The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-modulated Risk Model

Transcription

1 The Deompositions of the Disounted Penalty Funtions and Dividends-Penalty Identity in a Markov-modulated Risk Model Shuanming Li a and Yi Lu b a Centre for Atuarial Studies, Department of Eonomis The University of Melbourne, Australia b Department of Statistis and Atuarial Siene Simon Fraser University, Canada Abstrat In this paper, we study the expeted disounted penalty funtions and their deompositions in a Markov-modulated risk proess in whih the rate for the Poisson laim arrivals and the distribution of the laim amounts vary in time depending on the state of an underlying (external) Markov jump proess. The main feature of the model is the flexibility modeling the arrival proess in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Expliit formulas for the expeted disounted penalty funtion at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the laim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividendspenalty identity is derived when the model is modified by applying a barrier dividend strategy. Keywords: Markov-modulated risk model, expeted disounted penalty funtion, maximum surplus before ruin, maximum severity of ruin, dividendspenalty identity. 1

2 1 Introdution Asmussen (1989) proposed a Markov-modulated risk model in whih both the frequeny of the laim arrivals and the distribution of the laim amounts are influened by an external environment proess {J(t); t }. This model is a generalization of the lassial ompound Poisson risk model and the primary motivation for this generalization is the enhaned flexibility that it permits for the modeling of the laim arrival proess and the laim severity distribution assumed in the lassial risk proess. The impat of poor weather onditions on the finanial performane of automobile insurane portfolios, or of the outbreak of epidemis whih impat health insurane portfolios, is well known. See, for example, Asmussen (1989). Zhu and Yang (27) refer to states of the proess {J(t); t } as eonomi irumstanes or politial regime swithings. It is therefore appealing to inlude in the lassial risk proess assumptions whih permit variation in both laim frequenies and laim severities as a result of external environmental fators. The modeling framework that is advoated in this paper ahieves this. Suppose that {J(t); t } is a homogeneous, irreduible and reurrent Markov proess with finite state spae E = {1, 2,...,m}. Denote the intensity matrix of {J(t); t } by Λ =(α i,j ) m i,, with α i,i := α i for i E. Let π =(π 1,π 2,...,π m ) be the stationary distribution of {J(t); t }. Let N(t) be the number of laims ouring in (,t]. If J(s) =i for all s in a small interval (t, t + h], then the number of laims ouring in that interval, i.e., N(t + h) N(t), has a Poisson distribution with parameter λ i (> ). We assume further that given the proess {J(t); t }, the proess {N(t); t } has independent inrements. Then P ( N(t + h) =n +1 N(t) =n, J(s) =i for t<s t + h ) = λ i h + o(h). The proess {N(t); t } is alled a Markov-modulated Poisson proess, whih is a speial ase of a Cox proess. It also an be seen as a Poisson proess with its parameter driven by an external environment proess {J(t); t }. We also assume that, given J(t) = i, the laim amounts have distribution funtion F i (x), with density funtion f i (x) and finite mean µ i (i E). Moreover, we assume that premiums are reeived ontinuously at a positive onstant rate. The orresponding surplus proess {U(t); t } is given by N(t) U(t) =u + t X n, t, (1.1) n=1 where u is the initial surplus and X n is the amount of the n-th laim. We 2

3 also assume that the positive loading ondition holds, i.e., π i ( λ i µ i ) >. For notational onveniene, let P i ( ) =P( J() = i). Define T = inf{t : U(t) < } to be the time of ruin and let w(x, y), for x, y, be non-negative valued of penalty funtion. Define for δ, u and i, j E φ i,j (u) =E i [ e δt w(u(t ), U(T ) )I(T <,J(T )=j) U() = u ] to be the expeted disounted penalty (Gerber-Shiu) funtion at ruin if ruin is aused by a laim in state j, given the initial surplus u and the initial environment i E, for the surplus U(T ) before ruin and the defiit U(T ) at ruin, where I( ) is the indiator funtion. Then φ i (u) = φ i,j (u), u, i E, is the expeted disounted penalty funtion at ruin, given the initial surplus u and the initial environment i E. In partiular, when δ = and w(x, y) =1,φ i,j (u) simplifies to Ψ i,j (u) with the definition Ψ i,j (u) =P i ( T<,J(T )=j U() = u ), i,j E. Here Ψ i,j (u) is the ruin probability if ruin is aused by a laim in state j given that the initial state is i and hene Ψ i (u) = Ψ i,j (u), u, i E, is the probability of ruin given that the initial state is i, and orrespondingly Φ i (u) =1 Ψ i (u) is the non-ruin probability given that the initial state is i. Models of this type have also been investigated by some authors, e.g., Reinhard (1984), Bäuerle (1996), Shmidli (1997), Wu (1999), Snoussi (22), Lu and Li (25), and Lu (26). Ng and Yang (26) give losed form solutions for the joint distribution of the surplus before and after ruin when the initial surplus is zero or when the laim amount distributions are phase-type distributed. Li and Lu (27) study the moments of the present value of the dividend payments and the distribution of the total dividends prior to ruin for the Markov-modulated risk model modified by the introdution of a barrier dividend. Albreher and Boxma 3

