an expression for the joint generating function of ν and S ν, the surplus amount at termination time. This expression is built upon the generating functions of the overshoot and undershoot of the surplus process. Furthermore, we offer explicit results for the case where the claim amounts and the claim inter-arrival times follow mixed Erlang Distributions. We finally propose and implement an algorithm for the numerical calculation of the distributions of interest via appropriate computer algebra software. KEY WORDS AND PHRASES: Renewal risk model; Two-sided first exit time; Number of claims to ruin; Exponentially tilted probability measure; Mixed Erlang distribution. AMS 2010 Subject classification: Primary 60G40, Secondary: 62P05; 91B30 1 Introduction We consider the renewal (Sparre-Andersen) risk model, in which insurance claim amounts X 1, X 2,... appear according to a renewal process {N(t), t 0} with claim inter-arrival times T 1, T 2,..., i.e. the number of claims up to time t is equal to N(t) := sup{n : n T i t}. The claim amounts X 1, X 2,... are non-negative independent and identically distributed (i.i.d.) random variables (r.v.s) with common distribution F while the claim inter-arrival times T 1, T 2,... are non-negative i.i.d. r.v.s, independent also of X i s, with common distribution K. The insurer s surplus at time t is given by N(t) U(t) := u + ct X i, 1

2 where u 0 is the initial surplus and c > 0 denotes the premium income rate (per unit of time). One of the main subjects in actuarial risk theory is the study of time τ to ruin, i.e. the first time that the surplus process U := {U(t), t 0} of an insurance company (in a certain portfolio) becomes negative (i.e. τ := inf{t : U(t) < 0}). An important relevant quantity which also provides useful information is the number of claims until ruin. More specifically, as Egidio dos Reis (2002) and Frostig et al. (2012) mention, since ruin occurs at a claim instant, we can alternatively consider the waiting time until ruin in terms of the number of claims until ruin. Therefore, apart from the probability of ruin and the distribution of the time τ to ruin, insolvency of an insurance company can also be usefully measured by the number of claims that occur until the surplus process gets ruined. In the last decade, many researchers studied the distribution of the number of claims until ruin, mainly in connection to the distribution of τ. More specifically, concerning the classical Poisson risk model (i.e. {N(t), t 0} is a Poisson process), Stanford and Stroinski (1994) presented a recursive method to calculate the probability of ruin at claim instants, Egidio dos Reis (2002) derived the Laplace transform of the number of claims until ruin, while Dickson (2012) obtained an expression for the joint density of the number of claims until ruin and the time to ruin. In the context of the more general renewal risk model, Landriault et al. (2011) derived an expression of the distribution of the number of claims until ruin by assuming exponential claims, whereas Frostig et al. (2012) obtained the joint transform of the time to ruin and number of claims until ruin by considering phase-type distributions for both claims and inter-arrival times. Zhao and Zhang (2013) presented a formula for the joint density of the number of claims until ruin and the time to ruin, by assuming infinitely divisible inter-arrival times and Erlang claim sizes. Recently, Li and Lu (2014) considered a risk model with interest and obtained the moments and the distribution function of the number of claims until the surplus drops below a specific level, when the claim amounts are exponentially distributed. In several occasions, apart from the time to ruin, the insurer may also be interested in the time until the surplus process reaches a given upper barrier b (e.g. safety level). Problems related to first exit times in risk models including an upper barrier have been studied, under various assumptions, by Gerber (1990), Perry et al. (2005), Li (2008) and Dickson and Li (2013). Apart from the one-sided risk model that was mentioned above, Li and Lu (2014), studied also a two-sided first exit time problem (again in a model with interest), by considering the number of claims until the exit time, and obtained explicit results when the claim amounts are exponentially and Erlang(2) distributed. In the same paper, an approach for the case when the distribution of the claim amounts has a rational Laplace transform is also presented. The aim of this article is to study the distribution of the number of claims until a two-sided first exit time in the renewal risk model (without interest) after assuming that the insurer s surplus is inspected only at claim arrival times. More specifically, we are interested in the distribution of the number ν of claims until the first time that the surplus process U passes below zero or is above an upper barrier b (u < b) immediately after the payment of a claim. It is worth mentioning that the problem we study can also be viewed as a two-sided first exit time problem of a discrete time random walk. This follows from the assumption that the surplus process, which is modeled by a renewal risk model, can be inspected only on claim arrival times. Similar exit time problems for specific classes of random walks have been studied by several researchers in the past. A classical reference on the asymptotic analysis of random walks is the monograph of Borovkov and Borovkov (2008). Also, exact approaches for the two-sided exit time problem can be found back to the 60 s in Kemperman (1963), whereas for some recent works, see 2

