The distributions of some quantities for Erlang(2) risk models

where (0) = ( 1 (0); 2 (0)) with i (0) = 1 i (0):Now let u = e 2 (u) = ( (u) ^g(u; r)) ; (1 ) (u) + ^g(u; r)cr cr crand let I be the 2 2 identity matrix. Note that > (0) = [I (0)]1 > ;where 1 = (1; 1): Then the distribution of N can be re-expressed asPr(N = 0) = 1 u 1 > ;Pr(N = n) = u [ (0)] n 1 [I (0)] 1 > ; n = 1; 2; : : : :This shows that N follows a discrete phase-type distribution with representation( u ; (0)) :Example 4 Let p(x) = e x . Then it is well-known, e.g. Grandell (1991),that (u) = (1 R=)e Ru where R is the unique negative solution ofand froms 1 (u) = 2 (u) 2= 2 c c 2 s + c d du 2(u)(see, for example, Dickson and Li (2012)) we obtain 1 (u) = (1+cR=) 2 (u).As g i (u; y) = i (u) e y , we have8 < cr cr(+r) i(u); j = 1; ij (u) = : 1i(u); j = 2:cr + cr(+r)When = 2; = 1 and c = 1:2 we have Pr(N = 0) = 1 (u), and forn = 1; 2; 3; : : :Pr(N = n) = 0:1698(0:8302 n 1 ) (u).4.3 Unconditional distributionsWe can use the same ideas as in Section 4.2 to …nd the distribution of thej-th period of negative surplus for j = 2; 3; 4; : : :. Let a j be a vector given bya j = e 2 (u) (0) j 2 :Then for i = 1; 2 the i-th element of a j is the probability that the (j 1)-thupcrossing of the surplus process through 0 occurs with i phases until thenext claim. Thus, with probability (a j ) 1 1 (0) the density of T j is f (1)T 2(t) and15