4 (25) study the expeted disounted penalty funtion in a semi-markovian dependent risk model in whih at eah instant of a laim, the underlying Markov hain jumps to a new state and the distribution of laim depends on this state. This model inludes, as speial ases, the lassial risk model, and the Sparre Andersen model with phase-type inter-arrival times, as well as models with ausal dependene between the laim amounts distribution and the laim inter-arrivals distribution. In this paper, we study the expet disounted penalty funtions and their deompositions for the Markov-modulated risk model. Expliit formulas for the expeted disounted penalty funtion at ruin if ruin is aused by a laim in state j, given initial surplus u and initial environment i, are derived when the initial surplus is zero or when all the laim amount distributions are from the rational family. The distributions of the maximum surplus before ruin and the maximum severity of ruin are studied through the ruin probabilities and their deompositions. The dividends-penalty identity is also derived for the model modified by the introdution of a barrier dividend. 2 Expeted disounted penalty funtions 2.1 A system of integro-differential equations Using the same arguments as in Ng and Yang (26), we obtain the following integro-differential equations for φ i,j (u) by onditioning on the events ouring in a small interval [,h], for i E, [ u ] φ i,i (u) =(λ i + δ) φ i,i (u) λ i φ i,i (u x) f i (x)dx + ω i (u) α i,k φ k,i (u), and for i j, u φ i,j(u) =(λ i + δ) φ i,j (u) λ i φ i,j (u x) f i (x)dx (2.1) α i,k φ k,j (u), (2.2) where ω i (u) = w(u, x u)f u i(x)dx. Let ˆφ i,j (s), ˆfi (s), and ˆω i (s) be the Laplae transforms of φ i,j,f i and ω i, respetively. Taking Laplae transforms on both sides of Eqs. (2.1) and (2.2), we 4

5 have, for i, j E, [ s λ i + δ + λ ] i ˆf i (s) ˆφ i,j (s)+ 1 α i,k ˆφk,j (s) =φ i,j () λ i ˆω i(s)i(i = j). (2.3) Further for simpliity, define S i (s) =s δ/ (λ i /)(1 ˆf i (s)), for i E. Then Eq. (2.3) an be rewritten in the following matrix form: A(s) ˆφ(s) =φ() ˆω(s), where A(s) = diag(s 1 (s),s 2 (s),...,s m (s)) + Λ/, φ(u) =(φ i,j (u)) m i,, ˆφ(s) = (ˆφ i,j (s)) m i,, and ˆω(s) = diag(λ 1ˆω 1 (s)/, λ 2ˆω 2 (s)/,...,λ mˆω m (s)/). It follows that ˆφ(s) =[A(s)] 1 [φ() ˆω(s)] = A (s)φ() A (s)ˆω(s) det[a(s)] where A (s) is the adjoint matrix of A(s)., (2.4) 2.2 The initial values for φ() Using the same arguments as in Albreher and Boxma (25), we an show that the harateristi equation det[a(s)] = has exatly m roots with positive real parts, say, ρ 1,ρ 2,...,ρ m, whih play an important role in determining the initial values for φ i,j (). We assume that ρ 1,ρ 2,...,ρ m are distint in the sequel. Now we define the divided differenes of a matrix B(s), with respet to distint numbers r 1,r 2,..., reursively as follows: B[r 1,s] = B(s) B(r 1), s r 1 B[r 1,r 2,s] = B[r 1,s] B[r 1,r 2 ], s r 2 B[r 1,r 2,r 3,s] = B[r 1,r 2,s] B[r 1,r 2,r 3 ], s r 3 and so on. As for the divided differenes of a funtion (Gerber and Shiu (25)), we have the following formula for the (k 1)-th divided differene B[r 1,r 2,...,r k ]= k B(r j ) k,i =j (r j r i ). 5