3 Ezhov et al. (2007), Jacobsen (2011) and references therein. In this paper, we offer results (in the context of Sparre-Andersen risk model) that lead to the derivation of the distribution of the two-sided first exit time ν, provided that the distributions of the overshoot and undershoot of the surplus process are known. Moreover, we focus our study on the case where F and K belong to the mixed Erlang class of distributions. By taking into account that the mixed Erlang class is dense in the class of all non-negative distributions, our approach allows the approximation of any renewal risk model. The main technique we employ is based on the notion of tilted measures, elements of which appear in Boutsikas et al. (2011). Although, in general, the main emphasis is given on the derivation of theoretical results, explicit calculations that employ these results may not always be a trivial task. On the grounds of this observation, one of the main aims of this paper is to illustrate the applicability of our results by proposing and implementing an algorithm (via appropriate computer algebra software) for the numerical calculation of the distributions of interest (see the last section). The organization of the paper is as follows. In the Section 2 we present some preliminary results that are necessary for our exposition. In particular, we present some properties of the wellknown concept of exponential tilting (see e.g. Asmussen and Albrecher (2010)) that will be used extensively in the sequel. In the Section 3 we derive an expression for the joint generating function (g.f.) of ν, S ν that is built upon the moment generating functions of the overshoot and undershoot (deficit at ruin) of the surplus process (see Proposition 3). By recalling that there exists a technique that leads to the determination of the distributions of the overshoot and the undershoot when K, F belong to the mixed Erlang class of distributions (cf. Boutsikas and Politis (2015)), we subsequently focus our exposition on this particular case for K, F. After discussing two important properties of the mixed Erlang class (a density property and a generalized memoryless property) we study a particular and a general case. More specifically in Section 4 we consider the model where K and F are exponential distributions and obtain explicit results for the distribution of ν. Finally, in Section 5 we consider the more general case where the claim amounts and the interarrival times follow mixed Erlang distributions. In this last model we offer an expression for the joint g.f. of ν, S ν, as well as we describe and implement a technique (algorithm) that leads to the numerical computation of the distribution of ν. 2 Preliminaries - Exponentially tilted distributions In this section we give our notations and review some concepts and preliminary results that are necessary for our exposition. We consider the insurer s surplus process U = {U(t), t 0} described in the introduction and we assume that it is inspected only at claim arrival times. This process starts at U(0) = u and terminates at the first time that it falls below zero (ruin) or exceeds the barrier b (safety level exceedance) immediately after the payment of a claim. From a practical point of view, it might be preferable for an insurer to set this modified stopping rule (and not the usual inf{t : U(t) / [0, b]}) because it may not be appropriate to report the attainment of safety level when surplus reaches b from below (due to premium income) because a potentially large claim might be imminent. This is reasonable e.g. when the claim inter-arrival times follow a distribution with an increasing failure rate which stays near to 0 for an initial period. In this case a claim appearance is usually followed by an idle period with no claim arrivals and equivalently the probability of claim appearance increases after an idle period. In addition, it might be simpler for the insurer to inspect the surplus process only on claim arrival times and not continuously. Two random paths of the surplus process (inspected only at claim arrival times) are shown in Fig- 3

4 ures 1, 2 below; in Fig. 1 ruin occurs before reaching b, while in Fig. 2 the process exceeds b before ruin. Fig. 1. A path of U that leads to ruin Fig. 2. A path of U that ends above b Denote by R k := k T i, k = 1, 2,... the claim arrival times. The surplus process at claim arrival times can be expressed as S k = U(R k ) = u + cr k k X i = u + k (ct i X i ) = u + k Z i, k = 1, 2,... where Z i := ct i X i. The Z 1, Z 2,... are of iid rvs and G is their common distribution. Let now ν be the number of steps until the discrete time surplus process S := {S k, k = 1, 2,...} falls below zero (ruin) or exceeds a predefined upper level b (safety level), i.e. ν := inf{k : S k > b or S k < 0}. Obviously, ν is a stopping time (associated with the process S) that expresses the number of claims until the surplus process U passes below 0 or above b immediately after the payment of a claim. Note also that ν < a.s.. We denote by ψ := P(S ν < 0) = 1 P(S ν > b), the probability of ruin before S exceeds the upper barrier b. Perhaps a more appropriate notation for the above quantity would be ψ b (u), but in the sequel we shall use ψ for simplicity. We also denote by A(t) := E ( e t(sν b) S ν b ), B(t) := E ( e t( Sν) S ν 0 ) the moment generating functions of the overshoot and the undershoot of the process S above b and below 0 respectively. The probability ψ can be easily derived through the above g.f.s via Wald s Identity by using a standard technique (see e.g. Cox and Miller (1965) or Karlin and Taylor (1975), p.265). Specifically, we have the following simple lemma. Lemma 1. If there exists a value t 0 0 that satisfies E(e t 0Z ) = 1, then ψ = ebt 0 A(t 0 ) e ut 0 e bt 0 A(t0 ) B( t 0 ). Proof. By invoking Wald s (fundamental) Identity, E(E(e tz ) ν e t ν Z i ) = 1, and the fact that E(e t 0Z ) = 1 we get that E(e t 0(S ν u) ) = 1 and since E(e t 0(S ν u) ) = e t 0(b u) A(t 0 )(1 P(S ν < 0)) + e t 0u B( t 0 )P(S ν < 0) we easily derive the formula for ψ. 4

5 The above Lemma implies that when the distributions of the overshoot and undershoot are known, then the probability of ruin (before the surplus reach the safety level) is also known. It is remarkable that the same holds true for the distribution of the number of claims, ν, until the process S exits the interval [0, b]. This result will be shown in the next section via the notion of exponentially tilted distributions. In the rest of this section we briefly review some known properties related to this concept. As usual, we assume that all the above are defined on a probability space (Ω, F, P) and expectation with respect to (w.r.t.) P is denoted by E. In the same measurable space (Ω, F) we shall also consider another probability measure, to be denoted as P w. The subscript w implies that this measure depends on a real w that belongs to an interval W containing zero. Under P w, we assume that the claims X 1, X 2,... and the inter-arrival times T 1, T 2,... are again iid r.v. s, but now the X i s follow the w-exponentially tilted F (i.e. F w ), and the T i s follow the cw-exponentially tilted K (i.e. K cw ). Recall that, if H is the cumulative distribution function (c.d.f.) of a r.v. Y then the w-exponentially tilted H, denoted by H w, is the c.d.f. given by x H w (x) := ewt dh(t) ewt dh(t) = E(ewY I [Y x]), x R, E(e wy ) for all w such that E(e wy ) <. Note also that if the r.v. Y has density h, the density h w corresponding to H w is h w (x) = ewx h(x) E(e wy ), x R. Obviously, H 0 = H and h 0 = h. Hence, under P w, the c.d.f.s of X i s and T i s are P w (X i x) = F w (x) = E(e wx i I [Xi x]) E(e wx i ) and P w (T i x) = K cw (x) = E(ecwTi I [Ti x]) E(e cwt i ). (2.1) It follows that, under P w, the r.v.s Z i = ct i X i, i = 1, 2,... are again i.i.d. but now they follow the w-exponentially tilted G. Indeed, ˆ P w (Z i z) = P w (ct i X i z) = K cw ( z + x E(e cwt i I [Ti )df w (x) = z+x c ])e wx df (x) c E(e cwt i )E(e wx ) = E(ew(cT i X i ) I [cti X i z]) E(e w(ct i X i) ) = E(ewZ i I [Zi z]) E(e wz i ) = G w (z), z R. We shall denote by E w the expectation under the measure P w. If a r.v. Y is measurable w.r.t. the σ- algebra F i = σ(z 1, Z 2,...Z i ) for some i {1, 2,...} (i.e. Y = ζ(z 1, Z 2,..., Z i ) for some measurable function ζ) then E w (Y ) = = ˆ R i Y dg w (z 1 )... dg w (z i ) = i Z j ) ˆRi ζ(z 1, z 2,..., z i )e w i z j i dg(z 1 )... dg(z i ) E(ewZ j ) E(Y ew i. (2.2) E(ewZ j ) Note that, if Y depends only on Z i then the above reduces to E w (Y ) = E(Y ewz i ) E(e wz i ). (2.3) 5

