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

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1 Appld Mathmatc 3, 3(5: DOI:.593/.am 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 t gv cuv ad tgal quat f u pbablt f gald pc ud aumpt that bth quc f clam ad at f tt a hmgu Mav cha. Gald Ludbg qualt f u pbablt f th pc a dvd by ug cuv tchqu. Ftly, w gv a cuv quat f ft tm pbablty ad ultmat u pbablty. By ug th quat, w ca dv pbablty qualt f ft tm pbablty ad ultmat u pbablty. Th abv ult gv upp bud f ft tm pbablty ad ultmat u pbablty. A umcal xampl gv t llutat ult. Kywd Itgal quat, cuv quat, u pbablty, Hmgu Mav cha. Itduct F v a ctuy, th ha b a ma tt actuaal cc. Sc a lag pt f th uplu f uac bu fm vtmt cm, actua hav b tudyg u pblm ud mdl wth at f tt. F xampl, Tugl ad Sudt[],[] tudd th ffct f ctat at th u pbablty ud th cmpud P mdl. ag[3] tablhd bth xptal ad xptal upp bud f u pbablt a mdl wth ctat tt fc ad dpdt pmum ad clam. Ca[3],[4] vtgatd th u pbablt tw mdl, wth dpdt pmum ad clam ad ud a ft d autgv pc t mdl th at f tt. Ca ad Dc[5] btad Ludbg qualt f u pbablt tw dct- tm pc wth a Mav cha tt mdl ad dpdt pmum ad clam. I th pap, w tudy th mdl cdd by Ca ad Dc[5] t th ca hmgu Mav cha clam ad hmgu Mav cha at f tt ad dpdt pmum. Th ma dffc btw th mdl u pap ad th Ca ad Dc[5] that clam ad at f tt u mdl a aumd t fllw hmgu Mav cha. b X X W lt { } b pmum, { } Cpdg auth: quagmathftu@yah.cm (Quag Phug Duy Publhd l at Cpyght 3 Sctfc & Acadmc Publhg. All ght vd clam, I { I } b tt ad thy df pbablty pac ( Ω, AP,. T tablh pbablty qualt f u pbablt f th mdl, w tudy tw tyl f pmum cllct. O th had f th pmum a cllctd at th bgg f ach pd th th uplu pc { U } wth tal uplu u ca b wtt a U ( U I X (. whch ca b aagd a p p (,(. U u. ( I X ( I O th th had, f th pmum a cllctd at th d f ach pd, th th uplu pc { U } tal uplu u ca b wtt a whch quvalt t wth U ( U X ( I, (.3 p p,(.4 U u. ( I X ( I ( I thughut th pap, w dt xt ad t a b xt f a > b. t a W aum that: b

2 86 Quag Phug Duy: u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt Aumpt.. U U u > Aumpt.. X { X } a quc f dpdt ad dtcally dtbutd gatv ctuu adm vaabl wth th am dtbutv fuct F( x P( Ω: X( x. Aumpt.3. { } Mav cha, a hmgu ta valu a ft t f - gatv umb E { y, y,..., y } M wth y ad p P Ω : ( y ( y,( m N; y, y E m m M p, p. Aumpt.4. I { I } hmgu Mav cha, I ta valu a ft t f - gatv EI,,..., N wth I ad umb { } q P Ω : I ( X (,( m N;, E m m I q, q. N Aumpt.5. X, ad I a aumd t b dpdt. W df th ft tm ad ultmat u pbablt f mdl (. wth aumpt. t aumpt.5, pctvly, by ( ( u, y, P : U ( U ( u, ( y, I( Ω < (.5 ( ( uy,, lm ( uy,, P : U ( U ( u, ( y, I( Ω < (.6 Smlaly, w df th ft tm ad ultmat u pbablt f mdl (.3 wth aumpt. t aumpt.5, pctvly, by ( uy,, P : ( U ( U ( ux, ( y, I( Ω < (.7 ( uy,, lm ( uy,, P : ( U ( U ( u, ( y, I( Ω < (.8 I th pap, w dv pbablty qualt f ( uy,, ad ( uy,,. Th pap gad a fllw; Sct, w gv cuv ad tgal quat f ( uy,, ad ( uy,,. I Sct 3 w dv pbablty qualt f ( uy,, ad ( uy,, by a ductv appach. A umcal xampl gv t llutat th ult Sct 4. Fally, w cclud u pap Sct 5.. Itgal Equat f u Pbablt W ft gv cuv quat f ( uy,, ad a tgal quat f ( uy,,. Thm.. If mdl (. atf th aumpt. t.5 th f,, ad (,, ( (,, ( ( (. y u( u y u x y y df x F y u pq (,, ( (,, ( ( u y pq u x y y df x Fy u (. y u(

