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 America Joural of Applied Mahemaics ad Saisics, 015, Vol. 3, No. 5, Available olie a hp://pubs.sciepub.com/ajams/3/5/ Sciece ad Educaio Publishig DOI: /ajams-3-5- The Mome Approximaio of he Firs Passage Time for he Birh Deah Diffusio Process wih Immigrao o a Movig Liear Barrier Basel M. Al-Eideh * Deparme of Quaiaive Mehods ad Iformaio Sysem, Kuwai Uiversiy, College of Busiess Admiisraio, Safa, Kuwai *Correspodig auhor:basel@cba.edu.kw Received April 06, 015; Revised Sepember 01, 015; Acceped Sepember 0, 015 Absrac Today, he he developme of a mahemaical models for populaio growh of grea imporace i may fields. The growh ad declie of real populaios ca i may cases be well approximaed by he soluios of a sochasic differeial equaios. However, here are may soluios i which he esseially radom aure of populaio growh should be ake io accou. I his paper, we approximaig he momes of he firs passage ime for he birh ad deah diffusio process wih immigraio o a movig liear barriers. This was doe by approximaig he differeial equaios by a equivale differece equaios. A simulaio sudy is cosidered ad applied o some values of parameers which showed he capabiliy of he echique. Keywords: firs passage ime, birh-deah diffusio process, immigraio, differece equaios Cie This Aricle: Basel M. Al-Eideh, The Mome Approximaio of he Firs Passage Time for he Birh Deah Diffusio Process wih Immigrao o a Movig Liear Barrier. America Joural of Applied Mahemaics ad Saisics, vol. 3, o. 5 (015): doi: /ajams Iroducio Firs passage ime play a impora rule i he area of applied probabiliy heory especially i sochasic modelig. Several examples of such problems are he exicio ime of a brachig process, or he cycle leghs of a cerai vehicle acuaed raffic sigals. Acually he he firs passage imes o a movig barriers for diffusio ad oher markov processes arises im biological modelig (Cf. Ewes [8]), i saisics (Cf. Darlig ad Sieger [6] ad Durbi [7]) ad i egieerig (Cf. Blake ad Lidsey [4]). May impora resuls relaed o he firs passage ime have bee sudied from differe pois of view of differe auhors. For example, McNeil [13] has derived he disribuio of he iegral fucioal Tx Wx g{ X ( )} d, 0 where T x is he firs passage ime o he origi i a geeral birh deah process wih X(0) = x ad g(.) is a arbirary fucio. Also, Iglehar [10], McNeil ad Schach [14] have bee show a umber of classical birh ad deah processes upo akig diffusio limis o asympoically approach he Orsei Uhlebeck (O.U.). May properies such as a firs passage ime o a barrier, absorbig or reflecig, locaed some disace from a iiial sarig poi of he O.U. process ad he relaed diffusio process ad he relaed diffusio process such as he case of he firs passage ime of a Wieer process o a liear barrier is a closed form expressio for he desiy available is discussed i Cox ad Miller [5]. Also, ohers such as, Karli ad Taylor [11], Thomas [15], Ferebee [9], Tuckwell ad Wa [16], Alaweh ad Al- Eideh [1], Al-Eideh [,3] ec. have bee discussed he firs passage ime from differe pois of view. I paricular, Thomas (1975) describes some mea firs passage ime approximaio for he Orsei Uhledeck process. Tuckwell ad Wa [16] have sudied he firs-passage ime of a Markov process o a movig barriers as a firs-exi ime for a vecor whose compoes iclude he process ad he barrier. Alaweh ad Al-Eideh [1] have discussed he problem of fidig he momes of he firs passage ime disribuio for he Orsie-Uhlebeck process wih a sigle absorbig barrier usig he mehod of approximaig he differeial equaios by differece equaios. Also, Al-Eideh [,3] has discussed he problem of fidig he momes of he firs passage ime disribuio for he birh-deah diffusio ad he Wrigh-Fisher diffusio processes o a movig liear barriers ad o a sigle absorbig barrier, respecively usig he mehod of approximaig he differeial equaios by differece equaios. I his paper, we cosider he birh ad deah diffusio process wih immigraio ad sudy he firs passage ime for such a process o a movig liear barrier. More specifically, he mome approximaios are derived usig he mehod of differece equaios used i Al-Eideh [3] cosiderig he immigraio rae.