6 We also denote by A w (t) = E w ( e t(s ν b) S ν > b ) and B w (t) = E w ( e t( S ν) S ν < 0 ), the moment generating functions of the overshoot and the undershoot w.r.t. the measure P w. Lemma 1 can now be restated under the measure P w in the obvious way. The proof is analogous and thus omitted. Lemma 2. If there exists a real function χ 0 such that E w (e χ(w)z 1 ) = 1, w W, the probability ψ w of ruin under the measure P w, w W, is equal to ψ w = P w (S ν < 0) = e bχ(w) A w (χ(w)) e uχ(w) e bχ(w) A w (χ(w)) B w ( χ(w)). Obviously, for w = 0 we get ψ 0 = ψ, A 0 = A, B 0 = B, χ(0) = t 0 and thus the above reduces to Lemma 1. 3 On the number of claims until the surplus falls below 0 or exceeds b The following proposition shows that when the distributions of the overshoot (surplus above b) and undershoot (deficit at ruin) are known under P w,w W, then the joint g.f. E(r ν e xsν ) of ν, S ν is also known. Proposition 3. If there exists a real function ρ such that E(e ρ(r)z 1 ) = r 1 for r (0, 1) then (a) (b) (c) E ( r ν e xsν S ν < 0 ) = ψ ρ(r) ψ 0 e ρ(r)u B ρ(r) (ρ(r) x) E ( r ν e xsν S ν > b ) = 1 ψ ρ(r) e xb+ρ(r)(u b) A ρ(r) (x ρ(r)) 1 ψ 0 E(r ν e xsν ) = e xb+ρ(r)(u b) A ρ(r) (x ρ(r))(1 ψ ρ(r) ) + e ρ(r)u B ρ(r) (ρ(r) x)ψ ρ(r) for all r, x such that the above expectations exist. Proof. The stopping time ν is finite a.s. and thus I [ν=i] = I [ν< ] = 1 a.s. Hence, E(I [Sν<0]r ν e x(sν u) ) = E(I [Sν<0]r ν e x(sν u) I [ν=i] ) = = E(I [Si <0]r i e (x ρ(r))(s i u) Since r 1 = E(e ρ(r)z i ), the above expression can be written as E(I [Si <0]r i e x(si u) I [ν=i] )) i e ρ(r)z j I [ν=i] ). E(I [Sν<0]r ν e x(sν u) ) = e ρ(r)z 1 e ρ(r)z i E(Y i... ). E(e ρ(r)z 1 ) E(e ρ(r)z i ) 6

7 where Y i = I [Si <0]e (x ρ(r))(s i u) I [ν=i]. For every i {1, 2,...}, the r.v. Y i is measurable w.r.t. the σ-algebra F i and therefore, invoking relation (2.2), we deduce E(I [Sν<0]r ν e x(sν u) ) = E ρ(r) (Y i ) = E ρ(r) (I [Sν<0]e (x ρ(r))(sν u) ), and finally, E ( r ν e x(sν u) S ν < 0 ) = 1 ψ 0 E(I [Sν<0]r ν e x(sν u) ) = 1 ψ 0 E ρ(r) (I [Sν<0]e (x ρ(r))(sν u) ) which is relation (a). Similarly we get = ψ ρ(r) E ρ(r) (e (x ρ(r))(sν u) S ν < 0) = ψ ρ(r) e (ρ(r) x)u B ρ(r) (ρ(r) x) ψ 0 ψ 0 E ( r ν e x(sν u) S ν > b ) = 1 ψ ρ(r) 1 ψ 0 E ρ(r) (e (x ρ(r))(sν u) S ν > b) = 1 ψ ρ(r) 1 ψ 0 e (x ρ(r))(b u) A ρ(r) (x ρ(r)), which is relation (b). By combining (a) and (b) we easily derive (c). Remark 1. If we assume that both X i, T i have light-tailed distributions, i.e. there exists x 1 (0, ] such that E(e xx i ) < for x < x 1 while E(e xx i ) as x x 1, and similarly, there exists x 2 (0, ] such that E(e xt i ) < for x < x 2 while E(e xt i ) as x x 2, then the m.g.f. M Z (x) = E(e xz i ) is finite on ( x 1, x 2 /c) while M Z (x) as x x 1 and x x 2 /c. From the convexity of M Z and the fact that M Z (0) = 1 we deduce that, in this case, the equation E(e xz 1 ) = r 1, r (0, 1), has exactly two real solutions ρ 1, ρ 2 with ρ 1 ( x 1, 0) and ρ 2 (0, x 2 /c). We can choose either of these two solutions as ρ(r) in the above proposition (and of course both lead to the same results). From Proposition 3 we can derive the distribution of the number of claims ν until the discrete time surplus process S exits [0, b], in three cases: unconditionally (result (c)), given that ruin occurs (before reaching the safety level, result (a)), and given that the safety level has been reached (before ruin occurs, result (b)). Taking also into account Lemma 2, this derivation depends solely on the distribution of Z i s, and the distributions of the overshoot (excess of safety level, given that S ν > b) and the undershoot (deficit at ruin, given that S ν < 0) of S. In general, though, the distributions of the overshoot and undershoot are very difficult to determine. However, in cases where the distributions of T i, X i can be expressed via exponential distributions (e.g. as sum, or random sum, or mixture), we can employ a generalized form of the memoryless property and derive a representation for the moment generating functions A and B. In what follows we shall show how this technique can be used in the case where the distributions of X i and/or T i belong to the mixed Erlang class of distributions. This is an important wide class of distributions that has been studied by Willmot and Woo (2007). The c.d.f. of a mixed Erlang distribution (with parameters m, η and weights h k [0, 1], k = 1, 2,..., m, and m k=1 h k = 1) has the following form F ME (x) := m h k F Ek,η (x), x 0, (3.1) k=1 7