3 Appld Mathmatc 3, 3(5: Pf. Gv ( y E, I( E ( Ω. I Lt { A Ω : U ( u, ( y, ( y, I(, I( } A { Ω : X( < y u( }, A { Ω: X( y u( }. Fm (., w hav I addt, Lt { X },{ },{ I } U ( u( X ( y ad ( P Ω : U ( < A A P : ( U ( A A Ω < (.3 ( X ( X (, ( ( y, I ( I (. Thu, (.4 ad (. mply that P : ( U ( A A Ω < O th th had, (.5 mpl Thu, w hav Fm (.3, w hav P Ω : U ( < A A. (.4 b dpdt cp f { X }, { }, { } P : ( U ( A A Ω < I pctvly wth P Ω : u( X( y ( Im( ( Xm( m( ( Ip( m m p m ( U ( u( X( y, ( y, I( A P Ω : U ( ( Im( ( Xm( m( ( Ip( < m m p m U ( u( X ( y, ( y, I ( A (.5 ( ( u, y, P Ω : ( U ( < U ( u, ( y, I( ( uy,, pq P Ω : ( U ( < A pq P U A A PA Ω : ( ( <. ( P Ω : ( U ( < A A. PA ( (.6

4 88 Quag Phug Duy: u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt y u( (. P Ω : U ( < A A. P( A df( x Fm (.5, w hav P Ω : ( U ( < A A. P( A ( u( x y, y, df( x y u( Thf, (.6 wtt a y u( ( u, y, p q df( x ( u( x y, y, df( x y u( pq ( u( x y, y, df( x F y u( y u( (.7 Thu, th tgaal quat f ( uy,, Thm. fllw mmdatly fm th dmatd cvgc thm by lttg (.7. Th cmplt th pf Smlaly, th fllwg cuv quat f ( uy,, ad tgal quat f ( uy,, a hld. Thm.. If mdl (.3 atf aumpt. t.5 th, f,, y ( u ( u, y, pq ( u x( y, y, df( x F (.8 y ( u ad y u( ( u, y, p q ( u x( y, y, df( x F y ( u (.9 Nxt, w tablh pbablty qualt f u pbablt f mdl (. ad mdl ( Pbablty Iqualt f u Pbablt T tablh pbablty qualt f u pbablt f mdl (., w ft pf th fllwg Lmma. Lmma 3.. Lt mdl (. atfy aumpt. t.5 ad E( X If, ay y E, ( E Ω : ( y < EX ( ad th th xt a uqu ptv ctat Pf. Df ( < (,. P Ω:( X ( > ( y > (3. atfyg: ( ( X E Ω : ( y (3.

5 Appld Mathmatc 3, 3(5: t { ( X } f ( t E Ω : ( y ; t (, W hav t tx { } ( f( t E Ω : ( y. E g(. t ht ( Fm dct adm vaabl ad t ta valu E { y, y,..., y } t { } g ( t E : ( y M p ty Ω ha -th dvatv fuct ( tx I addt, ( tx h( t E f ( x dx wth f( x F' ( x tx h( t f ( x dx f ( x dx atfyg : th, (ay N N \{ } M. tx ad x f ( x dx x f ( x dx E ( X < (,. Th mpl that ht ( ha -th dvatv fuct (, wth, fuct (, wth, ad t ( X { } t ( X { } f ' ( t E ( X Ω : ( y f '' ( t E ( X Ω : ( y. Th mpl that ad ( { } By P( :( X( ( y P( Ω:( X( > δ > ( y >. Thu, f ( t ha -th dvatv f t a cvx fuct wth f ( (3.3 ' f ( E ( X Ω : ( y E ( Ω : ( y E ( X < (3.4 Ω > >, w ca fd m ctat δ > uch that Th, w ca gt that t { } ( X f ( t E Ω : ( y Imply t ( X { } { Ω:( X( > δ ( y } E Ω : ( y. { δ } t δ. P Ω:( X ( > ( y. lm f( t. (3.5 t Fm (3.3, (3.4 ad (3.5 th xt a uqu ptv ctat Th cmplt th pf. atfyg (3.. ( X Lt: m : E : ( y > Ω ( y E Ug Lmma 3. ad Thm., w bta a pbablty qualty f ( uy,, by a ductv appach. Thm 3.. If mdl (. atf aumpt. t.5, E( X f ay u >, y E ad EI < (, ad (3. th

