Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims

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1 ahmaca Ara, Vl 4, 4, 6, 6-63 Ru Prbably a Gralzd Rs Prcss udr Ras f Irs wh Hmgus arv Cha Clams Phug Duy Quag Dparm f ahmcs Frg Trad Uvrsy, 9- Chua Lag, Ha, V Nam Nguy Va Vu Tra Quc Tua Uvrsy Nguy Hg Nha Naal Ecmc Uvrsy Vu Ch Quag Ppl s Scury Acadmy ABSTRACT Th am f hs papr s gv rcursv ad gral quas fr ru prbabls f gralzd rs prcsss udr rs frc wh hmgus arv cha clams Gralzd Ludbrg quals fr ru prbabls f hs prcsss ar drvd by usg rcursv chqu W frs gv rcursv quas fr f m prbably ad a gral qua fr ulma ru prbably Thrm ad Thrm Usg hs quas, w ca drv prbably quals fr f m prbabls ad ulma ru prbably Thrm 3 ad Thrm 3 Th abv rsuls gv uppr buds fr f m prbably ad ulma ru prbably KEYWORDS: Igral qua, Rcursv qua, Ru prbably, Hmgus arv cha ahmacs Subc Classfcas: 6P5, 6G4, E5 Irduc Fr vr a cury, hr has b a mar rs acuaral scc Sc a larg pr f h surplus f surac busss frm vsm cm, acuars hav b sudyg ru prblms udr rs mdls wh ras f rs Fr xampl, Tugls ad Sud (995,997 sudd h ffcs f csa ra h ru prbably udr h cmpud Pss rs mdl Yag (999 sablshd bh xpal ad xpal uppr buds fr ru prbabls a rs mdl wh csa rs frc ad dpd prmums ad clams Ca (a, b vsgad h ru prbabls w rs mdls, wh dpd prmums ad clams ad usd a frs rdr aurgrssv prcss mdl h ras f rs Ca ad Dcs (4 bad Ludbrg quals fr ru prbabls w dscr- m rs prcss wh a arv cha rs mdl ad dpd prmums ad clams I hs papr, w sudy h mdls csdrd by Ca ad Dcs (4 h cas hmgus marv cha clams, dpd ras f rs ad dpd prmums Th ma dffrc bw h mdl ur papr ad h Ca ad

2 6 Phug Duy Quag, Nguy Va Vu, Nguy Hg Nha ad Vu Ch Quag Dcs (4 s ha clams ur mdl ar assumd fllw hmgus arv chas I hs papr, w sudy w syl f prmum cllcs O had f h prmums ar cllcd a h bgg f ach prd h h surplus prcss U wh al surplus u ca b wr as ( ( U U ( I X Y, ( whch ca b rarragd as ( ( U u ( I X Y ( I O h hr had, f h prmums ar cllcd a h d f ach prd, h h surplus prcss U wh al surplus u ca b wr as ( ( U (U X ( I Y, (3 whch s quval U u ( I X ( I Y ( I whr hrughu hs papr, w d x ad x f a b W assum ha: Assump b a ( ( U U u Assump X X s squc f dpd ad dcally dsrbud gav cuus radm varabls wh h sam dsrbu fuc F( x P( X x Assump 3 I I s squc f dpd ad dcally dsrbud gav cuus radm varabls wh h sam dsrbu fuc G( P( I Assump 4 valus a f s f - gav umbrs E y, b a Y Y s a hmgus arv cha such ha fr ay, Y as wh Y y ad p P Ym y Ym y,( m N ; y E whr p, p Assump 5 X,Y ad I ar assumd b dpd W df h f m ad ulma ru prbabls mdl ( wh assump assump 5, rspcvly, by ( u P (U U u,y y, (5 ( ( u lm ( u P (U U u,y y (6 Smlarly, w df h f m ad ulma ru prbabls mdl (3 wh assump assump 5, rspcvly, by