8 where F Ek,η denotes the c.d.f. of an Erlang distribution E k,η with shape parameter k, scale parameter η and density η k f Ek,η (x) := (k 1)! xk 1 e ηx, x 0. One important property of the above class (comprising of all F ME with m {1, 2,...}, η > 0 and weights h k ) is that it is dense in the class of all non-negative distributions. More specifically, any distribution H on [0, ) can be approximated by the following mixture of Erlang distributions, H (m) n (x) := H(0) + m k=1 p (m) k,n F E k,n (x) (3.2) for m large enough so that H( m ) 1 and p(m) n k,n := (H( k k 1 ) H( ))/H( m ). This property allows us to approximate, to any desired accuracy, any distributions for X i s and T i s by considering n n n appropriate mixed Erlang distributions. Another important property of this class is that it possesses a "generalized form of the memoryless property". In particular, the following result holds true. Lemma 4. (a) If the distribution K of the claim inter-arrival times (resp. F of the claim amounts) is a mixed Erlang distribution of the form (3.1) then the distribution of the overshoot S ν b, given that S ν > b (resp. the undershoot S ν, given that S ν < 0) is a also a mixed Erlang distribution of the same form, with parameters m, η/c (resp. m, η) and, in general, with a different mixing distribution (weights). For the first case in the above lemma (distribution of the overshoot) see Lemma 7 in Boutsikas and Politis (2015), while for the second case (distribution of the undershoot) see, for example, Willmot and Woo (2007). As already mentioned in the Introduction, in the sequel we shall study two cases for K and F. Initially we consider the simplest model where K and F are exponential distributions and finally we consider the case where the claim amounts and the inter-arrival times follow general mixed Erlang distributions. 4 The number of claims in the classical risk model We start by studying the simplest model, i.e. the classical (Poisson) collective risk model with exponential claims with mean µ. Here, the claim arrival process is a Poisson process with intensity. In this case, the density of the i.i.d. r.v.s Z i = ct i X i have the form g(z) = d dz P (ct i X i < z) = ˆ e c (z+x) e 1 µ x dx = cµ max{ z,0} and the respective moment generating function (m.g.f.) is E(e tz i ) = E(e tct i )E(e tx i ) = tc { c µ+c µ 1 µ+c 1/µ 1/µ + t, 1 µ < t < c. c e c z, z 0 µ e 1 µ z, z < 0, 8

9 Therefore, the density of Z i under measure P w, 1 < w < is given by µ c { cwµ+c g w (x) = ewx g(x) E(e wz ) = ( µ+c c w)e ( c w)x, x 0 µ cwµ (w µ+c µ )e(w+ µ )x, x < 0, and using (2.3) we derive the m.g.f. of Z i under P w, E w (e tz i ) = E(e(w+t)Z i ) E(e wz i ) = 1 (1 c t)(1 + µ t), w, w + t ( 1 µ, c ). cw wµ+1 It is clear that, under the measure P w, the distribution of Z i s can be considered as a mixture of two exponential distributions. In particular, the process S goes up or down on each step (i.e. Z i > 0 or Z i < 0) with probabilities cwµ+c and µ cwµ, while the length of each step is exponentially µ+c µ+c distributed with parameter w c and w + 1 respectively. Therefore, by invoking the memoryless µ property of the exponential distribution we have that, B w (t) = E w ( e t( S ν) S ν < 0 ) = w + 1 µ w + 1 µ t, t < w + 1 µ, (4.1) A w (t) = E w ( e t(s ν b) S ν > b ) = and hence, from Lemma 2, we derive that ψ w = P w (S ν < 0) = w c w t, t < c c w, (4.2) e bχ(w) A w (χ(w)) e uχ(w) ebχ(w) c w e uχ(w) e bχ(w) A w (χ(w)) B w ( χ(w)) = w+ 1 µ e bχ(w) c w w+ 1 µ w+ 1 µ c w, (4.3) where χ(w) = /c 2w 1/µ is the non-zero solution of the equation E w (e χ(w)z 1 ) = 1, provided that 2w /c 1/µ. Note that ψ 1/µ = 1, ψ /c = 0 and when 2w /c 1/µ the above reduces to ψ w = ( 1 + )(b u) + 2 µ c ( 1 + )b + 4. µ c Next, we employ Proposition 3 in order find the distribution of the number of claims ν until the surplus process S exits [0, b] for the first time. Specifically, we have the following result. Proposition 5. In the classical (Poisson) ruin model with exponential claims, the joint g.f. of ν, S ν is given by (a) (b) (c) E ( r ν e xsν S ν < 0 ) = ψ ρ(r) e ρ(r) + 1 ρ(r)u µ 1 ψ 0 + x µ E ( r ν e xsν S ν > b ) = 1 ψ ρ(r) 1 ψ 0 e xb+ρ(r)(u b) E(r ν e xsν ) = e xb+ρ(r)(u b) c ρ(r) ρ(r) c x c x (1 ψ ρ(r)) + e ρ(r) + 1 ρ(r)u µ c x ψ ρ(r) µ