6 9 Quag Phug Duy: u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt > u ( I ( uy,,. E Ω : I(, (3.6 x df ( x f,. F( Pf. Ftly, w hav ( x df x df x ( ( f f. > F( > F( F ay >, w hav F( x. df( x x. x df( x F(. x. (. df x. E X Th, f ay u >, y E ad EI, w ca wt ( df( x. (3.8 u y P Ω U > U u y I pq Fy u (3.9 (,, ( : ( (, (, ( ( Thu, cmbg (3.8 ad (3.9, w hav M N ( uy,, pq F( y u( M N pq y u( X E ( X u ( I E : ( y. E : I( Ω Ω u ( I E Ω : I(. (3. Applyg a ductv hypth, w aum f ay u >, y E ad EI, ( (,, u I uy E : I( Ω. (3. Th (3. mpl that (3. hld wth. F y E, E, u( x y > ad I ( ( Ω, w hav I ( u( x y, y, E Ω I ( ( ( : ( u x y I u x y

7 Appld Mathmatc 3, 3(5: f > ad F ay > : x df( x >. F( x ( x df( x df( x th F( F( f f > >, Ω : ( ( X E y x x ( x x df( x df( x F( F( df( x df( x. That F( F( u ( x y u ( x y > th ( (,, Thf, by Lmma 3., (., (3.7 ad (3., w gt u( xy u x y y. (3. ( u, y, p q F( y u( ( u( x y, y, df( x y u( Thu ( y u y u( u( x y x pq df( x df( x y u( y u( x p q df( x ( X u( I E Ω : ( y. E Ω : I( u( I Ω : ( E I ( (,, u I uy E Ω : I( Cqutly, f ay,,... (3. hld. Thf, (3.6 fllw by lttg (3.. Th cmplt th pf ma 3.. Lt w hav u ( I Au (, y,. E : I( Ω. Fm I ( ( Ω ad Au (, y,. E : I ( u u u Ω,

8 9 Quag Phug Duy: u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt Thf, upp bud f u pbablty (3.6 btt tha Smla t Lmma 3., w hav Lmma 3.. u. Lmma 3.. Aum that mdl (.3 atf aumpt. t.5 ad E( X ad EI, y ad E( X ( I Ω : ( y, I ( < ( < (,. If ay y E P X( I > Ω : ( y, I( >, (3.3 th th xt a uqu ptv ctat > atfyg: Lt X ( ( I E Ω : ( y, I ( X( I { ( } m > : E Ω : ( y, I ( ( y E, E I Nxt, w u Lmma 3. ad Thm. t gv a pbablty qualty f ( uy,, by a ductv appach. Thm 3.. If mdl (.3 atf aumpt. t.5, E( X E ad EI Pf. < (, ad (3.3 th, f ay ( u X( I ( uy,, E : ( y Ω E Ω : I( Smlaly wth Thm 3., w hav Th, f ay u >, y E ad EI x df ( x f >,. F( ad ay > y u( ( uy,, pq F( y u( x (3.4 F(.. df( x (3.5. X E. (3.6 y u( x p q df( x

9 Appld Mathmatc 3, 3(5: Hc y u( pq y u( x( y u( y ( u x( pq y ( u x( pq df( x df( x df( x ( u X( I E Ω : ( y. E Ω : I( ( u X( I ( uy,, E Ω : ( y. E Ω : I( Ud a ductv hypth, w aum that ( u X( I (3.7 ( uy,, E Ω : ( y. E Ω : I(. (3.8 Th, (3.7 mpl that (3.8 hld wth. y u( F y E, EI, x > ad I (,( Ω, w hav (( u x( y, y, ( u x( y X ( I E : ( y. E Ω Ω : I( ( u x( ( y I X( I E : ( y. E Ω Ω : I( Ω Ω ( ( X( I u x y E : ( y. E : I(. ( (. u x y ( y E, E, ( u x( y >, f > ad >. x df ( x F(, I Ω : (, ( ( X( I E y I

10 94 Quag Phug Duy: u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt F ay > : th > ( x ( x x x x df( x f F( W gt df( x df( x df( x df( x F( F( F( F( f > df( x x F( ( u x ( y ( u x ( y > th ( Thf, by Lmma 3., (.8, (3.5 ad (3.9, w gt ( u x( y ( u x( y, y,. (3.9 y u( ( u, y, p q F (( u x( y, y, df( x y u( Thu y u( y u( x ( u x( y pq df( x df( x y u( y u( ( ( y u x ( u x( y pq ( ( df x df x y u( y u( ( u x( y ( u x( y pq df( x df( x y u( y ( u x( p q df( x ( u X( I E : ( y Ω. E Ω : I(. ( ( (,, : (. u X I u y E Ω y E Ω : I( Cqutly, f ay,, (3.8 hld. Thf, (3.4 fllw by lttg (3.8.