6 For distint ρ 1,ρ 2,...,ρ m, sine ˆφ i,j (s) is finite for R(s), then A (ρ i )φ() = A (ρ i )ˆω(ρ i ), i =1, 2,...,m. Therefore A [ρ 1,ρ 2 ]φ() = (A ˆω)[ρ 1,ρ 2 ], where (A ˆω)[ρ 1,ρ 2 ] is the divided differene of the produt of matries A (s) and ˆω(s) with respet to ρ 1 and ρ 2, given by and reursively, (A ˆω)[ρ 1,ρ 2 ]=A (ρ 1 ) ˆω[ρ 1,ρ 2 ]+A [ρ 1,ρ 2 ] ˆω(ρ 2 ) (2.5) A [ρ 1,ρ 2,...,ρ i ]φ() = (A ˆω)[ρ 1,ρ 2,...,ρ i ], i =2, 3,...,m, where the matrix (A ˆω)[ρ 1,ρ 2,...,ρ m ] is given by the following formula: (A ˆω)[ρ 1,ρ 2,...,ρ m ]= A [ρ 1,...,ρ i ] ˆω[ρ i,...,ρ m ]. (2.6) Then we have the following result for φ() : φ() = {A [ρ 1,ρ 2,...,ρ m ]} 1 (A ˆω)[ρ 1,ρ 2,...,ρ m ]. (2.7) In partiular, when m =2, ( [ s α 1 +δ λ 1 A(s) = 1 ˆf1 (s) ] ) α 1 [ α 2 s α 2+δ λ 2 1 ˆf2 (s) ], and the adjoint matrix of A(s) is given as ( [ s α 2 +δ λ A 2 (s) = 1 ˆf2 (s) ] ) α 1 [ α 2 s α 1+δ λ 1 1 ˆf1 (s) ]. Then A [ρ 1,ρ 2 ] = diag ( 1+(λ 2 /) ˆf 2 [ρ 1,ρ 2 ], 1+(λ 1 /) ˆf 1 [ρ 1,ρ 2 ] ), and by Eq. (2.5), Eq. (2.7) simplifies to φ() = ˆω(ρ 1 )+{A [ρ 1,ρ 2 ]} 1 A (ρ 2 )ˆω[ρ 1,ρ 2 ] ( λ1 ) ˆω 1 (ρ 1 ) = + λ 2 ˆω 2 (ρ 1 ) λ 1 [S 2 (ρ 2 ) α 2 ]ˆω 1 [ρ 1,ρ 2 ] 1+ λ 2 ˆf2 [ρ 1,ρ 2 ] α 2 λ 1 2 ˆω 1 [ρ 1,ρ 2 ] 1+ λ 1 ˆf1 [ρ 1,ρ 2 ] α 1 λ 2 2 ˆω 2 [ρ 1,ρ 2 ] 1+ λ 2 ˆf2 [ρ 1,ρ 2 ] λ 2 [S 1 (ρ 2 ) α 1 ]ˆω 2 [ρ 1,ρ 2 ] 1+ λ 1 ˆf1 [ρ 1,ρ 2 ]. 6

7 Further, if w(x, y) = 1 and δ =, we have that ρ 1 = ρ, ρ 2 =,ω i (u) = F i (u), and φ i,j (u) =Ψ i,j (u) for i, j =1, 2. In this ase, we also get that S i () =, ˆω i [ρ, ]=[ˆω i (ρ) µ i ]/ρ and ˆf i [ρ, ] = ˆω i (ρ). Then Ψ() = ( λ1 ˆω 1 (ρ 1 ) λ 2 ˆω 2 (ρ 1 ) ) 1 2 ρ λ 1 α 2 [ˆω 1 (ρ) µ 1 ] 1 (λ 2 /)ˆω 2 (ρ) λ 1 α 2 [ˆω 1 (ρ) µ 1 ] 1 (λ 1 /)ˆω 1 (ρ) λ 2 α 1 [ˆω 2 (ρ) µ 2 ] 1 (λ 2 /)ˆω 2 (ρ) λ 2 α 1 [ˆω 2 (ρ) µ 2 ] 1 (λ 1 /)ˆω 1 (ρ) where Ψ() = ( Ψ i,j () ) 2 i, and ˆω i(ρ) =[1 ˆf i (ρ)]/ρ, for i =1, An expliit expression for φ(u) By applying the divided differenes repeatedly to the numerator of Eq. (2.4), we obtain the following expression for ˆφ(s): m ˆφ(s) = (s ρ [ ] i) A [ρ 1,ρ 2,...,ρ m,s]φ() (A ˆω)[ρ 1,ρ 2,...,ρ m,s]. det[a(s)] By Eq. (2.6), we have (A ˆω)[ρ 1,ρ 2,...,ρ m,s]=a [ρ 1,ρ 2,...,ρ m,s]ˆω(s) + A [ρ 1,...,ρ i ] ˆω[ρ i,...,ρ m,s]. Then ˆφ(s) an be rewritten as m ˆφ(s) = (s ρ i) [A [ρ 1,ρ 2,...,ρ m,s] ( φ() ˆω(s) ) det[a(s)] ] A [ρ 1,...,ρ i ] ˆω[ρ i,...,ρ m,s]. (2.8) The Laplae transform ˆφ(s) an be inverted for some speial laim amount distributions. Consider the ase where the laim amount distributions for m lasses are from the rational family, that is, their Laplae transforms an be expressed as a ratio of polynomials: ˆf i (s) = p(i) 1 (s) q (i) (s), N +,i J,. 7