10 for all r, x such that the above expectations exist, where ( ρ(r) := 1 2 c 1 1 µ µ + ) c c µ r, r (0, 1) (4.4) and ψ w is given by (4.3). Proof. First, we determine the solutions of the equation E(e ρ(r)z 1 ) = r 1 (ρ(r)µ + 1)( cρ(r)) = r 1, w.r.t. ρ(r) ( 1, ). One such solution is given by (4.4). Moreover, from equations (4.1) and (4.2) µ c we get A ρ(r) (x ρ(r)) = ρ(r) c x, x < c, B ρ(r)(ρ(r) x) = ρ(r) + 1 µ 1 + x, x > 1 µ, c µ and the proof is completed by directly applying Proposition 3. The pgf of ν follows from the above proposition by setting x = 0. The (conditional or unconditional) distribution of ν can be numerically evaluated for specific values of the parameters u, b, µ,, c with the aid of the relation d i P(ν = i) = 1 i! dr i E(rν ) r=0, i = 0, 1,... As an illustration, if u = 2, b = 5, µ = 0.5, = 1.5, c = 1 (and the premium loading factor is θ = c/(µ) 1 = 1/3), the probability of ruin before the surplus S exceeds b is ψ (cf. equation (4.3)) and the distribution of the number ν of claims is shown on the Figures 3 and 4. The values of P(ν = i), i = 1, 2,..., 40 were easily computed using the function SeriesCoefficient of the Wolfram Mathematica software package (represented as dots) and the results were verified by means of Monte Carlo simulation using 10 5 iterations (see barplots in the same figures). Fig. 3. The pmf of ν given that S ν > b Fig. 4. The pmf of ν given that S ν < 0 Finally, if we let b when µ < c (i.e. the expected claim payoffs is less than the premium income per unit time) then ψ ρ(r) = c ρ(r) e (b u)χ(ρ(r)) ρ(r)+ 1 µ c ρ(r) ρ(r)+ 1 µ ρ(r)+ 1 µ c ρ(r)e bχ(ρ(r)) 1, ψ 0 = 10 e b( c 1 µ ) µ c eu( c 1 µ ) e b( c 1 µ ) µ c c µ µ c eu( c 1 µ ),

11 since χ(ρ(r)) = c 2ρ(r) 1 µ = ( 1 µ + c ) c µ r > 0, and therefore, when there is no upper barrier, the joint g.f. of ν, S ν, given that ruin occurs, is given by E ( r ν e xsν S ν < 0 ) = c where ρ(r) = 1 2 ( ( ) c µ µ c 4 1 r c µ E (ν S ν < 0) = c + u c µ, µ e(ρ(r) ) c + 1 µ )u ρ(r) + 1 µ 1 µ + x, (4.5). From (4.5) we derive by differentiation that, Var (ν S ν < 0) = ((c2 + 2 µ 2 )u + (c + µ)cµ) (c µ) 3. The above formulae for the mean and variance were also derived by Dickson (2012), who also obtained the probability function of ν via a different method. 5 The number of claims for the general mixed Erlang case We finally consider the more general case where the claim amounts X i, i = 1, 2,... and the interarrival times T i, i = 1, 2,... follow general mixed Erlang distributions. As already mentioned previously (see Section 3), the mixed Erlang class is dense in the class of all non-negative distributions. Therefore we can employ the results of this section in order to approximate the distribution of ν for any renewal risk model with X i F 0, T i K 0 by choosing appropriate mixed Erlang distributions F and K that approximate F 0 and K 0 respectively (cf. (3.2)). However, under this setup, we cannot obtain closed form results for the pgf of ν since, as it is shown below, the whole technique depends on the determination of the roots of a polynomial of degree m. Nevertheless, we shall describe a procedure that can lead (with the aid of appropriate computing software such as Wolfram Mathematica or Matlab) to the numerical calculation of the distribution of ν when the model parameter values are given. In what follows we assume that the common distribution F of X i s and the common distribution K of T i s are finite mixtures of Erlang distributions of the form (see (3.1)) m 1 m 2 F (x) := d k F Ek,1/α (x), x 0, K(x) := h k F Ek,β (x), x 0 (5.1) k=1 where d k, h k [0, 1], m 1 k=1 d k = 1, (d m1 0) and m 2 k=1 h k = 1, (h m2 0). The m.g.f. of the distribution of Z 1 is given by E(e wz 1 ) = E(e wct 1 )E(e wx 1 ) = and hence the density of Z 1 under P w is m 2 E w (e tz 1 ) = E(e(w+t)Z 1 ) E(e wz 1 ) = h i (1 cw β )i m2 m 1 k=1 h i m1 (1 c β (w+t))i m2 11 d j (1 + αw) j, 1 α < w < β c, h i (1 c β w)i m1 d j (1+α(w+t)) j d j. (1+αw) j

