The Moment Approximation of the First Passage Time For The Birth Death Diffusion Process with Immigraton to a Moving Linear Barrier

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1 Rece Avaces i Auomaic Corol, oellig a Simulaio The ome Approximaio of he Firs Passage Time For The irh Deah Diffusio Process wih Immigrao o a ovig Liear arrier ASEL. AL-EIDEH Kuwai Uiversiy, College of usiess Amiisraio Dep. of Quaiaive ehos a Iformaio Sysem P. O. ox 5486, Safa 1355, Kuwai E-ail: basel@cba.eu.kw ASTRACT: Toay, he he evelopme of a mahemaical moels for populaio growh of grea imporace i may fiels. The growh a eclie of real populaios ca i may cases be well approximae by he soluios of a sochasic iffereial equaios. However, here are may soluios i which he esseially raom aure of populaio growh shoul be ake io accou. I his paper, we approximaig he momes of he firs passage ime for he birh a eah iffusio process wih immigraio o a movig liear barriers. This was oe by approximaig he iffereial equaios by a equivale ifferece equaios. Key Wors: Firs Passage Time, irh-deah Diffusio Process, Immigraio, Differece Equaios. 1. INTRODUCTION Firs passage ime play a impora rule i he area of applie probabiliy heory especially i sochasic moelig. Several examples of such problems are he exicio ime of a brachig process, or he cycle leghs of a cerai vehicle acuae raffic sigals. Acually he he firs passage imes o a movig barriers for iffusio a oher markov processes arises i biological moelig (Cf. Ewes (1979)), i saisics (Cf. Darlig a Sieger (1953) a Durbi (1971)). ay impora resuls relae o he firs passage ime have bee suie from iffere pois of view of iffere auhors. For example, cneil (197) has erive he isribuio of he iegral fucioal Wx Tx g{ X ( }, where T x is he firs passage ime o he origi i a geeral birh eah process wih X() x a g(.) is a arbirary fucio. Also, Iglehar (1965), cneil a Schach (1973) have bee show a umber of classical birh a eah processes upo akig iffusio limis ISN:

2 Rece Avaces i Auomaic Corol, oellig a Simulaio o asympoically approach he Orsei Uhlebeck (O.U.). ay properies such as a firs passage ime o a barrier, absorbig or reflecig, locae some isace from a iiial sarig poi of he O.U. process a he relae iffusio process a he relae iffusio process such as he case of he firs passage ime of a Wieer process o a liear barrier is a close form expressio for he esiy available is iscusse i Cox a iller (1965). Also, ohers such as, Karli a Taylor (1981), Thomas (1975), Ferebee (198), Tuckwell a Wa (1984), Al-Eieh (4), ec. have bee iscusse he firs passage ime from iffere pois of view. I paricular, Thomas (1975) escribes some mea firs passage ime approximaio for he Orsei Uhleeck process. Tuckwell a Wa (1984) have suie he firs-passage ime of a arkov process o a movig barriers as a firs-exi ime for a vecor whose compoes iclue he process a he barrier. Also, Al-Eieh (4), has iscusse he problem of fiig he momes of he firs passage ime isribuio for he birh-eah iffusio a he Wrigh-Fisher iffusio processes o a movig liear barriers usig he meho of approximaig he iffereial equaios by ifferece equaios. I his paper, we cosier he birh a eah iffusio process wih immigraio a suy he firs passage ime for such a process o a movig liear barrier. ore specifically, he mome approximaios are erive usig he meho of ifferece equaios use i Al-Eieh (4) cosierig he immigraio rae ε.. FIRST PASSAGE TIE OENT APPROXIATIONS Cosier he birh a eah iffusio Process wih immigraio X ( : wih ifiiesimal mea { } bx ε a variace sarig a some x >, where b a a are he rif a he iffusio coefficies respecively aε is he cosa X ( : immigraio rae. Also, { } is a arkov process wih sae space S, a saisfies he Io [ ) sochasic iffereial equaio ( bx( ) ax( W( ) X ( ε (1) Where { ( : } W is a saar Wieer process wih zero mea a variace. Assume ha he exisece a uiqueess coiios are saisfie (Cf. Gihma a Skoroho (197)). Le Y ( : be a movig liear barrier { } equaio such ha Y ( ) k. Or equivalely Y ( c k, wih Y ( c Now, eoe he firs passage ime of a process X ( o a movig liear barrier raom variables Y ( c k by he ISN:

