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

where R 1 and R 2 are the negative solutions of 2 2s = 2 ; (4.7)c c 2 s + anda 11 =a 12 =a 21 =a 22 = 2c 2 (r + ) 2 (3 + 2r) R 1 (r + 2)R 2 R 1; 2c 2 (r + ) 2 (3 + 2r) R 2 (r + 2)R 1 R 2; R 1;c 2 (r + ) R 2 R 1 2 2 R 2;c 2 (r + ) R 1 R 2with r > 0 being the positive solution of equation (4.7). The ultimate ruinprobability can be obtained asThenwhere(u) =FurtherZ 10g(u; y)dy = (a 11 + a 21 ) e R 1u + (a 12 + a 22 ) e R 2u ; u 0:g(u; y) =g(u; y)(u)= (u) e y + (1 (u)) 2 ye y ;(u) = a 11e R1u + a 12 e R 2u; u 0:(u) 2^g(u; r i ) = (u)r i + + (1 (u)) ; i = 1; 2;r i + so that formula (4.2) simpli…es toD (u) = (u) 12 r 1 + + 1 + 2 (1 (u)) 1r 2 + 2 (r 1 + ) + 12 (r 2 + ) 2 (u)+c (r 1 + )(r 2 + ) + 2 (1 (u))c (r 1 + )(r 2 + ) + 2 (1 (u))2 c (r 1 + ) 2 (r 2 + ) 2 (u) 2 (u)+2c (r 1 + ) 2 (r 2 + ) 2c (r 1 + )(r 2 + ) 2+ 3 (1 (u)) 3 (1 (u))2c (r 1 + ) 3 (r 2 + ) 2c (r 1 + )(r 2 + ) : 3 (4.8)12