Important result on the first passage time and its integral functional for a certain diffusion process


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1 Lcturs Mtmátics Volumn 22 (21), págins 5 9 Importnt rsult on th first pssg tim nd its intgrl functionl for crtin diffusion procss Yousf ALZlzlh nd Bsl M. ALEidh Kuwit Univrsity, Kuwit Abstrct. In this ppr w considr th gnrl birth nd dth procss nd study som importnt rsults on th first pssg tim nd its intgrl functionl, in prticulr w driv th symptotic distribution of th intgrl functionl. Finlly w driv th symptotic rgrssion qution of first pssg tim on its intgrl functionl. Ky words nd phrss. Birth & dth procss, rgrssion qutions Mthmtics Subjct Clssifiction. Primry 62J99. Scondry 62N99. Rsumn. En st rtículo s considrn l procso gnrl d nciminto y murt y s studin lgunos rsultdos importnts ntr l primr timpo d pso y su funcionl intgrl. En prticulr, s driv l distribución sintótic dl funcionl intgrl. Finlmnt, s driv l corrspondint cución d rgrsión sintótic dl primr timpo d cruc sobr su funcionl intgrl.
2 6 YOUSEF ALZALZALAH AND BASEL M. ALEIDEH 1. Introduction Mny importnt rsults rltd to rndom vribl nd thir intgrls hv bn studid from point of viws of diffrnt uthors. For xmpl, Puri (1966), (1968) hs invstigtd th joint distribution of th numbr of survivors X(t) in th procss nd th ssocitd intgrl t Y (t) = X(z)dz. In prticulr h hs obtind limiting rsults s t. McNil (197) hs drivd th distribution of th intgrl τx functionl W x g(x(t))dt, whrτ x is th first pssg tim to th origin in gnrl birthdth procss with X() = x nd g is n rbitrry function. Also Gni nd McNil (1971) hv studid th joint distribution of both {X(t),Y(t)} nd {T x,w x } for gnrl birthdth procsss. Also in prticulr, thy hv studid th joint distribution of {T x,w x } for diffusion procss whos bckwrd qution is givn by dt = βxα d2 f dx 2, whr f(y; x, t) =P (y <X(t) y + dy X() = x). Functionls of th form ris nturlly in trffic nd storg thory. In th prsnt ppr, w considr th most gnrl diffusion procss in which th drift condition is lso xists. W driv th symptotic W x distribution of nd som symptotic rsults on th joint distribution of {T x,w x }. Also, w driv th symptotic rgrssion qution (prdiction qution) of T x on W x. W follow th nlysis of Gni nd McNil (1971). 2. Joint Distribution of {T x,w x } Considr th birthdth diffusion procss whos bckwrd Kolmogorov qution is
3 IMPORTANT RESULT ON THE FIRST PASSAGE TIME... 7 whr dt = x dx + βxα d2 f dx 2, (1) f(y; x, t) =P (y <X(t) y + dy X() = x). In ordr to insur tht th first pssg tim is finit, w tk α<2, β is th diffusion condition nd is th drift condition. Now lt M x (θ, φ) =E( θt x φw x ) (2) whr R (θ), R (φ) >. Not tht M x is th joint Lplc trnsform of {T x,w x }. This stisfis th qution (θ + φx)m x (θ, φ) =xm x(θ, φ)+βx α M x (θ, φ), (3) whr M x nd M x indict first nd scond drivtiv with rspct to x rspctivly. This cn b usd to obtin th symptotic rgrssion qution of T x on W x, whn g(x) =x. Dfin K x (φ) := d dθ M x(θ, φ) = φu E[T x W x = u]dp (W x u). (4) Ltting θ =, thn K x (φ) stisfis th diffrntil qution ( ) ( ) φ K x(φ)+ x 1 α K x x 1 α K x (φ) = 1 β β β x α M x (,φ). (5) Also, using θ = nd th boundry conditions M (,φ) = 1 nd M (,φ) = in qution (3), w gt ( M x (,φ)= βφ βφ ) + 2 4β 2 + φβ β 2 + φβ +
4 8 YOUSEF ALZALZALAH AND BASEL M. ALEIDEH Also th solution of th diffrntil qution in (5), is givn by 1 K x (φ) = 1 x1 M x 2 4β 2 + 4φβ + 1 x1 M x 2 4β 2 + 4φ β 2 + 4φ 4β 2 β 2 4β 2 + 4φβ, whr M x is th solution in qution (6). Th invrsion of qution (6), using th bhvior of th Lplc trnsforms lim φm x(,φ) = lim f W φ x x (u), whr f Wx (u) is th probbility dnsity function of W x,forlrgx. Now ftr som mnipultion w gt, 2 x 2 x f Wx = 4β 4β u if u (6) if u< This implis tht W x hs n symptotic xponntil distribution with prmtr 2 x 4β. Similrly, th invrsion of qution (7), using th symptotic bhvior of th Lplc trnsforms lim φk x(φ) = lim E [T x W x = u] f Wx (u), φ x nd ftr som mnipultion for lrg vlus of x, wgt E [T x W x = u] f Wx (u) 2 x 4β u 2 16 u u. (7) Now dividing qution (9) into qution (8), w finlly obtin th symptotic rgrssion qution of T x on W x.i.. ( ) 2 4β E [T x W x = u] u 2 16 u, u, (8) for lrg vlu of x nd α =.
5 IMPORTANT RESULT ON THE FIRST PASSAGE TIME... 9 Rfrncs [1] Gni,J. & McNil, D.R. Joint Distributions of Rndom Vribls nd thir Intgrls for Crtin BirthDth nd Diffusion Procss. Adv. Appl. Prob. 3, (1971), págs [2] McNil, D.R. Intgrl functionls of birth nd dth procsss nd rltd limiting distributions. Ann. Mth. Sttist. 41, (197), págs [3] Puri, Prms S. On th homognous birth nd dth procss nd its intgrl. Biomtrik 53,(1966), págs [4] Puri, Prm S. Som furthr rsults on th birth nd dth procss nd its intgrl. Proc. Cmb. Phil. Soc. 64, (1968), págs (Rcibido n fbrro d 21) Yousf ALZlzlh Bsl M. ALEidh Dprtmnt Of Quntittiv Mthods And Informtion Systms Kuwit Univrsity, Kuwit P.O.Box 5486, Sft 1355, Kuwit
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