12 Denote by χ 1 (w), χ 2 (w),..., χ m (w) C the m = m 1 + m 2 1 non-zero solutions of the equation E w (e χz 1 ) = 1 or, equivalently, the m non-zero roots of the polynomial equation (w.r.t. χ) m 2 h i (1 c β (w + m 1 χ))m 2 i d j (1 + α(w + χ)) m 1 j = (1 c β (w + χ))m 2 (1 + α(w + χ)) m 1 m 2 m 1 h i (1 c β w)m 2 i d j (1+αw) m 1 j (1 c.(5.2) β w)m 2 (1+αw) m 1 Below (see Proposition 6) we derive an expression for the joint g.f. of ν, S ν that can lead to the computation of the distribution of ν when the model parameter values are given. Before presenting this result we introduce some necessary notations. Denote by a(w) the m 1 vector a(w) := [D 1 (w), D 2 (w),..., D m (w)] T, (5.3) and by B(w) the m m matrix A 11 (w) A 12 (w) A 1,m2 1(w) B 11 (w) B 12 (w) B 1,m1 1(w) C 1 (w) A 21 (w) A 22 (w) A 2,m2 1(w) B 21 (w) B 22 (w) B 2,m1 1(w) C 2 (w) B(w) :=... A m1 (w) A m2 (w) A m,m2 1(w) B m1 (w) B m2 (w) B m,m1 1(w) C m (w) (5.4) where A ij (w) := eχ i(w)(b u) (1 χ i(w) C i (w) := β/c w )j eχi(w)(b u) (1 χ i(w) β/c w )m 2 eχ i(w)(b u) (1 χ i(w) β/c w )m 2, B ij (w) := e χ i(w)u 1, D i (w) := 1 (1 + χ i(w) 1/α+w )j e χ i(w)u (1 + χ i(w) 1/α+w )m 1 e χi(w)u (1 + χ i(w) 1/α+w )m 1 for 1 α < w < β c. Denote also by κ i(w), i = 1, 2,..., m 2 1, π i (w), i = 1, 2,..., m 1 1, and a(w) the elements of the following m 1 vector, (5.5) k(w) := [κ 1(w), κ 2(w),..., κ m 2 1(w), π 1 (w), π 2 (w),..., π m1 1(w), a(w)] T := B 1 (w) a(w) (5.6) for all w ( 1 α, β c ) such that B 1 (w) exists. Let also κ i (w) := κ i(w)/a(w), i = 1, 2,..., m 2 1. In Proposition 6 we show that joint g.f. of ν, S ν can be expressed via the elements of the above vector k(w). Proposition 6. If ρ(r) ( 1, β ) α c is a real solution of E(eρ(r)Z 1 ) = r 1, with r (0, 1), or equivalently a real solution of, m 2 h i m 1 d j (1 c β ρ(r))i (1 + αρ(r)) = j r 1, (5.7) 12

13 such that B 1 (ρ(r)) exists, then the joint g.f. of ν, S ν is given by (a) (b) E ( r ν e xsν S ν < 0 ) = ψ ρ(r) ψ 0 m 1 e ρ(r)u E ( r ν e xsν S ν > b ) = 1 ψ ρ(r) 1 ψ 0 e xb+ρ(r)(u b) π i (ρ(r)) (1, ρ(r) x 1/α+ρ(r) )i m 2 κ i (ρ(r)) (1, x ρ(r) β/c ρ(r) )i for all r, x such that the above expectations exist, where ψ ρ(r) = (a(ρ(r)) + 1) 1, ψ 0 = (a(0) + 1) 1. The quantities κ i (ρ(r)), π i (ρ(r)), a(ρ(r)) and a(0) are given in (5.6) with w = ρ(r) and w = 0 respectively. Proof. By employing (2.1) we may easily verify that, under P w, the distribution of the r.v. s T 1, T 2,... is again a finite mixture of m 2 Erlang distributions with scale parameter β wc. Hence, according to Lemma 4, under P w, the overshoot S ν b S ν > b, follows a mixture of m 2 Erlangs with scale parameter (β wc)/c, for some (unknown) non-negative weights κ 1 (w), κ 2 (w),..., κ m2 (w) such that m 2 κ i(w) = 1. Thus A w (t) = E w (e t(sν b) S ν > b) = m 2 κ i (w) (1 t. (5.8) )i β/c w Similarly, under P w the distribution of the r.v. s X 1, X 2,... can be expressed as a mixture of m 1 Erlang distributions with scale parameter 1/α + w, and hence the undershoot S ν S ν < 0 follows a mixture of m 1 Erlangs with scale parameter 1/α + w, for some (unknown) non-negative weights π 1 (w), π 2 (w),..., π m1 (w) such that m 1 π i(w) = 1. Therefore, it follows that, B w (t) = E w (e t( Sν) S ν < 0) = m 1 π i (w) (1 t. (5.9) )i 1/α+w Next, we describe how we can obtain the above mentioned unknown weights κ i (w), i = 1, 2,..., m 2, π i (w), i = 1, 2,..., m 1 (and thus the distributions of the overshoot and undershoot). By invoking Wald s fundamental Identity E w (E w (e χz 1 ) ν e χ(sν u) ) = 1, and the fact that E w (e χ i(w)z 1 ) = 1, i = 1, 2,..., m, (cf. (5.2)) we get the following set of equations 1 = E w (e χ i(w)(s ν u) ) = e χ i(w)(b u) A w (χ i (w))(1 ψ w ) + e χ i(w)u B w ( χ i (w))ψ w, i = 1, 2,..., m. Using (5.8),(5.9), and by setting a(w) := ψw 1 of m linear equations m 2 e χ i(w)(b u) a(w)κ j (w) + (1 χ e χi(w)u i(w) )j β/c w 1, the above equations lead to the following system m 1 or equivalently (since m 2 κ j(w) = 1, m 1 π j(w) = 1), m 2 1 A ij (w)κ j(w) + m 1 1 π j (w) = 1 + a(w), i = 1, 2,..., m, (1 + χ i(w) )j 1/α+w B ij (w)π j (w) + C i (w)a(w) = D i (w), i = 1, 2,..., m. 13

