Renewal Risk Model, Gerber-shiu Penalty Function And The Bankruptcy Probability

This dissertation is devoted to dealing with ruin theory of renewal risk theory. The Gerber-Shiu discounted penalty function and ruin probability of renewal risk theory are gotten.At last the bound of ruin probability is made out.Sparre Andersen risk model is put forward based on the classic risk model by E.Sparre Andersen(E. Sparre Andersen [1]) in 1957,many actuarial literatures research it,such as Dickson [2], Gerber, Shiu [3].The Gerber-Shiu discounted penalty function is brought forward by Gerber, Shiu(Gerber, Shiu [4]) in 1998,and Gerber, Shiu [7] put the two results together. Cheng, Tang [5], Gerber, Shiu [6] [7], Li [8], Lin [9] research the Erlang(2) risk process.and Gerber, Shiu [3] not only extend a result in Li [8], but as a consequence,a closed-form expression of Erlang(n) (?)sk process is obtained for the discounted joint probability density of the defic(?) at ruin and the surplus just before ruin.Willmot, Dickson [10] research the discounted penalty function for stationary renewal risk model, and get an expression.Willmot [11] get a relation of the discounted penalty function between the ordinary renewal risk model and delayed or stationary renewal risk model. And Dickson, Drekic [12] get a distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin . Enlighted by these(?)iteratures, in this dissertation I'll consider the discounted penalty function and ruin probability of renewal risk model. The bound of ruin probability is researched at last.The thesis is divided into three sections according to contents.In chapter 1, Preface, we introduce the importance of the main contents of this paper.In chapter 2, the Gerber-shiu discounted penalty function and ruin probability are studied. We state the main results as follows:Theorem 2.2.1 For the delayed renewal risk process, if 8 = 0, the Gerber-Shiu discounted penalty function isJo Jo Jo"^' (1 + 9)8(0) Jo Jo Jo Jmax{0,u^yi-x) P(x - U + Vl + z)p(x + y)w(x,y)d8(z)dxdydHe(yl) -where,00 roo rt /(0)a;_^ + 2)roo poo rt f/r,COW Jo JO Jmax{0,t-x\ P[X -t +a:. v)dS(z)dxdyand7i =rt roo roo rt-yi Jq Jo Jo Jmax{O,tC^i + 0)5(0) Jo Jo JO Jmax{O,t-yi-x}p(x + y)w(x,y)d8(z)dxdydHc(yy),here, y[,yi is the amount of the first claim for the ordinary and delayed renewal risk model, respectively.Theorem 2.2.2 For stationary renewal risk process, if 8 = 0, the Gerber-Shiu discounted penalty function is1 pu /*OO poo pu — y\ r/' r\ ?)_! ■ \me>o{u) = ^{l +9}8(0) Jo Jo Jo Jna*{o.u-yl-x} T(x-u + y,+z)p(x + y)w(x, y)d5(z)dxdydHe(yi) + qo(u), where1 roo roo(l + 9)E(Y)]u Jt W yVl VxCorollary 2.3.1 For the delayed renewal risk process, the ruin probability ist* ('C\\ TP(\/\ /*^ /*'"** pOO pit — 7/1 C tr\ I . \/ \ 1 \ ) \ ) I I I I JV^'i^' — ** ^T % f" *J\ )(1 + 9)5(0) Jo Jo Jo Jmax{0.u-?x-x} ^P{x U + l)\ + z)roop(x + y)d8(z)dxdydile(yi) + / ea<"-l)/c0(t)dt,Juwherea f°° f°° fl^x f(O,x — t^ ' = FTTTx / / /C0{0) J0 J0 Jmax{O,t-x}+ —J—-7,,and/ ak\(O)E(V) fl f°° Ip(x + y)dS(z)dxdydHe(yi),Corollary 2.3.2 For the stationary renewal risk process, the ruin probability is^)^(O) Jo Jo Jo Jmax{Q,u-yi-x} P{x -p(x + y)dS(z)dxdydHe(yi)(1+9)6(0) Jo Jo Jo Jmax{Qtu-yi-x} P(x - U + yi + z)He\u)1 + 9In chapter 3, the stop-loss premiums by the property of random walk and compound geometric distribution n renewal risk model are considered, and the bounds of ruin probability under the condition of E(X2) < oo are got at last.Theorem 3.3.1 For any x > 0, the bounds of stop-loss premiums are(*F(x) + E(S)F(x)) , . 0(7rF(x) + E(S)F(x) + E(S2)F(x)/(2x))< Ks(x) < ---------------1 + E(S)F(x)/x Theorem 3.4.1 For any x;> \ the bounds of ruin probability are_<sub><sub><sub><f>F^x lf (F(x) + E(S)F(x)/x)1 ^ + cjyp^x) - %AX* 1 + E(S)F(x)/x...