8 where q (i) is a polynomial of degree, while p (i) 1 is a polynomial of degree 1or less; all have leading oeffiient 1 and satisfy p (i) 1 () = q(i) (). Further, equation q (i) (s) = has roots with only negative real parts. To obtain expressions whih an be inverted easily, we multiply both numerator and denominator of Eq. (2.8) by m q(i) (s), yielding {[ ] m ˆφ(s) = (s ρ i) m [φ() det[a(s)] A [ρ m 1,ρ 2,...,ρ m,s] q (i) ] q(i) (s) ˆω(s) (s) [ m ] m } q (i) (s) A [ρ 1,...,ρ i ]ˆω[ρ i,...,ρ m,s]. (2.9) First, we look at the denominator in (2.9), denoted by D(s): m D(s) = det[a(s)] q (i) (s), whih is learly a polynomial of degree m + m with the leading oeffiient 1, and therefore equation D(s) = has m + m roots in the omplex plane. By the fat that equation det[a(s)] = has exatly m roots, ρ 1,ρ 2,...,ρ m, with positive real parts, we an rewrite D(s) as D(s) = m K m (s ρ i ) (s + R i ), where K m = m, and all R i s have positive real parts by the definition of the rational distribution. For simpliity, we further assume that these R i s are distint. Consequently, Eq. (2.9) an be expressed as follows: {[ ] 1 m (φ() ˆφ(s) = Km (s + R A [ρ 1,ρ 2,...,ρ m,s] q (i) ) (s) ˆω(s) i) [ m ] m } q (i) (s) A [ρ 1,...,ρ i ]ˆω[ρ i,...,ρ m,s]. (2.1) It is easy to see that the elements in matrix A [ρ 1,ρ 2,...,ρ m,s] m q(i) (s) are polynomials of degrees whih are less than K m, and all A [ρ 1,...,ρ i ] for i E 8

9 are onstants. Then we have the following partial frations: A [ρ 1,ρ 2,...,ρ m,s] m Km l=1 (s + R l) q(i) (s) = K m l=1 M (l) s + R l m q(i) K (s) m Km (s + R i) =1+ n l, s + R l where M (l) =(m (l) i,j )m i,, for l =1, 2,...,K m, are oeffiient matries with while n l is the oeffiient given by M (l) = A [ρ 1,ρ 2,...,ρ m, R l ] m n l = Km ν=1,ν =l (R ν R l ) l=1 q(i) ( R l ), m q(i) ( R l ) Km ν=1,ν =l (R ν R l ), l =1, 2,...,K m. Thus by partial fration Eq. (2.1) an be expressed as { K m 1 ˆφ(s) = M (l)[ φ() ˆω(s) ] m } n l A [ρ 1,...,ρ i ]ˆω[ρ i,...,ρ m,s] s + R l l=1 A [ρ 1,...,ρ i ]ˆω[ρ i,...,ρ m,s]. (2.11) To obtain the expliit Laplae inverse of (2.11), we introdue an operator T r for a matrix B(y) with respet to a omplex number r, tobe T r B(y) = y e r(x y) B(x)dx, r C,y. (2.12) Here B(y) is a matrix with eah element being an integrable real-valued funtion of y. The omposition operators of T r an be defined reursively, for example, T r1 T r2 B(y) =T r2 T r1 B(y) = T r 1 B(y) T r2 B(y) r 2 r 1, r 1 r 2 C,y. This operator has been used for the integrable real-valued funtion in some papers, see, for example, Dikson and Hipp (21) and Li and Garrido (24). Similar to 9

10 (1.1) in Gerber and Shiu (25), the following result holds for the relationship between the operator T r and the orresponding divided differene: ( m ) T ri B() = ( 1) m 1 ˆB[r1,r 2,...,r m ]. (2.13) Further by the definition of operator T r in (2.12), we have that T s T r B() = e sx [T r B(x)]dx, whih shows that the Laplae inverse of matrix T s T r B() is T r B(x). In general, we have the following formula for the Laplae inverse of matrix [T s ( m T r i )B](): [ ( m ) ] ( m ) L 1 T s T ri B() = T ri B(x). (2.14) Thus it follows from Eq. (2.11) and by (2.13) and (2.14), the expliit Laplae inversion of ˆφ(s) is given by ( m ) { K m φ(u) = ( 1) m i A [ρ 1,...,ρ i ] T ρk ω(u)+ e R l u M (l) φ() [ e R l u ω(u) n l m k=i where in above formula is the onvolution operator. l=1 ( m ) ]} ( 1) m i A [ρ 1,...,ρ i ] T ρk ω(u), k=i 3 The maximum surplus before ruin For b>u, define ( ξ i,j (u; b) =P i sup U(t) <b,t<,j(t )=j ) U() = u, i,j E, t T to be the probability that ruin ours from initial surplus u without the surplus proess reahing level b prior to ruin if ruin is aused by a laim in state j given that the proess starts from initial state i. Alternatively, ξ i,j (u; b) is the probability that ruin ours in state j from initial state i in the presene of an absorbing barrier at b. Obviously, ξ i,j (u; b) = for b u. Then ξ i (u; b) = ξ i,j (u; b), i E, 1

