A Note on Positive Supermartingales in Ruin Theory. Klaus D. Schmidt

Transcription

1 A Note o Positive Supermartigales i Rui Theory Klaus D. Schmidt

2 1 A ote o positive supermartigales i rui theory Klaus O. SeHMIOT Semiar für Statistik, Uiversität Maheim, A 5, Maheim, West Germay I this ote we give a elemetary proof of Kolmogorov's iequality for positive supermartigales. As a applicatio we obtai a Ludberg type iequality for a class of surplus processes with i.i.d. icremets for which a adjustmet coefficiet eed ot exist. Keywords: Adjustmet coefficiet, Kolmogorov's iequality, Ludberg's iequality, positive supermartigales, rui theory.

3 2-1. Itroductio All radom variables cosidered i this paper are defied o a fixed probability space (Q,F,p) For a subset A of Q, let X A deote its idicator fuctio Q : --? {0,1}. Let G be a itegrable radom variable. G has a adjustmet coefficiet if there exists some RE: (0,00) satisfyig E[e- RG ] = 1 ; ecessary ad sufficiet coditios for a adjustmet coefficiet of G to exist have bee give by Mammitzsch (1986). Cosider ow u E:(0,00), a sequece {G } of i.i.d. radom variables havig the same distributio as G, ad the surplus process {U }, give by U := u + L G k k=1 for all E:JN. If G has a adjustmet coefficiet, the the probability of rui satisfies Ludberg's iequality P ( if JNU < 0 ) -Ru < e Gerber (1973,1979) has show that Ludberg's iequality ca be obtaied from Kolmogorov's iequality for positive supermartigales. Ufortuately, however, the traditioal proofs of Kolmogorov's iequality ivolve a otrivial property of supermartigales, ad it appears that this fact makes the supermartigale approach appear much less attractive tha it iso

4 3 - I this ote we give a etirely elemetary proof of Kolmogorov's iequality for positive supermartigales. As a immediate applicatio, we obtai a Ludberg type iequality for a class of surplus processes for which a adjustmet coefficiet of G eed ot exist.

5 4-2. Kolmogorov.s iequality Let {X } be a sequece of itegrable radom variables. For each E ln, let F deote the a-algebra geera ted by {X 1'...,X }. A mappig -r : g ~ ln U {co} is a stoppig time if {-r=}e F holds for all E ln, ad it is bouded if SUPg -r(w) < co Let T deote the collectio of all bouded stoppig times for {F }. For -re T, defie X-r co The X is a itegrable radom variable satisfyig -r EX = -r co ote that all sums exted oly over a fiite umber of terms sice -r is bouded. The followig result is well-kow i the theory of asymptotic martigales; see e.g. Gut ad Schmidt (1983) ad the refereces give there: 2. 1 Lemma. The iequality P ( supln IX I ~ e: holds for all e: E (O,co) Proof. Let us assume that the X are all positive. For all E ln, defie sets := : = -1 {X ~ e:} Q{X k < e:} g 'U Bk k=1

6 - 5 - ad a stoppig time 1: E T by {1:=k} J Bk := L BUC lettig if ke{1,.,-1} if k = The we have E [ ] ri I: EX B < E X k=1-1: k hece <, by the mootoe covergece theorem, ad thus < which yields the assertio. o The sequece {X } is a supermartigale if E[x A X < E[xAXl holds for all E JN ad A E F 2.2. Lemma. If {X } is a supermartigale, the EX1: < EX 1 holds for all 1:E T Proof. Choose E E satisfyig 1: < The we have E[X{1:=k}Xkl + E[X{1:>k+1}Xk+11 < E[X{1:=k} X kl + E[X{1:>k+1}Xkl = E[X{1:>k}Xkl for all k E {1,,}, ad thus, by iductio, EX 1: = = as was to be show. o Tho followig result is Kolmogorov's iequality for positive supermartigales:

7 the Theorem. If {X } is a positive supermartigale, P ( supm X ~ e: < holds for all e: E (0,00) This follows from Lemmas 2.1 ad 2.2. We remark that Theorem 2.3 is usually deduced from the otrivial fact that a positive supermartigale {X } satisfies EX~ ~ EX 1 for arbitrary stoppig times ~ i see e.g. Neveu (1972).

8 Ludberg's iequality We ow retur to the surplus process {U } 3.1. Theorem. The iequality -pu P( ifmu < 0 < e holds for all p E (0,00) satisfyig E [e-pg] < 1. Proof. For all E m, defie The we have X := k=1 -pgk e for all E m ad A E F Therefore, {X } is a positive supermartigale, ad Theorem 2.3 yields = P( supm I: (-G k ) > u ) k=1 < < e -pu E [e -pg1 ] -pu e as was to be show. o Defie ow I (G) := { t E JR I E [etg] < 00 } ad J(G) := { t E JR I E [etg] < 1 } As a cosequece of Theorem 3.1 we obtai the followig result:

9 Corollary. If if I(G) < 0 < EG, the P( if m U ~ 0 ) ~ ifj(g) e tu < 1. Proof. The assumptio o G implies the existece of some te (-co,o) satisfyig E[e tg ] < 1 ; see Mammitzsch (1986). The assertio ow follows from Theorem 3.1. o 3.3. Corollary. the P ( if m U ~ 0 If G has a adjustmet coefficiet R, -RG < e This follows from Corollary 3.3. We remark that the hypothesis of Corollary 3.2 does ot imply that G has a adjustmet coefficiet; see Mammitzsch (1986).

10 9 Refereces Gerber, H.U. (1973). Martigales i risk theory. Mitt. Verei. Schweiz. Vers. Math. 2i, Gerber, H.U. (1979). A Itroductio to Mathematical Methods i Risk Theory. Irwi, Homewood (Illiois). Gut, A., ad Schmidt, K.D. (1983). Amarts ad Set Fuctio Processes. Lecture Notes i Mathematics, vol Spriger, Berli - Heidelberg - New York. Mammitzsch, V. (1986). A ote o the adjustmet coefficiet i rui theory. Isurace Math. Ecoom. ~, Neveu, J. (1972). Martigales a Temps Discret. Masso, Paris.