14 where A ij (w), B ij (w), C i (w), D i (w), are given in (5.5), and κ i(w) = a(w)κ i (w). Therefore, we have a system of m = m 1 + m 2 1 linear equations which can be easily solved w.r.t. κ i(w), π i (w) and a(w). This solution can be written in the compact form k(w) = B 1 (w) a(w), where k(w), a(w) and B(w) are given in (5.6),(5.3),(5.4). The proof is now completed by employing Proposition 3 along with relations (5.8), (5.9). Observe that the m.g.f. M Z (x) = E(e xz i ) is finite on ( 1 α, β c ) while M Z(x) as x 1 α and x β c. Hence, according to Remark 1, the equation (5.7), has exactly two real solutions ρ 1, ρ 2 with ρ 1 ( 1, 0) α and ρ 2 (0, β ). We can choose either of these solutions as ρ(r). c Note that results similar to Proposition 6 can be also derived via the methodology described by Jacobsen (2011) for the two sided exit time of a random walk, where the distributions of the downward and upward jumps (increments) belong to a class containing mixtures of exponentials. Apart from its theoretical interest, it is now essential to verify the applicability of the above result. In what follows we shall describe and implement an algorithm that leads to the numerical calculation of the probabilities P(ν = t S ν < 0) for t = 1, 2,..., n, when the mixed Erlang distributions of X i, T i and the values of the parameters u (initial surplus), c (premium income rate) and b (safety level) are given. One of the difficulties of this task regarding the determination of the probability distribution of ν from the expression of E (r ν S ν < 0) (cf. Proposition 6 with x = 0), arises from the fact that this expression does not have a closed form w.r.t. r, since it is written as a function of the roots of specific polynomials that depend on r. Therefore, we cannot directly determine the probability function of ν from the coefficients of the series expansion of its g.f. via repeated differentiation (e.g. using the function SeriesCoefficient of Wolfram Mathematica software). Instead, we need to apply numerical differentiation (see steps 8,9 below). The steps for the computation of P(ν = t S ν < 0), t = 1, 2,..., n, via Proposition 6, given the values of the parameters u, c, b, α, β, d k, k = 1,..., m 1, h k, k = 1,..., m 2, are the following: STEP 1. Set w = 0 in the equation (5.2) and find numerically its m non-zero roots, χ 1 (0),..., χ m (0) C. STEP 2. Form B(0) and a(0) (cf. (5.4), (5.3)) for w = 0, and calculate k(0) = B 1 (0) a(0) (cf. (5.6)). Set ψ 0 = (a(0) + 1) 1 where a(0) is the last element of k(0) and ψ 0 is the probability of ruin before the surplus reaches the safety level. STEP 3. Choose a small ε (e.g ). STEP 4. Set r = ε in the equation (5.7) and find numerically a root, ρ(r). STEP 5. Set w = ρ(r) in the equation (5.2) and find numerically its m non-zero roots, χ 1 (ρ(r)),..., χ m (ρ(r)) C. STEP 6. Form B(ρ(r)) and a(ρ(r)) (cf. (5.4), (5.3) for w = ρ(r)) and calculate k(ρ(r)) = B 1 (ρ(r)) a(ρ(r)) (cf. (5.6)). Set ψ ρ(r) = (a(ρ(r)) + 1) 1 where a(ρ(r)) is the last element of k(ρ(r)). STEP 7. Calculate E (r ν S ν < 0) by using the formula (a) in Proposition 6, for x = 0. 14

15 STEP 8. Repeat the steps 4-7 for r = ε, 2ε,..., nε and denote by g 0,i, i = 1, 2,..., n + 1 the computed values of E (r ν S ν < 0), i.e. where g 0,1 = E(0 ν S ν < 0) = 0. g 0,i := E(((i 1)ε) ν S ν < 0), i = 1, 2,..., n + 1, STEP 9. (Numerical differentiation) compute recursively the following quantities: d 1,i = d 0,i+1 d 0,i, i = 1, 2,..., n ε d 2,i = d 1,i+1 d 1,i, i = 1, 2,..., n 1, 2ε... d n,i = d n 1,i+1 d n 1,i, i = 1. nε Finally, the quantities d 1,1, d 2,1,..., d n,1 are the computed values of P(ν = t S ν < 0) for t = 1, 2,..., n, respectively. Numerical Example 1. As already mentioned in the introduction, Dickson (2012) derived (via a different methodology) a formula for p n (u), the probability of ruin at the nth claim, assuming exponential inter-claim times (classical model) and b = (i.e. no upper barrier). This formula (cf. relation (10) in Dickson (2012)) is expressed via sums of integrals involving f j, j = 1, 2,..., n 1, where f j denotes the j-fold convolution of the pdf of the claim amounts. Note that similar results were also derived by Landriault et al. (2011) considering a renewal risk model with exponential claims. We observe that p n (u) is actually equal to P(ν = n, S ν < 0) when b. Hence it would be interesting to numerically compare the value of p n (u) calculated by Dickson s formula, with the value of p n,b (u) := P(ν = n, S ν < 0) calculated by the procedure described above, as b increases. In the following Table 1 we present these values (first 5 digits) by assuming c = 1, u = 4, Erlang(3) claim amounts (with scale parameter 1) and exponential inter-arrival times with mean 4.5 (left side) and 1.5 (right side). We also include the values of ψ 0 and θ = ce(t i )/E(X i ) 1, i.e. the probability of ruin before the surplus crosses b and the premium loading factor respectively. n ψ 0 n = 1 n = 2 n = 3 n = 4 n = 5 ψ 0 n = 1 n = 2 n = 3 n = 4 n = 5 p n,6 (u) p n,10 (u) p n,15 (u) p n(u) E(T i ) = 4.5 (θ = 1/2), E(T i ) = 1.5 (θ = 1/2) Table 1. The probability of ruin at the nth claim, for b = 6, 10, 15 and. As it was expected, p n,b (u) p n (u) as b and it is remarkable that p n,b (u) p n (u) even for relatively small values of b. Note that Dickson s method may lead to exact results when f j has a known closed form (as in the above case) whereas our methodology offers, in general, approximate results but with arbitrary precision (see also the following example). All values were easily calculated via Wolfram Mathematica Software. Numerical Example 2. In this example, we implement the above algorithm in a renewal model with mixed Erlang K and F and specific values of the model parameters using again the Wolfram 15