2 America Joural of Applied Mahemaics ad Saisics 185. Firs Passage Time Mome Approximaios Cosider he birh ad deah diffusio Process wih X ( ) : 0 wih ifiiesimal mea immigraio bx ad variace sarig a some x 0 > 0, where b ad a are he drif ad he diffusio coefficies respecively ad is he cosa immigraio rae. Also, X ( ) : 0 is a Markov process wih sae space S 0, ad saisfies he Io sochasic differeial equaio dx ( ) bx ( ) d ax ( ) dw ( ) (1) Where W ( ) : 0 is a sadard Wieer process wih zero mea ad variace. Assume ha he exisece ad uiqueess codiios are saisfied (Cf. Gihma ad Skorohod). Le Y ( ) : 0 be a movig liear barrier equaio such ha Y() c k, wih Y(0) k. Or dy () equivalely c. d Now, deoe he firs passage ime of a process X () o a movig liear barrier Y() c k by he radom variables TY if{ 0: X ( ) c k} () wih probabiliy desiy fucio ck d g ; x0 px0, x; d Here p ( x 0, x ; ) is he probabiliy desiy fucio of X () codiioal o X (0) = x 0 Le M x0, Y ; ; = 1,,3,, be he -h mome of he firs passage ime T Y, i.e. M x0 E( TY ), 1,,3,..., (3) I follows from he forward Kolmogorov equaio ha he -h mome of T Y mus saisfy he ordiary differeial equaio 0, ; 0, ;, ;, ; M x Y bx M x Y cm x Y M x Y Or equivalely bx M x0 M x0 c M x0 Y M1 x0 Y, ;, ; Where M x0, Y ; ad M x0, Y ; derivaives of 0, ; x x Y 0 (4) (5) are he firs M x Y wih respec o x, wih appropriae boudary codiios for =1,,3,. Noe ha M0 x0 1. Now, rewrie he equaio i (5), we obai M x0 b c M1 x0 Y M x0 Y a, ;, ; (6) Le be he differece operaor. The we defied he M x0, Y ; as follows: firs order differece of, ;, ;, ; M x0 Y M1 x0 Y M x0 Y (7) (Cf. Kelly ad Peerso [1]). Noe ha equaio (6) ca be approximaed by M x0 Y M1 x0 Y b c M x0 a, ;, ; By applyig equaio (7) o equaio (8) we ge : M x0y ; M1 x0 b c M x 0, Y ; a b c M 1 x 0, Y ; a (8) (9) Now, we will use he marix heory o solve he differeial equaio defied i equaio (9). If we le M x0 M1 x0, M x0, The we ge Where d M x0 AM x0 (10) b c b c 0 0 a a b c b c 0 a a A 3 b c b c 0 a a 4 b c 0 0 a Now le This imply dm x0 R x0 (11), ;, ; d M x0 Y dr x0 Y Apply o equaio (10), we ge (1) d R x0 0 A R x 0. M x I 0 M x 0 0 Where I is he ideiy marix ad 0 is he zero marix. (13)

3 186 America Joural of Applied Mahemaics ad Saisics Thus, he soluio of he sysem of equaio i (13) is he give by 0 A * 0 D 0 0 R x R x e. M x M x 0 0 (14) Where D = [ d ij ] ; i, j 1 is he diagoal marix wih eries dij c k x0 ; j i 0 ; Oherwise (15) X( 1), X( k),..., wih 1 1, 1,.... Therefore, he followig graph, Figure 1, represes he simulaed process whe X0 (1) 50 ad he parameers are se o a 0.5, b 0.0 ad 0.0. Ad A a ij ; i, j 1 is he marix wih eries by i c k l ; ji1 x0 b c c k x0 ; j i aij a b c c k x0 ; j i 1 a 0 ; Oherwise Noe ha he marix B e where 3 0 B D B B B e I B...! 3! (16) A is defied 0 This series is coverge sice i is a cauchy operaor of equaio (.6) (Cf. Zeifma [17]). 