11 is the probability that ruin ours without the surplus proess reahing level b prior to ruin from initial state i and initial surplus u. For u b and i, j E, define χ i,j (u; b) to be the probability that the surplus proess attains level b at state j from initial state i and initial surplus u without first falling below zero. Clearly, χ i,j (b; b) =I(i = j) for i, j E. Then χ i (u; b) = χ i,j (u; b), i E, is the probability that the surplus proess attains level b from initial state i and initial surplus u without first falling below zero. Sine eventually either ruin ours without the surplus attaining level b or the surplus attains level b, then we have χ i (u; b) =1 ξ i (u; b) for i E. Let χ(u; b) = ( χ i,j (u; b) ) m. It follows from Li i, and Lu (27) that χ(u; b) =v(u)[v(b)] 1, u b, (3.1) where v(u) =(v i,j (u)) m i, is an m m matrix with v i,j (u) being the solution of the following system of integro-differential equations: u v i,j(u) =λ i v i,j (u) λ i v i,j (u x) f i (x)dx α i,k v k,j (u), (3.2) with the boundary onditions v i,j () = I(i = j) for i, j E. By onsidering whether or not the surplus reahes b (> u) before ruin, we have Ψ i,j (u) =ξ i,j (u; b)+ χ i,k (u; b)ψ k,j (b), i,j E, or in matrix notation, Ψ(u) =ξ(u; b) +χ(u; b)ψ(b), (3.3) where ξ(u, b) = ( ξ i,j (u; b) ) m with ξ(b; b) = and being the m m zero matrix. i, In partiular, when m =1, the model simplifies to the lassial risk model and sine χ(u; b) =1 ξ(u; b), then Eq. (3.3) gives Ψ(u) Ψ(b) ξ(u; b) =, u b. 1 Ψ(b) This formula an be found in Dikson and Gray (1984). For m N +, we will show in the next setion that χ(u; b) an be expressed in terms of the ruin probability matrix Ψ(u) and therefore the distribution of the maximum surplus before ruin given in (3.3) an also be expressed in terms of the ruin probability matrix. 11

12 4 The maximum severity of ruin In this setion, we allow the surplus proess to ontinue if ruin ours, and onsider the insurer s maximum severity of ruin from the time of ruin until the time that the surplus returns to level. Sine we assume that the positive loading ondition holds, it is ertain that the surplus proess will attain this level after ruin. For the lassial risk model, Piard (1994) gives an expliit expression in terms of the ruin probability for the distribution of the maximum severity of ruin. Li and Dikson (26) study the distribution of the maximum severity of ruin for the Sparre Andersen risk model with Erlang inter-arrival times. For u, we define T to be the time of the first uprossing of the surplus proess through level after ruin, i.e., T = inf{t : t>t,u(t) }, and define M u = sup{ U(t),T t T} to be the maximum severity of ruin. Let H i,j (z; u) =P i ( Mu z,t <,J(T )=j ), z, i,j E, denote the distribution funtion of the maximum severity of ruin if ruin is aused by a laim in state j given that the proess starts from initial state i and H i (z; u) = H i,j (z; u), i E, is the distribution funtion of the maximum severity of ruin given that the proess starts from initial state i. If the surplus proess starts with an initial surplus u and initial state i, then the maximum severity of ruin will be no more than z if ruin ours (by a laim in state j) with a defiit y z and if the surplus does not fall below z from the level y. The probability of the latter event is χ j (z y; z) sine attaining level from level y without falling below z is equivalent to attaining level z from level z y without falling below. Thus H i,j (z; u) = where g i,j (u, y) = G i,j (u, y)/ y with g i,j (u, y)χ j (z y; z)dy, G i,j (u, y) =P i ( T<,J(T )=j, U(T ) y ), u,y,i,j E, 12

13 being the probability that ruin ours by a laim in state j and the defiit at ruin is at most y given that the initial state is i. Therefore H i (z; u) = or in matrix notation, H i,j (z; u) = g i,j (u, y)χ j (z y; z)dy, H(z; u) = g(u, y)χ(z y; z)1dy, (4.1) where H(z; u) = ( H 1 (z; u),h 2 (z; u),...,h m (z; u) )T is an m 1 olumn vetor, g(u, y) = ( g i,j (u, y) ) m is an m m matrix, and 1 =(1, i, 1,...,1)T is an m 1 olumn vetor. For the lassial risk model (m = 1), Piard (1994) shows that the integral in (4.1) an be expressed in terms of the ruin probability. Now we aim at alulating the integral in (4.1) for m N +. To ahieve this, first we need to express χ(u; b) in terms of Ψ(u). Let Ψ(u) =( Ψ i,j (u)) m i, = I Ψ(u) with I being the m m identity matrix. Setting δ = and w(x, y) = 1 in Eqs. (2.1) and (2.2), we have for, i, j E, that Ψ i,j(u) =λ i Ψi,j (u) λ i u Ψ i,j (u x) f i (x)dx α i,k Ψk,j (u)+α i,j. (4.2) Let ˆ Ψ i,j (s) = e su Ψ i,j (u)du be the Laplae transform of Ψ i,j (u). Taking Laplae transforms on both sides of Eq. (4.2) yields, for i, j E, that [ s λ ( i 1 ˆf ) ] i (s) ˆ Ψ i,j (s)+ 1 Eq. (4.3) an be expressed in the following matrix form: α i,k ˆ Ψ k,j (s) = Ψ i,j () + α i,j s. (4.3) A(s) ˆ Ψ(s) = Ψ() + Λ s, (4.4) where ˆ Ψ(s) = (ˆ Ψi,j (s) ) m, and A(s) is defined in Setion 2 with δ =. Sine i, χ(u; b) =v(u)[v(b)] 1 with ˆv (s) =(ˆv i,j (s)) m i, =[A(s)] 1, then we have ˆ Ψ(s) =ˆv (s) Ψ() + ˆv(s) s Λ. (4.5) 13

