Let the surplus of an insurance company to be modelled as N1 (t) N2 (t)

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U(t)=u+ct? X1,i ? X2,i, U(0)=u?0, i=1 i=1

where N1(t) and N2(t) are Poisson processes, with parameters ?1 and ?2, respec- tively, and {X1,i}i?1, {X2,i}i?1 are independent and identically distributed ran- dom variables with common distribution function FX(x). Assume that {X1,i}i?1, {X2,i}i?1, N1(t) and N2(t) are independent.

5. Let the surplus of an insurance company to be modelled as Ni(t ) N2 (t ) U(t ) = utct - > Xli - > X2,i, U(0) = u20, 1= 1 where Ni(t) and N2(t) are Poisson processes, with parameters , and 12, respec- tively, and {Xi,i}izl, {X2/}izi are independent and identically distributed ran- dom variables with common distribution function Fx(x). Assume that {X1,}iz1, {X2i}>1, Ni(t) and N2(t) are independent. (a) Let o(u) = P(T = 00|U(0) = u) =1-(u). (i) Show that the survival probability, o(u), satisfies the following integro- differential equation co' ( u ) = (1 + 12)o(2) - (21+ 12 ) / 4(u - x)fx (x ) dx , where fx(x) is the (common) density function of X, and X2, respectively. (ii) Let A1 = 3, 12 = 7, c = 5 and Fx(x) = 1 -e-5x, x 2 0. Show that 8(u) = 1 - --31 (b) Let 12 = 0, i.e. N2(t) = 0 in [0, t) and consider the finite time survival probability, i.e. Q(u, t) = P(U(s) 20,0 o o(h)/h = 0. (ii) Show that o(u, t) satisfies the following partial integro-differential equation Can o(u, t) - 25(u, t) = And(u, t) - M1 / o(u - x, t) fx(x)da