16 Mathematica software package. Specifically, we assume that the initial surplus u = 2, the premium income rate c = 1 and the safety level b = 4. We also assume that the claim amounts X 1, X 2,... follow a mixed Erlang distribution (cf. (5.1)) with parameters α = 1/5, m 1 = 5 and weights (d 1, d 2, d 3, d 4, d 5 ) = ( 9 45, 1 45, 1 45, 9 45, ), (E(X i ) = 7/9) while the inter-arrival times follow a mixed Erlang distribution (cf. (5.1)) with parameters β = 4, m 2 = 5 and weights (h 1, h 2, h 3, h 4, h 5 ) = ( 1 15, 2 15, 3 15, 4 15, 5 15 ), (E(T i ) = 11/12). The above densities are shown in Figures 5, 6. Fig. 5. The density of claim amounts Fig. 6. The density of inter-arrival times The premium loading factor is θ = ce(t i )/E(X i ) 1 = 5/28. Four simulated paths of the surplus process under the above assumptions are given in the following Figure 7. Fig. 7. Four simulated paths until the surplus process drops below 0 or exceeds the safety level. Following the steps described previously we find that the probability of ruin (before reaching the safety level) is ψ The Monte Carlo estimate, based on 10 7 random paths, of the same probability is (± for a 99% c.i.). The values of d t,1 = P(ν = t S ν < 0) for t = 1, 2,..., 36, computed by the above algorithm (with numeric differentiation step ε = 10 8 ), are shown in the following Table 2. The first column of the table gives the value of t, the second column ( prob. ) consists of the value of P(ν = t S ν < 0) that was computed by the above algorithm, the third column ( MC estim. ) displays the Monte Carlo estimated value of P(ν = t S ν < 0) (based on 10 7 random paths) and the last column ( 99% c.i. ) gives a 99% confidence interval for the same probability, which has been obtained via the simulated sample. 16

17 t prob. MC estim. 99% c.i. t prob. MC estim. 99% c.i. t prob. MC estim. 99% c.i ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 2. Computed and MC estimated values of pmf of ν given that S ν < 0. Finally, the above computed and estimated values of the probability function of ν (given that ruin occurs before the surplus reaches b) are depicted in Figure 8, where the computed values are represented as dots and the Monte Carlo estimated values are shown by the barplot. Fig. 8. The pmf of ν given that S ν < 0 All the computations and simulations were performed using the Wolfram Mathematica software package (v.9). The roots were extracted using the function NSolve after setting a very high working precision (500 digits) so that the matrix inversions and especially the numeric differentiation does not suffer from significant numerical errors. By setting lower working precision, e.g. 100 digits, we discovered that were able to compute accurately only the first 5 values of the probability function of ν, while using 200 digits we could obtain accurately only the first 14 values. The amount of time required for the computation of the probability function of ν in the above table was less than 5 minutes, while (using the same machine) the estimation of the same quantities through MC simulation required about one hour. References [1] Asmussen S., Albrecher H. (2010) Ruin Probabilities, 2nd ed., World Scientific, Singapore. [2] Borovkov A.A. and Borovkov K.A. (2008). Asymptotic Analysis of Random Walks. Encyclopedia of Mathematics and its Applications, 118, Cambridge University Press. [3] Boutsikas M.V. and Politis K. (2015) Exit times, overshoot and undershoot for a surplus process in the presence of an upper barrier. Methodology and Computing in Applied Probability (to appear). 17

18 [4] Boutsikas M.V., Rakitzis A.C. and Antzoulakos D.L (2011) On the relation between the distributions of stopping time and stopped sum via Wald s Identity with applications. arxiv: [5] Cox, D.R. and Miller, H.D. (1965) The theory of stochastic processes. Wiley. [6] Dickson D.C.M. (2012) The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model. Insurance: Mathematics and Economics, 50, [7] Dickson D.C.M. and Li S. (2013). The distribution of the time to reach a given level and the duration of negative surplus in the the Erlang(2) risk model. Insurance: Mathematics and Economics, 52, [8] Egidio dos Reis A.D. (2002) How many claims does it take to get ruined and recovered? Insurance: Mathematics and Economics, 31, [9] Ezhov I.I., Kadankov V.F. and Kadankova T.V. (2007) Two-boundary problems for a random walk. Ukrainian Mathematical Journal, 59, [10] Frostig E., Pitts S.M. and Politis K. (2012) The time to ruin and the number of claims until ruin for phase-type claims. Insurance: Mathematics and Economics, 51, [11] Gerber, H.U. (1990) When does the surplus reach a given target? Insurance: Mathematics and Economics, 9, [12] Jacobsen M. (2011) Exit times for a class of random walks: exact distribution results. Journal of Applied Probability, Spec. Vol 48A, [13] Karlin S. and Taylor H.W. (1975) A First Course in Stochastic Processes, second ed. Academic Press, California. [14] Kemperman J.H.B. (1963) A Wiener-Hopf type method for a general random walk with a twosided boundary. Annals of Mathematical Statistics, 34, [15] Landriault D., Shi T. and Willmot G.E. (2011) Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions. Insurance: Mathematics and Economics, 49, [16] Li S. (2008) The time of recovery and the maximum severity of ruin in a Sparre Andersen model. North American Actuarial Journal, 12, [17] Li S. and Lu Y. (2014) On the time and the number of claims when the surplus drops below a certain level, Scandinavian Actuarial Journal, DOI: / [18] Perry D, Stadje W. and Zacks S. (2005) A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary. Methodology and Computing in Applied Probability, 7, 51 62, 2005 [19] Stanford D.A. and Stroinski K.J. (1994) Recursive Methods for Computing Finite-Time Ruin Probabilities for Phase-Distributed Claim Sizes. ASTIN Bulletin, 24, [20] Willmot G.E. and Woo J-K. (2007) On the Class of Erlang Mixtures with Risk Theoretic Applications. North American Actuarial Journal, 11, [21] Zhao C. and Zhang C. (2013) Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang(n) claims. Journal of Computational and Applied Mathematics, 244,