14 Inverting (4.5) yields ( u Ψ(u) =v(u) Ψ() + ) Λ v(x)dx. (4.6) Taking derivatives with respet to u on both sides of (4.6) we obtain the following first order differential equation for the matrix v(u): v (u)+v(u) Λ[ Ψ()] 1 = Ψ (u)[ Ψ()] 1, u, with boundary ondition v() = I. Solving it gives v(u) = Ψ(u)[ Ψ()] 1 u Ψ(x)[ Ψ()] 1 Λ[ Ψ()] 1 e Λ[ Ψ()] 1 (u x) dx. (4.7) For simpliity, let Ψ(u) = Ψ(u)[ Ψ()] 1 and =(Λ/)[ Ψ()] 1. Then Eq. (4.7) an be rewritten as v(u) = Ψ(u) u Ψ(x) e (u x) dx. (4.8) Sine χ(u; b) =v(u)[v(b)] 1 with v(u) being given by (4.8), then Eq. (4.1) an be rewritten as H(z; u) = g(u, y)v(z y)[v(z)] 1 1 dy = g(u, y) Ψ(z y)[v(z)] 1 1 dy ( y ) g(u, y) Ψ(z y x) e x dx dy[v(z)] 1 1 = g(u, y) Ψ(z y)dy [v(z)] 1 1 ( x ) g(u, y) Ψ(z y x)dy e x dx [v(z)] 1 1. (4.9) For further evaluation, note that Ψ i,j (u + z) = g i,k (u, y)ψ k,j (z y)dy + or in matrix form Ψ(u + z) = g(u, y)ψ(z y)dy + 14 z z g i,j (u, y)dy, g(u, y)dy,

15 whih is equivalent to I Ψ(u + z) = g(u, y)[i Ψ(z y)]dy + z g(u, y)dy. (4.1) By using the formula that g(u, y)dy = Ψ(u), Eq. (4.1) an be simplified to and onsequently, g(u, y) Ψ(z y)dy = Ψ(u + z) Ψ(u), g(u, y) Ψ(z y)dy = Ψ(u + z) Ψ(u). (4.11) Finally, by substituting (4.8) and (4.11) into (4.9) we have [ } H(z; u)={ Ψ(u + z) Ψ(u) Ψ(u + z x) Ψ(u)] e x dx [ 1 Ψ(z) Ψ(z x) e dx] x 1, where Ψ(u) = Ψ(u)[ Ψ()] 1 =[I Ψ(u)][I Ψ()] 1. Remarks: 1. As in the lassial risk model, H(z; u) depends only on the ruin probability matrix Ψ(u). 2. When m =1, the model redues to the lassial risk model; Λ =, and then =. In this ase, H(z; u) simplifies to H(z; u) =[ Ψ(u + z) Ψ(u)][ Ψ(z)] 1, whih an be found in Piard (1994). Here Ψ(u) is the survival probability for the lassial risk model. 5 The dividends-penalty identity In this setion, as in Lin et al. (23) and Gerber et al. (26), we derive the dividends-penalty identity for the Markov-modulated risk model. Now we onsider the surplus proess (1.1) modified by the payment of dividends aording to a barrier strategy: when the surplus exeeds a onstant barrier b ( u), dividends 15

16 are paid ontinuously so the surplus stays at level b until a new laim ours. Let U b (t) be the surplus proess with initial surplus U b () = u under the above barrier strategy and define T u,b = inf{t :U b (t) < } to be the time of ruin. Let δ> be the fore of interest for valuation and define D u,b = Tu,b e δt dd(t), u b, to be the present value of all dividends paid until the time of ruin T u,b given that the initial surplus is u, where D(t) is the aggregate dividends paid by time t. Define V i,j (u; b) =E i [ Du, b I(J(T u,b )=j) ], u b, i, j E, to be the expeted present value of the dividend payments before ruin if ruin is aused by a laim in state j given the initial state is i. Then V i (u; b) = V i,j (u; b), u b, i E, is the expeted present value of the dividend payments before ruin given that the initial state is i. Let V(u; b) = ( V i,j (u; b) ) m be an m m matrix. It follows from Li and Lu i, (27) that V(u; b) =v δ (u)[v δ (b)] 1, u b, (5.1) where v δ (u) =(v i,j (u; δ)) m i, is an m m matrix with v i,j (u; δ) satisfying the system of homogenous integro-differential equations u v i,j(u; δ) =(λ i + δ) v i,j (u; δ) λ i v i,j (u x; δ) f i (x)dx α i,k v k,j (u; δ), (5.2) with boundary onditions v i,j (; δ) =I(i = j) for i, j E. For the modified surplus proess {U b (t); t }, we define φ i,j (u; b) for u b and i, j E to be the expeted disounted penalty funtion at ruin if ruin is aused by a laim in state j, given initial surplus u and initial environment i E. In the definition of φ i,j (u; b), we use the same δ and penalty funtion w(x, y) as in φ i,j (u). Then φ i (u; b) = φ i,j (u; b), u, i E, 16