3 Ru Prbably a Gralzd Rs Prcss 63 ( u P (U U u,y y, (7 ( u lm ( u P (U U u,y y (8 ( I hs papr, w drv prbably quals fr (u ad (u Th papr s rgazd as fllws: sc, w frs gv rcursv quas fr ( ( (u ad (u ad a gral qua fr (u ad (u W h ( drv prbably quals fr (u ad apprach Fally, w cclud ur papr Sc 4 Igral Equa fr Ru Prbabls (u sc 3 by a ducv ( ( W frs gv a rcursv qua fr (u ad a gral qua fr (u Thrm If mdl ( sasfs h assumps 5 h fr =,, ( ( ( u p ( x h df( x dg( F( h dg( h ad ( ( ( u p ( x h df( x dg( F( h dg( h whr h y u( Prf Gv Y y E, frm (, w hav L, (9, ( U U ( I X Y u( I X y ( ( ( B U u,y y,a Y y, (, A X Y u( I A X Y u I Thus, w hav PU B A A P (U B A A, ( ad L X, Y, I P U B A A ( b dpd cps f X, Y, I wh X X,Y Y y,i I Thus, ( ad ( mply ha fr rspcvly

4 64 Phug Duy Quag, Nguy Va Vu, Nguy Hg Nha ad Vu Ch Quag ( ( P (U B A A P (U B A A P u( I X y ( I ( X Y ( I p B A A p ( ( P U ( I ( X Y ( I p U u( I X y,y y B A (3 p Tha, (5 mply ( u P (U U u,y y Thus, w hav ( u pp (U B A p P (U B A A P( A B A P (U B A A P( A B A (4 Frm (, w hav h ( P U B A A P( A B A df( x dg(, whr h y u( Frm (3, w hav ( ( P U B A A P( A B A x h, y df( x dg( h Thrfr, (4 s wr as h ( ( ( u p ( x h df( x dg( df( x dg( h ( p ( x h df( x dg( F( h dg( (5 h ( Thus, frm h dmad cvrgc hrm, h garal qua fr (u Thrm fllws mmdaly by lg (5 Ths cmpls h prf Smlarly, h fllwg rcursv quas fr (u ad a gral qua fr (u hld Thrm If mdl (3 sasfs h assumps 5 h fr =,, ( ( ( u p (( u x ( y df( x dg( F( h dg( h ad, (6

5 Ru Prbably a Gralzd Rs Prcss 65 ( ( ( u p (( u x ( y df( x dg( F( h dg( h y u( whr h, (7 3 Prbably Iqualy fr Ru Prbabls T sablsh prbably quals fr ru prbabls f mdl (, w frs prf h fllwg Lmma E X (, Lmma 3 L mdl ( sasfy assumps 5 ad Ay y E, f ad PY X Y y E(YY y E X (8 h, hr xss a uqu psv csa R sasfyg: Prf Df W hav Frm R ( Y X E Y y (9 ( Y f ( E X Y y ; (, Y X f ( E Y y E g ( h( Y s dscr radm varabls ad as valus E y y y,,, h Y y g( E Y y p has -h drvav fuc N N \ I add, x h( f ( x dx wh f ( x F '( x sasfyg : x h( f ( x dx f ( x dx x ad x f ( x dx x f ( x dx E X (,, (ay Ths mpls ha h ( has -h drvav fuc, wh, Thus, f( has -h drvav fuc, wh, ad ' ( ( ( Y f E Y X X Y y ( Y X f ( E ( Y X Y y '' Whch mpls ha

6 66 Phug Duy Quag, Nguy Va Vu, Nguy Hg Nha ad Vu Ch Quag ad f( s a cvx fuc wh f( ( ' By P( Y X Y y P( Y X Y y f ( E ( Y X Y y E Y Y y E( X (, w ca fd sm csa such ha Th, w ca g ha ( Y X ( Y X f( E Y y E Y y Y X Y y P( ( Y X Y y Imply lm f ( ( Frm (, ( ad ( suy ra hr xss a uqu psv csa R sasfyg (9 Ths cmpls h prf L: R ( Y X R m R : E Y y ( y E Us Lmma 3 ad Thrm, w w ba a prbably qualy fr ( ( uy, by a ducv apprach Thrm 3 If mdl ( sasfs assumps 5 E X (, ad (8 h fr ay u ad y E, whr ( Prf Frsly, w hav ( u, y E R u( I, (3 R Rx df( x f, (4 F ( R R x R R df( x df( x df( x f f f F( F( F( Fr ay, w hav R Rx df( x R Rx ( ( F df x F ( R Rx df( x (5 R R x R R X df( x E (6

