I-POLYA PROCESS AND APPLICATIONS Leda D. Minkova

Transcription

1 The XIII Inenaonal Confeence Appled Sochasc Models and Daa Analyss (ASMDA-009) Jne 30-Jly 3, 009, Vlns, LITHUANIA ISBN L Sakalaskas, C Skadas and E K Zavadskas (Eds): ASMDA-009 Seleced papes Vlns, 009, pp Inse of Mahemacs and Infomacs, 009 Vlns Gedmnas Techncal Unvesy, 009 I-POLYA PROCESS AND APPLICATIONS Leda D Mnkova Facly of Mahemacs and Infomacs, Sofa Unvesy S KlOhdsk E-mal: leda@fmn-sofabg Absac: The Inflaed-paamee negave bnomal pocess (I - Pólya pocess) as a mxed Pólya - Aeppl pocess s defned Some basc popees ae gven We consde he sk model n whch he conng pocess s he I - Pólya pocess I s called I - Pólya sk model The jon pobably dsbon of he me o n and he defc afe n occs s sdded The pacla case of exponenally dsbed clams s gven Keywods: Pólya - Aeppl pocess, n pobably, I-Pólya pocess Inodcon The Pólya-Aeppl pocess as a genealzaon of he homogeneos Posson pocess s nodced n [6] I s a homogeneos pocess and he pobably ha wo o moe clams can ave smlaneosly s geae hen zeo The nmbe of ems n he me neval [0,] has a Pólya - Aeppl dsbon, e λ e, k 0, k P( N k ) λ λ k [ ( ) ] ( ) λ ρ k e ρ, k,, K,! whee λ > 0 and ρ [0,) ae paamees We say also ha he dsbon () s IPo( λ, ρ ) (Inflaed- λ paamee Posson dsbon) The mean nmbe of avals s E( N λ) ρ The me nl he fs epoch T and ne-aval mes T, T 3, K ae mally ndependen andom vaables T s exponenally dsbed wh paamee λ, exp( λ ) T, T 3, K ae dencally exponenally dsbed wh paamee λ and mass a zeo eqal o ρ Ths means ha clams can () ave smlaneosly wh pobably ρ, fo,3, K Fo moe deals elaed o he nepeaon of he Pólya - Aeppl pocess and he addonal paamee ρ he eade can see he pape [6] If ρ 0 he conng pocess defned by () becomes he sal homogeneos Posson pocess In hs pape we sppose ha λ s he ocome of a andom vaable Λ Followng he well known emnology [] he eslng pocess s a mxed Pólya - Aeppl pocess The pobably dsbon Λ s called mxng dsbon We may nepe () as he condonal dsbon of N, gven he ocome Λ λ We wll defne a genealzaon of he classcal Pólya pocess as a mxed Pólya - Aeppl pocesses The new genealzed Pólya pocess s called I - Pólya pocess Some popees of he defned pocess ae gven We consde a sk model wh I - Pólya conng pocess A specal case of exponenally dsbed clams s sded The jon pobably dsbon of he me o n and he defc afe n occs s oban 35

2 I-POLYA PROCESS AND APPLICATIONS Genealzed Pólya pocess Le he dsbon of he mxng andom vaable Λ be gamma one wh paamees and Is pobably densy fncon s gven by λ λ e, Γ( ) > 0, λ > 0, whee Γ s he Γ -fncon, s called he shape paamee and he scale paamee In hs case N has he followng pobably mass fncon:, m 0 P( N m) m m () m [( ρ) ] ρ, m,, K Ths s js he Inflaed- paamee negave bnomal dsbon wh paamees, ρ and, say INB(,ρ, ) see [5] In he case of ρ 0 () concdes wh he sal negave bnomal dsbon If he nmbe of evens p o me s negave bnomal dsbed, he conng pocess s known as he Pólya pocess, see [], p4 Ths movaes he followng defnon Defnon The conng pocess, { N ( ), 0} s sad o be an Inflaed - paamee Pólya pocess o I - Pólya pocess, f sas a zeo, N (0) 0 and fo each > 0, he dsbon of N () s gven by () Popees of he I-Pólya pocess Denoe S n T T K Tn, n,, K, he wang me nl he n h even The basc popees of he I-Pólya pocess ae gven by he nex heoem Theoem Le N () has he INB(,ρ, )dsbon () Then () The me nl he fs epoch T s Paeo dsbed The ne-aval mes T, T 3, K ae zeo wh pobably ρ and wh pobably ρ Paeo dsbed () The wang me nl he n h even has he followng pobably densy fncon (pdf) f S n n n n ( ) ( ρ) ρ (3) 0 Fom he Theoem follows ha he dsbon of T s Paeo wh paamees and and pobably dsbon fncon FT ( ), 0 (4) The dsbon of T s Paeo wh mass a zeo eqal o ρ and pobably dsbon fncon FT ( ) ( ρ), 0 (5) 353