17 is the expeted disounted penalty (Gerber-Shiu) funtion at ruin, given initial surplus u and initial environment i. In partiular, when δ = and w(x, y) =1, φ i,j (u; b) simplifies to Ψ i,j (u; b) with the definition Ψ i,j (u; b) =P i ( Tu,b <,J(T u,b )=j U() = u ), i,j E, where Ψ i,j (u; b) is the ruin probability if ruin is aused by a laim in state j given that the initial state is i and hene Ψ i (u; b) = Ψ i,j (u; b), i E, is the probability of ruin given that the initial state is i. Φ i (u; b) =1 Ψ i (u; b) is the non-ruin probability. To onstrut the dividends-penalty identity, we define τ b to be the first time that the surplus U b (t) reahes b, and for δ>, define L i,j (u; b) =E i [ e δτ b I ( τ b <T u,b,j(τ b )=j ) U b () = u ], u b, i, j E. L i (u; b) an be interpreted as the expeted present value of one dollar payable at the time of reahing the barrier b in state j without ruin ouring, given that the initial environment state is i and the initial surplus is u. Alternatively, it an be viewed as the Laplae transform of the time to reah the dividend barrier b without ruin ouring, with respet to the parameter δ. Using the same arguments as in Li and Lu (27), we an show that L i,j (u) for i, j E and u<bsatisfy the following system of homogenous integrodifferential equations u L i,j(u; b) =(λ i + δ) L i,j (u; b) λ i L i,j (u x; b) f i (x)dx α i,k L k,j (u; b), with boundary onditions L i,j (b; b) =I(i = j). Let L(u; b) = ( L i,j (u; b) ) m be an m m matrix. Following from the fat i, L(b; b) =I, we have L(u; b) =v δ (u)l(; b) =v δ (u)[v δ (b)] 1. (5.3) Using the same arguments as in Gerber et al. (26), we have for u<b that φ i,j (u; b) φ i,j (u) = L i,k (u; b)[φ k,j (b; b) φ k,j (b)], (5.4) 17

18 with boundary onditions φ i,j (b ; b)=, i,j E. Formula (5.4) an be obtained by reasoning as in Gerber et al. (26): onsider a partiular sample path of the surplus proess starting at u<band in state i. The penalties at ruin if the ruin is aused by a laim in state j, with and without the dividend barrier, an be different only if the surplus reahes the level b in state k before ruin for k E. The boundary onditions φ i,j(b ; b) = an be explained by heuristi reasoning as in Gerber et al. (26). In matrix notation, Eq. (5.4) and its boundary onditions an be expressed as φ(u; b) φ(u) = L(u; b)[φ(b; b) φ(b)], φ (b ; b) =, where φ(u; b) = ( φ i,j (u; b) ) m, i, φ (b ; b)= ( φ i,j(b ; b) ) m, and is the m m i, zero matrix. Then φ(b; b) φ(b) = [L (b; b)] 1 φ (b) = [ v δ(b)[v δ (b)] 1] 1 φ (b) = v δ (b)[v δ (b)] 1 φ (b). (5.5) Finally by Eqs. (5.1), (5.3), and (5.5) we have the following dividends-penalty identity in matrix form: φ(u; b) = φ(u) L(u; b)v δ (b)[v δ(b)] 1 φ (b) = φ(u) v δ (u)[v δ (b)] 1 v δ (b)[v δ (b)] 1 φ (b) = φ(u) V(u; b)φ (b), u b. (5.6) We remark that φ(u) an be obtained expliitly for rational laim amounts as in Setion 2. In partiular, when δ = and w(x, y) = 1, Eq. (5.6) simplifies to Ψ(u; b)=ψ(u) V(u; b)ψ (b), u b. (5.7) Eq. (5.7) an be used to show that ruin is ertain under the barrier strategy, i.e., Φ i (u; b) = for i E and u b. Let Φ(u) =(Φ 1 (u), Φ 2 (u),...,φ m (u)) T and Φ(u; b) =(Φ 1 (u; b), Φ 2 (u; b),...,φ m (u; b)) T be m 1 olumn vetors. Then Φ(u; b) = 1 Ψ(u; b)1 = 1 Ψ(u)1 + V(u; b)ψ (b)1 = Φ(u) V(u; b)φ (b), u b. (5.8) 18

