Ruin Probability-Based Initial Capital of the Discrete-Time Surplus Process

Transcription

1 Ru Probablty-Based Ital Captal of the Dsrete-Tme Surplus Proess by Parote Sattayatham, Kat Sagaroo, ad Wathar Klogdee AbSTRACT Ths paper studes a surae model uder the regulato that the surae ompay has to reserve suffet tal aptal to esure that ru probablty does ot exeed the gve quatty a We prove the exstee of the mmum tal aptal To llustrate our results, we gve a example approxmatg the mmum tal aptal for expoetal lams KEYwORdS Ital aptal, surae, lam proess, ru probablty 74 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

2 Ru Probablty-Based Ital Captal of the Dsrete-Tme Surplus Proess Itroduto I reet years, rs models have attrated muh atteto the surae busess, oeto wth the possble solvey ad the aptal reserves of surae ompaes The ma terest from the pot of vew of a surae ompay s lam arrval ad lam sze, whh affet the aptal of the ompay I ths paper, we assume that all the proesses are defed a probablty spae (W, F, Pr Clams happe at the tmes T, satsfyg T T T The th lam arrvg at tme T auses the lam sze X ow let the ostat represet the premum rate for oe ut tme; the radom varable T desrbes the flow of aptal to the busess by tme T, ad X desrbes the outflow of aptal due to paymets for lams ourrg [, T ] Therefore, the quatty U u, U u T X ( s the surer s balae (or surplus at tme T,,, 3,, wth the ostat U u as the tal aptal We osder the dsrete-tme surplus proess ( the stuato that the possble solvey (ru a our oly at lam arrval tmes T,,, 3, Thus, the model beomes U u, U u X ( for all,, 3, The geeral approah for studyg ru probablty the dsrete-tme surplus proess s through the so-alled Gerber-Shu dsouted pealty futo; for example, Pavlova ad Wllmot (4, Dso (5, ad L (5a, 5b All of these artles study (or alulate the ru probablty as a futo the tal aptal u I ths paper, we shall wor the opposte dreto, e, we study the tal aptal for the dsrete-tme surplus proess as a futo of ru probabltes Ma results Let {U, } be a surplus proess (as Seto that s drve by the lam proess {X, } We osder the fte-tme ru probabltes of the dsrete-tme surplus proess Equato ( wth the depedet ad detally dstrbuted (d lam proess {X, } We let F X (x be the dstrbuto futo of X, e, FX ( x Pr { X x} ( The premum rate s alulated by the expeted value prple, e, ( θ E[ X ] ( where q > whh s the safety loadg of surer Let u be a tal aptal For eah,, 3,, we let ϕ : Pr{ U, U, U3,, U U u} (3 deote the survval probablty at the tmes Thus, the ru probablty at oe of the tme,, 3,, s deoted by Φ ϕ (4 Defto Let {U, } be a surplus proess whh s drve by the lam proess {X, } ad > be a premum rate Gve a (, ad {,, 3, } Let a tal aptal u, f F (u a the u s alled a aeptable tal aptal orrespodg to (a,,, {X, } Partularly, f u* m { u: Φ (5 u exsts, u* s alled the mmum tal aptal orrespodg to (a,,, {X, } ad s wrtte as u*: MIC ( α,,, { X, } (6 VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 75

3 Varae Advag the See of Rs Ru ad survval probablty We defe the total lam proess by S : X X X (7 for all,, 3, Lemma Let {,, 3, } ad > be gve If {X, } s a d lam proess, the ϕ (u s reasg ad rght otuous ad F (u s dereasg ad rght otuous u Proof The survval probablty at the tme as metoed (3 a be expressed as follows where ϕ Pr { S u, S u,, S u } Pr { S u } I E S u { } E I { S u } I A, x A ( x, x A ( (8 for all A Se I{ S S u }( ω I ( ω, ( u forall ω Ω, ϕ E I( S, ( u ( 9 For eah a R ad u, we obta, u a I(, ] ( a u, u< a, the (-, ] (a - u s reasg ad rght otuous u Ths mples that P (-, ] (a - u s also reasg ad rght otuous u, moreover, ths boudg futo s detally equal to, where a R,,, 3,, Therefore, by the mootoe overgee theorem, we have lm ϕ lm ( v E I( S v, ( v u v u E lm I( (, S v v u E I( (, S u ( u ϕ ( Therefore, ϕ (u s reasg ad rght otuous Moreover, we a olude that F (u - ϕ (u s dereasg ad also rght otuous Theorem Let {,, 3, } ad > be gve If {X, } s a d lam proess, the lm ϕ ad lm Φ ( u u Proof Frst, we wll show the followg propertes { X u } { S u } Let w ( {X u } be gve For eah {,, 3,, }, we have X (w u ad S( ω X ( ω u u (3 That s, w {S u } Therefore, ( follows ext, se the proess {X, } s d, the Pr { X u } { X u } ( Fu ( Pr (4 By Equato (8, we have ϕ ( u Pr { S u } (5 By (, (4 ad (5, we obta ( Fu ( ϕ( u (6 76 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