7 Ru Prbably a Gralzd Rs Prcss 67 Th, fr u ad y E, ( ( ( ( u, y P( U U u, Y y Thus, frm (6 ad (7, w hav p F( h dg( (7 ( R y u( RX ( u, y p F( h dg( E p dg( RX Ry Ru ( E p dg( R X RY Ru ( I E E Y y E R ( Y X Ru ( I Ru ( I E Y y E E (8 Udr a ducv hyphss, w assum fr ay u ad y E, ( ( u, y E Frm (8 mpls (9 hlds wh Fr y E, x h ad I, w hav whr R u( I (9 ( ( ( ( ( R x u y I R x u y ( x h, y E f R R x df( x F ( (, R Y E X Y y ad R ( ( R R R x R x R x R x df( x df( x df( x df( x Ay : F F F F h W g f R ( ( ( ( R Rx df( x F ( R Rx df( x f F ( R x u( y R x u( y h ( R x u( y (x h (3 Thrfr, by Lmma 3, (9, (5 ad (3, w g ( ( ( u, y p ( x h, y df( x dg( F( h dg( h h R xu( y R y u( Rx p df( y dg( df( x dg( h h Ry Ru ( R x R x p dg( df( x df( x h

8 68 Phug Duy Quag, Nguy Va Vu, Nguy Hg Nha ad Vu Ch Quag Ry ( R u R ( x p dg df( x E R ( Y X Ru ( I Ru ( I E Y y E E Hc, fr ay,, (9 hld Thrfr, (3 fllws by lg (9 Ths cmpls h prf Ru ( I Rmar 3 L A( u, x E Frm I ad, w hav Ru Ru Ru A( u, x E Ru Thrfr, uppr bud fr ru prbably (3 s br ha Smlarly, w hav Lmma 3 Lmma 3 Assum ha mdl (3 sasfs assumps 5 ad E X (, Ay y E, f E Y X( I Y y ad PY X( I Y y (3 Th, hr xss a uqu psv csa R sasfyg: R Y X ( I E Y y L: R ( Y X ( I (3 R m R : E Y y ( y E Us Lmma 3 ad Thrm, w w ba a prbably qualy fr ( ( uy, by a ducv apprach Thrm 3 If mdl (3 sasfs assumps 5 E X (, ad (3 h Fr ay u ad y E, whr ( RY R( ux( I ( u, y E Y y E R Rx df( x f, F ( Prf Smlarly, w hav ad ay, w hav Th, fr u ad y (33 (34 R Rx F( df( x (35 E,

9 Ru Prbably a Gralzd Rs Prcss 69 ( ( ( ( u, y P( U U u, Y y p F( h dg( (36 Thus, frm (35 ad (36, w hav h y u( ( R Rx ( u, y p F( h dg( p df( x dg( ( h y u R ( ( x h y u x R p df( x dg( p df( x dg( y u( whr h Tha, fr h (37 y u( x( h R R y u( x( R y ( ux( df( x df( x df( x (38 Frm (37 ad (38, w hav ( R y ( ux( ( u, y p df( x dg( RY R( ux( I E Y y E (39 Udr a ducv hyphss, w assum fr ay u ad y E, ( RY R( ux( I ( uy, E Y y E Frm (39 mpls (4 hlds wh Fr y E, x h ad I, w hav RY R( ( ( ux y I RX( I E Y y E RY R( ( ux y R X( I E Y y E ( ( RY ( ( RX( I E Y y E ( RY R ( ux( y X( I (( u x( y, y E Y y E whr Ay h R u x y R u x y f R R x df( x F ( ( (, R Y X E I Y y ad R ( ( R R R x R x R x R x df( x df( x df( x df( x : F F F F ( ( ( R ( (4