19 Lu and Li (25) show that Φ i (u) for i E satisfies the following system of integro-differential equations u Φ i (u) =λ i Φ i (u) λ i Φ i (u x, b) f i (x)dx α i,k Φ k (u). (5.9) The solutions of the above system of equations are uniquely determined by the initial onditions Φ i () for i E. It follows from (5.2) that m v i,j(u)φ j () for i E has the same initial values as Φ i (u) and satisfies the system of integro-differential equations (5.9) when δ =. Then we onlude that Φ i (u) = m v i,j(u)φ j (), or Φ(u) =v (u)φ(), and Eq. (5.8) simplifies to Φ(u; b) = v (u)φ() V(u; b)v (b)φ() = v (u)φ() v (u)[v (b)] 1 v (b)φ() =, u b, where is the zero m 1 olumn vetor. Conluding remarks By using matrix notation, we have shown how the evaluation of Gerber-Shiu s expeted disounted penalty funtion for the lassial risk model an be extended to a Markov-modulated risk model with laim amount distributions belonging to the rational family. We have generalized the results on the maximum surplus before ruin and maximum severity of ruin for the lassial risk model to those for the Markov-modulated risk model. The dividends-penalty identity was first given in Lin et al. (23) for the lassial ompound Poisson risk model. Gerber et al. (26) extend the identity to the risk model with independent and stationary inrements. The matrix form dividends-penalty identity is derived in this paper for the Markov-modulated risk model. All the results obtained in Setion 3-5 an be evaluated by the expeted disounted penalty funtions and the (deomposed) ruin probabilities studied in Setion 2. Finally, all the results obtained in this paper an be extended to the ase where premium rate varies aording to the state of the external environment proess {J(t); t }. Aknowledgment The authors would like to thank Prof. David Dikson, Dr. David Pitt, and an anonymous referee for their valuable omments and suggestions to improve the paper and larify its presentation. 19

20 Referenes [1] Albreher, H. and Boxma, O.J. (25) On the disounted penalty funtion in a Markov-dependent risk model. Insurane: Mathematis and Eonomis, 37, [2] Asmussen, S. (1989) Risk theory in a Markovian environment. Sandinavian Atuarial Journal, [3] Asmussen, S., Frey, A., Rolski, T. and Shmidt V. (1995) Does Markovmodulation inrease the risk? ASTIN Bulletin, 25, [4] Bäuerle, N. (1996) Some results about the expeted ruin time in Markovmodulated risk models. Insurane: Mathematis and Eonomis, 18, [5] Dikson, D.C.M. and Gray, J. (1984) Approximations to ruin probability in the presene of an upper absorbing barrier. Sandinavian Atuarial Journal, [6] Dikson, D.C.M. and Hipp, C. (21) On the time to ruin for Erlang(2) risk proess, Insurane: Mathematis and Eonomis, 29, [7] Gerber, H.U., Lin, X.S., and Yang, H. (26) A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bulletin, 36, [8] Gerber, H.U. and Shiu, E.S.W. (25) The time value of ruin in a Sparre Andersen model. North Amerian Atuarial Journal, 9(2), [9] Li, S. and Garrido, J. (24) On a lass of renewal risk models with a onstant dividend barrier. Insurane: Mathematis and Eonomis, 35, [1] Li, S. and Dikson, D.C.M. (26) The maximum surplus before ruin in an Erlang(n) risk proess and related problems. Insurane: Mathematis and Eonomis, 38, [11] Li, S. and Lu, Y. (27) Moments of the dividend payments and related problems in a Markov-modulated risk model. North Amerian Atuarial Journal 11(2), [12] Lu, Y. (26) On the severity of ruin in a Markov-modulated risk model. Sandinavian Atuarial Journal,

21 [13] Lu, Y. and Li, S. (25) On the probability of ruin in a Markov-modulated risk model. Insurane: Mathematis and Eonomis, 37, [14] Ng, A.C.Y. and Yang, H. (26) On the joint distribution of surplus before and after ruin under a Markovian regime swithing model. Stohasti Proesses and their Appliations, 116, [15] Piard, P. (1994) On some measures of the severity of ruin in the lassial Poisson model. Insurane: Mathematis and Eonomis, 14, [16] Reinhard, J. M. (1984) On a lass of semi-markov risk models obtained as lassial risk models in a Markovian environment. ASTIN Bulletin, 14, [17] Shmidli, H. (1997) Estimation of the Lundberg oeffiient for a Markov modulated risk model. Sandinavian Atuarial Journal, [18] Snoussi, M. (22) The severity of ruin in Markov-modulated risk models. Bulletin of the Swiss Assoiation of Atuaries, [19] Wu, Y. (1999) Bounds for the ruin probability under a Markovian modulated risk model. Commun. Statist. -Stohasti Models, 15(1), [2] Zhu, J. and Yang, H. (27) Ruin theory for a Markov regime-swithing model under a threshold dividend strategy. Insurane: Mathematis and eonomis, doi:1.116/j.insmatheo SHUANMING LI Centre for Atuarial Studies Department of Eonomis The University of Melbourne Vitoria 31 Australia shli@unimelb.edu.au Fax: YI LU Department of Statistis & Atuarial Siene Simon Fraser University 8888 University Drive Burnaby, BC, CANADA V5A 1S6 yilu@sfu.a 21