4 Ru Probablty-Based Ital Captal of the Dsrete-Tme Surplus Proess Se (F(u as u, the ϕ (u as u Thus, we olude that ϕ (u, ad F (u - ϕ (u as u Ths s the proof Corollary 3 Let a (,, {,, 3, } ad > be gve If {X, } s a d lam proess, the there exsts ũ suh that, for all u ũ, u s a aeptable tal aptal orrespodg to (a,,, {X, } Proof We osder by ase Case : F ( a Se F (u s dereasg, the F (u F ( for all u Case : F ( > a By Theorem, we have F (u as u Thus, there exsts ũ > suh that F (ũ < a Se F (u s dereasg, we olude that F (u F (ũ < a for all u ũ Reursve formula of ru probabltes From Theorem ad Corollary 3, we ow that the small ru probablty a be otrolled by a large tal aptal I ths part, we shall desrbe the upper boud of ru probablty wth egatve expoetal I order to prove the followg lemma, we shall use a equvalet defto of the ru probablty whh s gve as follows: Φ Pr max X u > (7 Theorem 4 Let {,, 3, }, > ad u be gve If {X, } s a d lam proess, the the ru probablty at oe of the tmes,, 3,, satsfes the followg equato Φ Φ Φ ( u x dfx ( x where F (u u (8 Proof We wll prove (8 by duto We start wth Se F (u for all u, the u Φ ( u x df ( x (9 X Ths proves (8 for ow assume that (8 holds for The Φ X u > Pr max Pr( X > u Pr max X X ux u >, u Φ x X u > Pr max df ( x X u Φ X ( Pr max > u x df ( x X u Φ X ( Pr max > u x df ( x X u Φ X Pr max > u x df ( x u X Φ Φ ( u x df ( x whh proves (8 for ad oludes the proof Corollary 5 Let {,, 3, }, > ad u be gve If {X, } s a d lam proess, the the ru probablty at oe of the tmes,, 3,, satsfes the followg equato: Φ, Φ Pr ( X u, Φ Φ Θ where X ( u u x Θ ( u Φ ( u x v df ( X v df ( X x for all, 3, 4, VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 77

5 Varae Advag the See of Rs Proof Let, by Theorem 4, we obta u Φ Φ Φ ( u x df ( x X u Φ Φ ( u x u x Φ u x v dfx ( v ( dfx ( x u Φ Φ ( u x df ( x u u x Φ ( u x v dfx ( v dfx ( x u u x Φ Φ ( u x v dfx ( v df ( x X ( ( Ths ompletes the proof Corollary 6 Let {,, 3, } ad u Assume that {X, } s a sequee of expoetal dstrbuto wth testy l >, e, X has the probablty desty futo f(x le -lx The obtaed ru probablty s the followg reursve form Φ, Φ Φ ( u λ ( u (! X e λ ( u ( for all,, 3,, where the tal aptal u ad premum rate > E[X ] /l Proof We wll prove ( by duto We start wth, F (u - Pr {X u } - ( - e -l(u e -l(u Ths proves ( for ext we assume that ( holds for From Theorem, we have u Φ Φ Φ ( u x df ( x X u Φ Φ ( u x ( u x λ ( u ( x (! λ ( u ( x e dfx ( x ad Φ λ ( u ( x e df ( x Θ u x u x ( λ ( ( (! u u X ( u x λ ( u ( x (! λ ( u ( x λx e λe dx e λ λ ( ( (! u u ( u ( x ( u ( x ( dx e λ λ ( ( (! u u (( u ( x ( ( u ( x dx ( u λ ( u (! e ( λ ( u ( whh proves ( for ad ompletes the proof 3 Exstee of mmum tal aptal A quatty a, whh was dsussed the prevous seto, a be desrbed as the most aeptable probablty that the surae ompay wll beome solvet As a result of Corollary 3, we obta that {u : F (u a} s a o-empty set Ths meas that we a always hoose a tal aptal whh maes the value of ru probablty ot exeed a Se the set {u : F (u a} s a fte set, the there are may aeptable tal aptal orrespodg to (a,,, {X, } I ths seto, we wll prove the exstee of MIC ( α,,, { X, } m { u : Φ u (3 Lemma Let a, b ad a be real umbers suh that a b If f s dereasg ad rght otuous o [a, b] ad a [ f(b, f(a], the there exsts d [a, b] suh that d m { x [ ab, ]: f( x (4, 78 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