10 63 Phug Duy Quag, Nguy Va Vu, Nguy Hg Nha ad Vu Ch Quag W g f R Rx df( x F ( R Rx df( x f F ( R ( ( u x y R ( u x( y h ( R ( ux( y (( u x( y, y (4 Thrfr, by Lmma 3, (6, (35 ad (4, w g ( ( ( u, y p (( u x( y, y df( x dg( F( h dg( h h y u( R ( ux( y R Rx p df( y dg( df( x dg( h h y u( x( R ( ux( y R p df( y dg( df( x dg( h Frm (38 ad (4, w hav (4 ( R y ( ux( ( u, y p df( x dg( RY R( ux( I E Y y E Hc, fr ay,, (4 hld Thrfr, (33 fllws by lg (4 Ths cmpls h prf RY R( ux( I Rmar 3 L B( u, y E Y y E Frm I, X ad, w hav RY Ru( I RX( I B( u, y E Y y E R Y R u R X ( I E E X x Ru RY X( I Ru Ru E X x Ru Thrfr, uppr bud fr ru prbably (33 s br ha 4 Cclus Our ma rsuls hs papr, Thrm ad Thrm gv rcursv qua fr ( ( uy, ad ( ( uy, ad gral qua fr ( ( uy, ad ( ( uy, ; Thrm 3 ad Thrm 3 gv prbably quals fr ( ( uy, ad ( ( uy, by a ducv apprach

11 Ru Prbably a Gralzd Rs Prcss 63 Acwldgms Th auhrs wuld l ha h Edr ad h rvwrs fr hr hlpful cmm a arlr vrs f h mauscrp whch has ld a mprvm f hs papr Rfrcs []Albrchr, H (998 Dpd rss ad ru prbabls surac IIASA Irm Rpr, IR-98-7 [] Asmuss, S ( Ru prbabls, Wrld Scfc, Sgapr [3] Ca, J ( Dscr m rs mdls udr ras f rs Prbably h Egrg ad Ifrmaal Sccs, 6, [4] Ca, J ( Ru prbabls wh dpd ras f rs, Jural f Appld Prbably, 39, 3-33 [5] Ca, J ad Dcs, D C (4 Ru Prbabls wh a arv cha rs mdl Isurac: ahmacs ad Ecmcs, 35, [6] Nyrh, H (998 Rugh dscrps f ru fr a gral class f surplus prcsss Adv Appl Prb, 3, 8-6 [7] Prmslw, S D (99 Th prbably f ru a prcss wh dpd crms Isurac: ahmacs ad Ecmcs,, 99-7 [8] Rls, T, Schmdl, H, Schmd, V ad Tugls, J L(999 Schasc Prcsss fr Isuarac ad Fac Jh Wly, Chchsr [9] Shad, ad Shahumar, J (994, Schasc Ordrs ad hr Applcas Acadmc Prss, Sa Dg [] Sud, B ad Tugls, J L (995 Ru smas udr rs frc, Isurac: ahmacs ad Ecmcs, 6, 7- [] Sud, B ad Tugls, J L (997 Th adusm fuc ru smas udr rs frc Isurac: ahmacs ad Ecmcs, 9, [] Xu, L ad Wag, R (6 Uppr buds fr ru prbabls a aurgrssv rs mdl wh arv cha rs ra, Jural f Idusral ad aagm pmza, Vl N,65-75 [3] Yag, H (999 N xpal buds fr ru prbably wh rs ffc cludd, Scadava Acuaral Jural,, [4] Wllms, G E, Ca, J ad L, XS ( Ludbrg Apprxmas fr Cmpud Dsrbu wh Isurac Applcas Sprgr Vrlag, Nw Yr Rcvd: Ju, 4

Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims and Homogenous Markov Chain Interests

Ruin Probability in a Generalized Risk Process under Rates of Interest with Homogenous Markov Chain Claims and Homogenous Markov Chain Interests Appld Mathmatc 3, 3(5: 85-97 DOI:.593/.am.335.5 u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt Quag Phug Duy Dpatmt f Mathmatc, Fg Tad Uvty, Ha, Vt Nam Abtact Th am f th pap