3 L D Mnkova Kolmogoov fowad eqaons In [4] he I - Pólya pocess s defned as a genealzed pe bh pocess The anson pobables ae gven by he followng poslaes: P ( N( h) n N( ) m) k (ρ) [(ρ) ] h o( n m, k k n m k, k,, K, (ρ) [(ρ) ] h o( fo evey m 0,, K, whee o ( h) 0 as h 0 If ( ) P( N( ) m), m 0,,, K, we oban he followng Kolmogoov fowad eqaons: P m P0 ( ) P0 ( ), Pm ( ) Pm ( ) ( ρ) fo m,, K The solon of (6) wh condons P0 (0) and Pm (0) 0, m,,k s gven by () 3 Applcaon o Rsk Theoy m k ( ρ) Consde he sandad sk model { X ( ), 0}, defned on he complee pobably space ( Ω, F, P) and gven by N ( ) k P mk ( ), 0 X ( ) c Z k, 0 (7) k Hee c s a posve eal consan epesenng he sk pemm ae The seqence { Z k} k of nonnegave mally ndependen dencally dsbed andom vaables s ndependen of he conng pocess N ( ), 0 The clam szes { Z k} k ae dsbed as he andom vaable Z wh dsbon fncon F, F(0) 0 and mean vale µ We consde he sk model (7), whee N () s I - Pólya pocess and wll call hs pocess I - Pólya sk model The elave safey loadng θ s defned by c( ρ) µ c( ρ) θ, µ µ µ and n he case of posve safey loadng θ > 0, c > ( ρ) Le τ nf{ : X ( ) < } wh he convenon of nf be he me o n of an nsance company havng nal capal 0 We denoe by Ψ( ) P( τ < ) he n pobably and Φ ( ) Ψ( ) he non-n pobably In he followng we se he noaon of [3] Le G ( be he jon pobably dsbon of he me o n τ and he defc n po o n D, e P( τ, D (8) and lm y Ψ( ) Usng he poslaes, we have (6) 354

4 I-POLYA PROCESS AND APPLICATIONS whee ( ρ) h ch, h ( ) ρ k k k k ( F ( ch F ( ch) ) o( ch k ch x, df ( x) 0 * ( x), k,, K s he dsbon fncon of Z Z K Z F k Reaangng he ems, leng k k * k H ( x) ( ρ) ( ρ) F ( x) k h we oban he followng dffeenal eqaon (, ) (, ) ( ) [ ( ) ( )] G y G x y dh x H y H c 0 H s he non defecve pobably dsbon fncon of he clams wh and 0 The fncon (x) In ems of he safey loadng H (0) 0, H ( ) ρ (, ) (, ) ( ) [ ( ) ( )] ( ) G y µ θ G x y dh x H y H 0 Theoem The fncon G (0, s gven by y 0, [ H ( )] d c (9) 0 Theoem 3 The n pobably wh no nal capal sasfes µ Ψ (0) (0) ( ρ) c 3 Exponenally dsbed clams Le s consde he case of exponenally dsbed clam szes, e F( ) e, 0, µ > 0 In hs case, x x (ρ) (ρ) µ µ ρ h( x) e and H ( x) e, x 0 µ The jon pobably dsbon of he me o n and he defc a n s gven by µ e c( ρ) The n pobably n he exponenal case s µ y ρ µ (ρ) c ( ) µ µ ρ e ρ µ c( ) µ µ ( ) e ρ Ψ c( ρ) 4 Acknowledgmen Ths pape s paally sppoed by Sofa Unvesy gan /

5 Refeences L D Mnkova [] Gandell J(997) Mxed Posson Pocesses, Chapman & Hall, London [] Johnson NL, Koz S and Kemp AW (99) Unvaae Dscee Dsbons, Wley Sees n Pobably and Mahemacal Sascs nd edon [3] Klgman S A, Panje H and Wllmo G (998) Loss Models Fom Daa o Decsons, John Wley & Sons, Inc [4] Mnkova LD (00) Inflaed-paamee modfcaon of he pe bh pocess, Comp Rande Blg Acad Sc, 54(), 7 [5] Mnkova LD (00) A Genealzaon of he Classcal Dscee Dsbons, CommnSas - Theoy and Mehods, 3(6), [6] Mnkova LD (004) The Pólya-Aeppl pocess and n poblems, JAMSA, 004(3),