3. Simmulaio Sudy I his secio we will cosider he simmulaig he birh ad deah diffusio process wih immigraio X ( ) : 0 as cosidered i equaio (1) as well as approximaig he momes of he firs-passage ime for such a process usig he firs ad he secod order differece operaors o he differeial equaio i (9). For simulaio of he process X ( ) : 0 we used he followig discree approximaio. For ieger values k 1,,3,..., ad 1,,3,..., defie * k 1 * k 1 * k X ( ) X( ) bx ( ) 1 * k ax ( ) Z ( k 1) (17) where Zk ( ) is a idepede sequece of sadard ormal radom variables. For each se of posiive iegers k, 1,..., k, he sequece of radom vecrors * * 1 k ( X ( ),..., X ( )) coverges i disribuio o ( X ( 1),..., X ( k )). As a example is chose o be 50, i.e. each ui of ime is broke io 50 seps for he purpose of simulaig Figure 1. The Simulaed Process X() whe a=0.5, b=0.0 ad ε=0.0 Now for approximaig he momes of he firspassage ime for such a process usig he firs ad he secod order differece operaors o he differeial equaio i (9), we defie he operaors as follows: Le be he secod order differece operaors. The we defied he secod order differeces of M x0, Y ; ad as follows: 0, ; 0, ; M x Y M x Y M x M x (Cf. Kelley ad Peerso [1]). Noe ha equaio (9) ca be approximaed by M x0 M1 x0 b c M x 0, Y ; a b c M 1 x 0, Y ; a By applyig equaio (18) o equaio (19) we ge:, ;, ; M x Y M x Y M x0 Y M1 x0 Y b c M x 0, Y ; a b c M 1 x 0, Y ; a, ;, ; Now rewriig equaio (0) we ge: M x 0 b c M1 x0 a b c 1 M x0 a M 1 x 0, Y ; (18) (19) (0) (1)

4 Through equaio (1), he firs mome M 1 x 0 ad he secod mome M x Y of he firs passage 0, ; ime ca be approximaed by 1 0, ; b c M x Y America Joural of Applied Mahemaics ad Saisics 187 a () ad M x 0 b c b c (3) M1x0 1 a a Followig he above example of such a process show i Figure 1 assumig differe values for he parameer c where c 0, 0.0, 0.1, 0.5, 1 ad 5, we se he followig Figures for he firs mome M 1 0, ; he secod mome ime. x Y ad M x0, Y ; of he firs passage Figure 5. The Secod Mome of he Firs-Passage Time M () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0.0 Figure 6. The Firs Mome of he Firs-Passage Time M 1 () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0.1 Figure. The Firs Mome of he Firs-Passage Time M 1 () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0 Figure 7. The Secod Mome of he Firs-Passage Time M () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0.1 Figure 3. The Secod Mome of he Firs-Passage Time M () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0 Figure 4. The Firs Mome of he Firs-Passage Time M 1 () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0.0 Figure 8. The Firs Mome of he Firs-Passage Time M 1 () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0.5

5 188 America Joural of Applied Mahemaics ad Saisics Figure 9. The Secod Mome of he Firs-Passage Time M () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=0.5 Figure 13. The Secod Mome of he Firs-Passage Time M () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=5 The above figures of he firs moe M 1 x 0 of he firs-passage ime of he process show he growh expoeially for he suggesed values of he parameer c eve whe c 0 ad coverges as he ime icreased. Bu o he corary he secod mome M x0, Y ; of he firs-passage ime of he process show declie for he same suggesed values of