3 Rece Avaces i Auomaic Corol, oellig a Simulaio T Y if{ : X ( c k} () wih probabiliy esiy fucio g ( ; x ) - c k p ( x, x ; Here p ( x, x ; is he probabiliy esiy fucio of X ( coiioal o X () x Le ( x, Y ; ; 1,,3,, be he -h mome of he firs passage ime T Y, i.e. ( x E( ), T Y ; 1,,3,, (3) I follows from he forwar Kolmogorov equaio ha he -h mome of T Y mus saisfy he oriary iffereial equaio c ( x, ( bx ) ( x, ( x, Y ; ε Or equivalely 1 bx ε c 1 Where ( x, Y ; a ( x are he firs erivaives of ( x, Y ; wih respec o x ( x x Y), (4) (5), wih appropriae bouary coiios for 1,,3,.Noe ha x 1 ( )., Now, rewrie he equaio i (5), we obai b ε a ( x, Y ; ( x, Y ; 1 ( x, Y ; (6) Le be he ifferece operaor. The we efie he firs orer ifferece of as follows: ( x Y ;) ( x, Y ;) ( x, Y (7), 1 ; (Cf. Kelley a peerso (1991) ). Noe ha equaio (6) ca be approximae by a 1 (8) y applyig equaio (7) o equaio (8) we ge : 1 a a ( x 1 (9) Now, we will use he marix heory o solve he iffereial equaio efie i equaio (9). If we le x x,, x,, ( ) [ ( ) ( ) ] The we ge, 1 A (1) ISN:

4 Rece Avaces i Auomaic Corol, oellig a Simulaio Where a A Now le This imply 3 ( x, Y ; 4 ( x, Y ; R (11) ( x, R( x, (1) Apply o equaio (1), we ge R ( x, I A R( x ) ( ),. x, (13) Where I is he ieiy marix a is he zero marix. Thus, he soluio of he sysem of equaio i (13) is he give by R x ( x, (, e * A D R( x ) ( ),. x, (14) Where D [ ij ] ; i, j 1 is he iagoal marix wih eries ( c k x ) ; j i ij ; Oherwise (15) A [ a ]; i, j 1 wih eries a ij A ij is he marix i c k l ; j i 1 x ( c kx) ; j i ( c kx) ; j i 1 ; Oherwise Noe ha he marix A is efie by D e I! 3 3! (16) e where This series is coverge sice i is a cauchy operaor of equaio (.6) (Cf. Zeifma (1991)). 3. CONCLUSION I coclusio he avaage of his echique is o use he ifferece equaio o approximae he oriary iffereial equaio sice i is he iscreizaio of he ODE. Also, he sysem of he soluios i equaio (14) gives a explici soluio o he firs passage ime momes for he birh a eah iffusio process wih immigraio o a movig liear barriers. This icreases he applicabiliy of he iffusio process i sochasic moelig or i all area of applie probabiliy heory. Noe ha i case of immigraio whe ε, we obaie he same resuls as i Al-Eieh (4). ISN:

5 Rece Avaces i Auomaic Corol, oellig a Simulaio REFERENCES [1].. Al-Eieh, Firs-passage ime mome approximaio for he birheah iffusio process o a movig liear barriers. J. Sa. & aag. Sysems, Vol.7 (4), No.1, [] D. R. Cox a H. D. iller, The heory of sochasic processes. ehue, Loo (1965). [3] D. Darlig a A. J. F. Sieger, The firs - passage problem for a coiuos arkov process. A. ah. Sais. 4 (1953), [4] J.Durbi, ouary-crossig probabiliies for he rowia moio a Poisso processes a echiques for compuig he power of he Kolmogorov-Smirov es. J. Appl. Prob. 8 (1971), [5] W. J. Ewes, ahemaical Populaio Geeics. Spriger-Verlag, erli (1979). [6]. Ferebee, The age approximaioo oe-sie rowia exi esiies. Z.Wahrscheilichkeish 61 (198), [9] W.G. Kelly a A.C. Peerso, Differece Equaios : A Iroucio wih Applicaios. Acaemic Press, New York (1991). [1] D. R. cneil, Iegral fucioals of birh a eah processes a relae limiig isribuios. A. ah. Sais. 41 (197), [11] D. R. cneil a S. Schach, Ceral limi aalogues for arkov populaio processes. J. R. Sais. Soc.,35 (1973),1-3. [1]. U. Thomas, Some mea firs passage ime approximaios for he Orsei Uhlebeck process.j. Appl. Prob. 1 (1975),6-64. [13] H. C. Tuchwell a F. Y.. Wa, Firs-passage ime of arkov processes o movig barriers. J. Appl. Prob. Vol. 1 (1984), [14] A. I. Zeifma, Some esimaes of he rae of covergece for birh a eah processes. J. Appl. Prob. 8 (1991), [7] D. L. Iglehar, Limiig iffusio approximaio for he may server queuea he repairma problem. J. Appl. Prob. (1965), [8] S. Karli a H.. Taylor, A Seco Course i Sochasic Processes. Acaemic press. New York (1981). ISN: