First-Passage Time moment Approximation For The Birth Death Diffusion Process To A General moving Barrier

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1 Ieriol Jourl of emics Sisics Iveio IJSI E-ISSN: P-ISSN: Volume 5 Issue 9 Jury 8 PP-4-8 Firs-Pssge Time mome pproimio For Te Bir De Diffusio Process To Geerl movig Brrier Bsel. l-eie KuwiUiversiy College of Busiess miisrio Dep. of Quiive eos Iformio Sysem Sf355Kuwi Correspoig uor: Bsel. l-eie BSTCT: Te evelopme of memicl moels for populio grow of gre imporce i my fiels. Te grow eclie of rel populios c i my cses e well pproime y e soluios of socsic iffereil equios. However ere re my soluios i wic e esseilly rom ure of populio grow soul e ke io ccou. Tispper focusses i pproimig e momes of e firs pssge ime for e ir e iffusio process o geerl movig rriers. Tis ws oe y pproimig e iffereil equios y equivle ifferece equios. KEWODS: Firs Pssge Time Bir-De Diffusio Process Differece Equios Geerl ovig Brrier De of Sumissio: 3--7 De of ccepce: I. INTODUCTION Firs pssge ime ply impor rule i e re of pplie proiliy eory especilly i socsic moelig. Severl emples of suc prolems re e eicio ime of rcig process or e cycle legs of ceri veicle cue rffic sigls. cully e e firs pssge imes o movig rriers for iffusio oer mrkov processes rises i iologicl moelig Cf. Ewes 979 i sisics Cf. Drlig Sieger 953 Duri 97. y impor resuls rele o e firs pssge ime ve ee suie from iffere pois of view of iffere uors. For emple cneil 97 s erive e isriuio of e iegrl fuciol W T g{ X } were T is e firs pssge ime o e origi i geerl ir e process wi X = g. is rirry fucio. lso Igler 965 cneil Scc 973 ve ee sow umer of clssicl ir e processes upo kig iffusio limis o sympoiclly pproc e Orsei Uleeck O.U.. y properies suc s firs pssge ime o rrier sorig or reflecig loce some isce from iiil srig poi of e O.U. process e rele iffusio process e rele iffusio process suc s e cse of e firs pssge ime of Wieer process o lier rrieris close form epressio for e esiy ville is iscusse i Co iller 965. lso oers suc s Krli Tylor 98 Toms 975Fereee 98 Tuckwell W 984l-Eie 4 ec.ve ee iscusse e firs pssge ime from iffere pois of view. I priculr Toms 975 escries some me firs pssge ime pproimio for e Orsei Uleeck process.tuckwell W 984 ve suie e firs-pssge ime of rkov process o movig rriers s firs-ei ime for vecor wose compoes iclue e process e rrier. lso l- Eie 4 s iscusse e prolem of fiig e momes of e firs pssge ime isriuio for e ir-e iffusio e Wrig-Fiser iffusio processes o movig lier rriers usig e meo of pproimig e iffereil equios y ifferece equios. I is pper we cosier e ir e iffusio process suy e firs pssge ime for suc process o geerl movig rrier. ore specificlly e mome pproimios re erive usig e meo of ifferece equios. 4 Pge

2 Firs Pssge Timemome pproimio For Te Bir De.. II. FIST PSSGE TIE OENT PPOXITIONS Cosier e ir e iffusio Process X : wi ifiiesiml me vrice srig some > were re e rif e iffusio coefficies respecively. lso X : is rkov process wi se spce S sisfies e Io socsic iffereil equio X X X W Were W : is sr Wieer process wi zero me vrice. ssume e eisece uiqueess coiios re sisfie Cf. Gim Skoroo 97. Le : e geerl movig rrier equio suc wi. Or equivlely Noweoe e firs pssge ime of process X o geerl movig rrier y e romvriles if{ : X } T wi proiliy esiy fucio g = - p Here p is e proiliy esiy fucio of X coiiol o X = = 3 e e - mome of e firs pssge ime T i.e. Le E = 3 3 T I follows from e forwr Kolmogorov equio e - mome of T mus sisfy e oriry iffereil equio 4 Or equivlely 5 Were wi pproprie oury coiios for =3.Noe. re e firs erivives of wi respec o Now rewrie e equio i 5 we oi 6 Le e e ifferece operor. Te we efie e firs orer ifferece of Cf. Kelley peerso s follows: 5 Pge

3 Firs Pssge Timemome pproimio For Te Bir De.. 6 Pge Noe equio 6 c e pproime y 8 By pplyig equio 7 o equio 8 we ge : 9 Now we will use e mri eory o solve e iffereil equio efie i equio 9. If we le Te we ge Were 4 3 Now le Tis imply pply o equio we ge 3. I Were Iis e ieiy mri is e zero mri. Tus e soluio of e sysem of equio i 3 is e give y

4 Were D = [ ] j ij Firs Pssge Timemome pproimio For Te Bir De.. i is e igol mri wi eries e D * 4. 5 ij j i Oerwise i j ij is e mri wi eries ij i l j i Oerwise j i j i 6 Noe e mri B e were e B D 3 B B I B! 3! B is efie y... Tis series is coverge sice i is cucy operor of equio.6 Cf. Zeifm 99. III. CONCLUSION I coclusio e vge of is ecique is o use e ifferece equio o pproime e oriry iffereil equio sice i is e iscreizio of e ODE. lso e sysem of e soluios i equio 4 gives eplici soluio o e firs pssge ime momes for e ir e iffusio processo geerl movig rriers. Tis icreses e ppliciliy of e iffusio process i socsic moelig or i ll re of pplie proiliy eory.so i cse of movig lier rier we c i cse of movig cos rrier we c we go e sme resul s i l-eie 4. EFEENCES []. B.. l-eie Firs-pssge ime mome pproimio for e ir-e iffusio process o movig lier rriers. J. S. &g. SysemsVol.7 4 No []. D.. Co H. D. iller Te eory of socsicprocesses. eue Loo 965. [3]. D. Drlig. J. F. Sieger Tefirs - pssge prolem for coiuos rkov process... Sis [4]. J.Duri Boury-crossig proiliies for e Browi moio Poisso processes eciques for compuig e power of e Kolmogorov-Smirov es. J. ppl. Pro [5]. W. J. Ewes emicl Populio Geeics. Spriger-Verlg Berli 979. [6] B. Fereee Te ge pproimioo oesie Browi ei esiies. Z. Wrsceilickeis [6]. D. L. Igler Limiig iffusio pproimio for e my server queue e repirm prolem. J. ppl. Pro [7]. S. Krli H.. Tylor Seco Course i SocsicProcesses. cemic press. New ork Pge

5 Firs Pssge Timemome pproimio For Te Bir De.. [8]. W.G. Kelly.C. Peerso Differece Equios : Iroucio wi pplicios. cemic Press New ork 99. [9]. D.. cneil Iegrl fuciols of ir e processes rele limiig isriuios... Sis []. D.. cneil S. Scc Cerl limi logues for rkov populio processes. J.. Sis. Soc. B [].. U. Toms Some me firs pssge ime pproimios for e Orsei Uleeck process.j. ppl. Pro []. H. C. Tucwell F... W Firs-pssge ime of rkov processes o movig rriers. J. ppl. Pro.Vol [3].. I. Zeifm Some esimes of e re of covergece for ir e processes.j. ppl. Pro Bsel. l-eie. Firs-Pssge Time mome pproimio For Te Bir De Diffusio Process To Geerl movig Brrier. Ieriol Jourl of emics Sisics Iveio IJSI vol. 5 o. 9 7 pp Pge