11 Appld Mathmatc 3, 3(5: ma 3.. Lt ( u X( I B( u, y, E : ( y. E : I( Ω Ω I (, X (,( Ω ad Fm, w hav u( I X( I Ω Ω B( u, y, E : ( y. E : I ( u X ( I E Ω: ( y. E Ω : I ( E Ω : y E Ω :. X ( I u ( ( I u u. Hc, upp bud f u pbablty (3.4 btt tha u. 4. A Numcal Illutat I th ct w gv a umcal xampl t llutat th bud f ( uy,, dvd Sct 3. X X b a quc f dpdt ad dtcally dtbutd -gatv ctuu adm vaabl Lt { } wth th am dtbutv fuct Lt { },5x ( ( F x x. b a hmgu Mav cha uch that f ay, havg a dtbut: ad matx P p x,3,7 P,,8 Lt { } gv by 3 P,4,6 I I b a hmgu Mav cha uch that f ay, havg a dtbut: ad matx Q [ q ] x gv by, 5,75 Q,6, 4 Th, w hav ( E.,3 3., 7, 4 E ( 3., 3.,8, 6; EX ( 4,5 I,,5 P,35,65 ta valu {,3} E wth I ta valu {,;,5} E I wth I

12 96 Quag Phug Duy: u Pbablty a Gald Pc ud at f Itt wth Hmgu Mav Cha Clam ad Hmgu Mav Cha Itt Thf I th th had, ad E ( y < E( X, y E (4. ( > >, P ( X P X ( Cmbg (4., (4. ad (4.3 mply that Lmma 3. hld. Nxt, w lv quat (3.. Ftly, w hav ( X X E y E y E (,. X (,5,5,5 x dx (, E,5 ad > 3 > (4. E X < (, (4.3 E. P. P 3,3,7 3 3 E P P ,,8 pctv quat (3. f,, by 3,3,7 4 (4.4,,8 4 Ug Mapl, w fd pctv t f (3. f,, by,33878;, 84 Hc, { } m,, 84. W ca apply th ult f Thm 3. f ( uy,, u( I 3 (4.5 (4.6 ( uy,, E I gu (, ( EI u ( I gu ( ;, E I,,u,5 u. PI, I,. PI,5 I,,u,5 u, 5,75 u ( I gu ( ;,5 E I,5,u,5 u. PI, I,5. PI,5 I,5,6, 4,u,5 u

13 Appld Mathmatc 3, 3(5: Tabl hw valu upp bud gu (, ( ux,, f a ag f valu f u Tabl. Upp bud gu (, f ( uy,, f EFEENCES [] Albch, H. (998 Dpdt ad u pbablt uac. IIASA Itm pt, I u gu ( ;, gu ( ;, Cclu Ou ma ult th pap a Thm. ad Thm. gvg cuv quat f ( uy,, ad ( uy,, ad tgal quat f ( uy,, ad ( uy,, ; Thm 3. ad Thm 3. gvg pbablty qualt f ( uy,, ad ( uy,, by a ductv appach. I addt, a umcal xampl gv llutatg Thm 3.. ACKNOWLEDGMENTS Th auth thaful t th f f pvdg valuabl uggt t mpv th qualty f th pap. I addt, th auth wuld l t xp h c gattud t Pf Bu Kh Dam f may ctfc uggt dug th ppaat f th pap. [] Amu, S. ( u pbablt, Wld Sctfc, Sgap. [3] Ca, J. ( Dct tm mdl ud at f tt. Pbablty th Egg ad Ifmatal Scc, 6, [4] Ca, J. ( u pbablt wwth dpdt at f tt, Jual f Appld Pbablty, 39, [5] Ca, J. ad Dc, D. CM (4 u Pbablt wth a Mav cha tt mdl. Iuac: Mathmatc ad Ecmc, 35, [6] Nyh, H. (998 ugh dcpt f u f a gal cla f uplu pc. Adv. Appl. Pb., 3, 8-6. [7] Pmlw, S. D. (99 Th pbablty f u a pc wth dpdt cmt. Iuac: Mathmatc ad Ecmc,, [8] l, T., Schmdl, H., Schmdt, V. ad Tugl, J. L.(999 Stchatc Pc f Iuaac ad Fac. Jh Wly, Chcht. [9] Shad, M. ad Shathuma, J. (994, Stchatc Od ad th Applcat. Acadmc P, Sa Dg. [] Sudt, B. ad Tugl, J. L (995 u tmat ud tt fc, Iuac: Mathmatc ad Ecmc, 6, 7-. [] Sudt, B. ad Tugl, J. L. (997 Th adutmt fuct u tmat ud tt fc. Iuac: Mathmatc ad Ecmc, 9, [] Xu, L. ad Wag,. (6 Upp bud f u pbablt a autgv mdl wth Mav cha tt at, Jual f Idutal ad Maagmt ptmat, Vl. N., [3] ag, H. (999 N xptal bud f u pbablty wth tt ffct cludd, Scadava Actuaal Jual,, [4] Wllmt, G. E, Ca, J. ad L, X.S. ( Ludbg Appxmat f Cmpud Dtbut wth Iuac Applcat. Spg Vlag, Nw.