河北工业大学 硕士学位论文 带分红的Sparre Andersen模型的期望折扣罚函数 姓名：李拴柱 申请学位级别：硕士 专业：应用数学 指导教师：刘国欣 20071201 ? ?Sparre Andersen? ? ? ?Sparre Andersen? ?Shuanming Li?Jos e Garrido20?b? ? ?Elang(n)?(?n? ?)? ? ?-?. ?.? ? ?Sparre Andersen?- ? ?. ?:Sparre Andersen?-? ? i ?Sparre Andersen? THE PENALTY FUNCTION OF SPARRE ANDERSEN MODEL WITH A THRESHOLD DIVIDEND STRATEGY ABSTRACT In this paper,we consider a Sparre Andersen risk model with a dividend barrier. The complete dividend notion proposed by Shuanming Li and Jos e Garrido20?with the assumption that the surplus is above b is extended to the situation of dividend by proportion. Then we discussed the bound- ary condition which the expected discounted penalty function satisfi es,when claim waiting times are generalized Elang(n) distributed(i.e.,convolution of n exponential distributions with possibly diff erent parameters).Moreover, we give the integro-diff erential equation which the expected discounted penalty function satisfi ed. This paper includes four chapters. In the fi rst chapter, the relative back- ground on this paper is given. The preparatory knowledge underlying this paper and the model we consider in this paper are introduced in the second one. The third chapter is the main body of this paper, in which the ex- pected discounted penalty function of a Sparre Andersen risk model with a threshold dividend barrier is considered,and shown that the expected dis- counted penalty function satisfi es a certain integro-diff erential equation. In the last chapter ,the main results of this paper are given. KEY WORDS:Sparre Andersen model, expected discounted penalty function, integro-diff erential equation, time of ruin, defi cit at ruin, surplus before ruin ii ? ? ?.? ?.? ?.? ?Lundberg8?Cram er17?.? ?Poission?(?)? ? ?,?4?24. ? ? 24 ?.? ? ?.? ? Gerber?Shiu12? ? ?.?Gerber ?Shiu14?Lin et al.21?Lin?Pavlova22?.?Gerber?Shiu13? ?Sparre Andersen?. De Finetti6?.? ?.? ?.?.?Gerber9 10 11? Albrecher?Kainhofer1?Gerber?Shiu14?Lin?Pavlova22?. ? ? ? ? ?Lin et al.21? ? ? ?.?Lin ?Pavlova22? ?. ?Lin?Pavlova22?Li?Garrido20? ?Sparre An- dersen?.? ?-?. ?b = ?Sparre Andersen? Gerber?Shiu13? = c1(c1?)? 1 ?Sparre Andersen? ? ?Sparre Andersen?Li?Garrido20? ?. ?Gerber-Shiu? (u) = EeT(U(T),|U(T)|)I(T 0, x2 0? ?x1? ?x2?I(E)?E? ?. ?(x1,x2)?.? ? ? ? ? ? ?. ?.? ?Sparre Andersen?-? ?. 2 ? ? 2-1? ?Tr? ?. ?Tr?f? Trf(x) = Z x er(yx)f(y)dy,r C.(1) ?C?Tr? TrTs= TsTr, (1I D)T1= 1, Tr( ) = (Tr) + Tr(0)(Tr). ?I, D?. ?h(s)?r1,r2,r3?k?hr1,r2,rk,s? ? h(s)=h(r1) + (s r1)hr1,s, hr1,s=hr1,r2 + (s r2)hr1,r2,s, hr1,r2,s=hr1,r2,r3 + (s r3)hr1,r2,r3,s, hr1,rk1,s=hr1,rk + (s rk)hr1,rk,s. ?h(s)?n?hr1,rn,s?h(s)?sn?.? ? hr1,r2,rk = k X j=1 h(rj) Qk i=1,i6=j(rj ri) . 3 ?Sparre Andersen? Gerber?Shiu151613?Vi? Erlang(n)?(?n?)?Sparre Andersen?-? Vi?k(t)? k(s) = Z 0 estk(t)dt = Qn i=1i Qn i=1(i+ s) ,n Z+,i 0.(2) ?Erlang(n)?1?2? n? k(t) = n X i=1 n Y j=1,j6=i j j i ieit.(3) ?n 2?Erlang(n)?k(t)? k(l)(0) = 0,l = 0,1,2, ,n 2. ?Vi?Erlang(n)?Gerber?Shiu15? ?(u)?-? n Y j=1 (1 + j )I c j D(u) = Z u 0 (u x)p(x)dx + (u).(4) ?(u) = R u (u,xu)p(x)dx?I?D?. Gerber?Shiu ?13?16?-?