c eve whe c 0 ad coverges as ime icreased oo. 4. Coclusio Figure 10. The Firs Mome of he Firs-Passage Time M 1 () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=1 I coclusio he advaage of his echique is o use he differece equaio o approximae he ordiary differeial equaio sice i is he discreizaio 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 ad deah diffusio process wih immigraio o a movig liear barriers. This icreases he applicabiliy of he diffusio process i sochasic modelig or i all area of applied probabiliy heory. For furher sudy I hik i is possible o se up a exac or a approximaed echique o predic he parameers of he suggesed process ad he ope a possibiliy o be applied o a real life problem. Suppor This work was suppored by Kuwai Uiversiy, Research Gra No. [IQ 03/1]. Figure 11. The Secod Mome of he Firs-Passage Time M () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=1 Figure 1. The Firs Mome of he Firs-Passage Time M 1 () of he Process X() whe a=0.5, b=0.0, ε=0.0 ad c=5 Refereces [1] A. J. Alaweh ad B. M. Al-Eideh, Mome approximaio of he firs-passage ime for he Orsei-Uhlebeck process. Ier. Mah. J. Vol.1 (00), No.3, [] B. M. Al-Eideh, Firs-passage ime mome approximaio for he Righ-Fisher diffusio process wih absorbig barriers. Ier. J. Coemp Mah. Scieces, Vol.5 (010), No.7, [3] B. M. Al-Eideh, Firs-passage ime mome approximaio for he birh-deah diffusio process o a movig liear barriers. J. Sa. & Maag. Sysems, Vol.7 (004), No.1, [4] I. F. Blake ad W. C. Lidsey, Level crossig problems for radom processes. IEEE Tras. Iformaio Theory 19 (1973), [5] D. R. Cox ad H. D. Miller, The heory of sochasic processes. Mehue, Lodo (1965). [6] D. Darlig ad A. J. F. Sieger, The firs - passage problem for a coiuos Markov process. A. Mah. Sais. 4 (1953),

6 America Joural of Applied Mahemaics ad Saisics 189 [7] J.Durbi, Boudary - crossig probabiliies for he Browia moio ad Poisso processes ad echiques for compuig he power of he Kolmogorov-Smirov es. J. Appl. Prob. 8 (1971), [8] W. J. Ewes, Mahemaical Populaio Geeics. Spriger-Verlag, Berli (1979). [9] B. Ferebee, The age approximaio o oe-sided Browia exi desiies. Z. Wahrscheilichkeish 61 (198), [10] D. L. Iglehar, Limiig diffusio approximaio for he may server queuead he repairma problem. J. Appl. Prob. (1965), [11] S. Karli ad H. M. Taylor, A Secod Course i Sochasic Processes. Academic press. New York (1981). [1] W.G. Kelly ad A.C. Peerso, Differece Equaios : A Iroducio wih Applicaios. Academic Press, New York (1991). [13] D. R. McNeil, Iegral fucioals of birh ad deah processes ad relaed limiig disribuios. A. Mah. Sais. 41 (1970), [14] D. R. McNeil ad S. Schach, Ceral limi aalogues for Markov populaio processes. J. R. Sais. Soc. B, 35 (1973), 1-3. [15] M. U. Thomas, Some mea firs passage ime approximaios for he Orsei Uhlebeck process.j. Appl. Prob. 1 (1975), [16] H. C. Tuchwell ad F. Y. M. Wa, Firs-passage ime of Markov processes o movig barriers. J. Appl. Prob. Vol. 1 (1984), [17] A. I. Zeifma, Some esimaes of he rae of covergece for birh ad deah processes. J. Appl. Prob. 8 (1991),