6 Ru Probablty-Based Ital Captal of the Dsrete-Tme Surplus Proess Proof Let S: { x [ ab, ]: f( x Se a [ f(b, f(a], e, f(b a f(a, the b S That s, S s a o-empty set Se S s a subset of losed ad bouded terval [a, b], the there exsts d [a, b] suh that d f S ext, we osder by ase Case : d b We ow that b S, thus b m S Case : a d < b Se d f S, the there exsts d S suh that d d < d for all For eah > /(b - d, we have b d b d d < d < d < b Ths meas that d / (d, b [a, b] for all > /(b - d Se f s dereasg ad d S, we get f( d f( d α, e, d / S for all > /(b - d Se f s rght otuous at d, we have f( d lm f( d α Therefore, d S, e, d m S Ths ompletes the proof Theorem 7 Let a (,, {,, 3, }, ad > The there exst u* suh that Case : F ( > a, by Corollary 3, there exsts ũ > suh that F (ũ < a, e, a [F (ũ, F (] Se F (u s dereasg ad rght otuous, by Lemma there exsts u* [, ũ ] suh that u m { u: Φ α } m { u: Φ That s, u [, u ] u [, u MIC ( α,,, { X, } ext, we wll approxmate the mmum tal aptal MIC(a,,, {X, } by applyg the bseto tehque for the dereasg ad rght otuous futo Theorem 8 Let a (,, {,, 3, }, ad v, u suh that v < u Let {u } ad {v } be a real sequee defed by u v v v ad u f Φ v u v ad u u, f Φ, u v ( α u v ( >α for all,, 3, If F (u a < F (v, the ad lmu MIC ( α,,, { X, } (5 u MIC ( α,,, { X, } u v u MIC ( α,,, { X, } (6 Proof We osder by ase Case : F ( a, we have for all,, 3, MIC ( α,,,{ X, } Proof Obvously, {u } s dereasg ad {v } s reasg, moreover, v u for all,, VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 79

7 Varae Advag the See of Rs Table Mmum tal aptal MIC (a,,, {X, > } the dsrete-tme surplus proess wth expoetal lams (l a a a 3 q q 5 q q 5 q q , , , , Thus, {u } ad {v } are overget Se u v ( u v as, the there exsts u* [v, u ] suh that lm u lm v : u (7 Se F (u s dereasg ad F (v > a for all,, 3,, the F (u > a for all u < u* Se F (u s rght otuous ad F (u a for all,, 3,, the Therefore, Φ ( u* lm Φ ( u α (8 u* MIC ( α,,, { X, } (9 For eah,,,, we have v u* u Ths mples that u u* u u* u* v u v u v (3 Ths ompletes the proof 4 umeral results We provde umeral llustratos of the ma results We approxmate the mmum tal aptal of the dsrete-tme surplus proess ( by usg Theorem 8 the ase of {X, } a sequee of d expoetal dstrbuto wth testy l, by hoosg model parameter ombatos q ad 5, e, ad 5, respetvely; ad a,, ad 3 Table shows the approxmato of MIC(a,,, {X, } wth u 5 as metoed Theorem 8, hoosg v ad u, ad F (u s omputed from the reursve form ( Fgure shows the approxmato of MIC(a,,, {X, } for the varous values of a wth u 5 as metoed Theorem 8 Here we hoose v, u, ad parameter ombatos q, q 5, e, ad 5, respetvely 8 CASUALTY ACTUARIAL SOCIETY VOLUME 7/ISSUE

8 Ru Probablty-Based Ital Captal of the Dsrete-Tme Surplus Proess Fgure Mmum tal aptal MIC(a,,, {X, > } the dsrete-tme surplus proess wth expoetal lams (l, Mmum Ital Captal u u θ θ 5 Referees Dso, D C M, Isurae Rs ad Ru ew Yor: Cambrdge Uversty Press, 5 L, S, O a Class of Dsrete Tme Reewal Rs Models, Sadava Atuaral Joural 5a(4, pp 4 6 L, S, Dstrbutos of the Surplus before Ru, the Deft at Ru ad the Clam Causg Ru a Class of Dsrete Tme Rs Model, Sadava Atuaral Joural 5b (, pp 7 84 Pavlova, K P, ad G E Wllmot, The Dsrete Statoary Reewal Rs Model ad the Gerber-Shu Dsouted Pealty Futo, Isurae: Mathemats ad Eooms 35(: 4, pp α VOLUME 7/ISSUE CASUALTY ACTUARIAL SOCIETY 8