b levels ( a b 0) and first falling moment of the process into the delaying barrier

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1 Ordu Ünv Bl T Drg, Cl:6, Sı:, 6,-5/Ordu Unv J Sc Tch, Vol:6, No:,6,-5 THE AACE TRANSFORM OF A BOUNDARY FUNCTIONA OF THE SEMI- MARKOVIAN RANDOM WAK ROCESS WITH TWO DEAYING BARRIERS Slhn MADEN * Bhr G SHAMIOVA Dprmn of Mhmcs, Fcul of Ars nd Scncs, Ordu Unvrs, 5, Ordu, Tur Dprmn of Appld Mhmcs nd Crncs, Bu S Unvrs, AZ8, Bu, Arjn ABSTRACT In hs su, sp procss of sm-mrovn rndom wl wh dlng rrrs nd lvls nd frs fllng momn of h procss no h dlng rrr -lvl,, r mhmcll consrucd, n hs cs whn h rndom wl hppns ccordng o h plc s dsruon I s gvn smpl form of h plc rnsformon of h dsruon of h rndom vrl Also h smpl formuls for xpcon nd vrnc of r ond h mns of hs plc rnsformon AMS Sujc Clssfcon: 6A, 6J5, 6G5 Kwords: Sm-Mrovn rndom wl procss, plc dsruon, dlng rrr, xpcd vlu, vrnc, plc rnsformon mdn55@mncom

2 SMADEN, B G SHAMIOVA INTRODUCTION In rcn rs, rndom wls wh on or wo rrrs r ng usd o solv numr of vr nrsng prolms n h flds of nvnor, quus nd rll hors, mmcl olog c Mn good monogrphs n hs fld xs n lrur s rfrncs Borovov 976 Flr 968 nd c In prculr, numr of vr nrsng prolms of soc conrol, quus nd rll hors cn xprssd mns of rndom wls wh wo rrrs Ths rrrs cn rflcng, dlng, sorng, lsc, c, dpndng on concr prolms hnd For nsnc, s possl o xprss rndom lvls of soc n wrhous wh fn volums or quung ssms wh fn wng m or sojourn m mns of rndom wls wh wo dlng rrrs Furhrmor, h funconng of sochscs ssms wh spr qupmn cn gvn rndom wls wh wo rrrs, on of hm s dlng nd h ohr on s n p rrr I s nown h h mos of h prolms of soc conrol hor s ofn gvn mns of rndom wls or rndom wls wh dlng rrrss Rfrncs - c Numrous suds hv n don ou sp procsss of sm-mrovn rndom wl wh wo rrrs of hr prccl nd horcl mpornc Bu n h mos of hs suds h dsruon of h procss hs fr dsruon Thrfor h ond rsuls n hs cs r cumrsom nd h wll no usful for pplcons Borovov 975 Borovov 976 Flr 968 Khnv 98 nd In h followng, sm-mrovn rndom wl wh dlng rrrs nd lvls,, h hs dnumrl s spc s consrucd nd mn prol chrcrscs of oundr funconl of hs procss r consdrd

3 Th plc Trnsform Of A Boundr Funconl Of Th Sm-Mrovn Rndom Wl rocss Wh Two Dlng Brrrs CONSTRUCTION OF THE ROCESS Suppos h {, :,,} s squnc of dncll nd ndpndnl dsrud rndom vrls, dfnd on n prol spc, F,, such h s r posv vlud,, { } Also, h numrs,, nd r gvn Also l us dno h dsruon funcons of w nd w { w }, F x { w x}, R, x R B h mns of hs rndom vrls, w cn consruc h followng procss, f,,,, Ths procss forms sp procss of h sm-mrov rndom wl Now, w cn consruc h dsrd sochsc procss n h followng w w shll cll h sp procss of h sm-mrovn rndom wl wh dlng rrrs lvls whr, f,,,mx,,, mn us dno h frs fllng momn of h procss h lvl In hs cs, w cn wr nf : Also w g whn for vr nd no h dlng rrr 5

4 SMADEN, B G SHAMIOVA 6 No h s mporn from scnfc nd prccl pon of vw nd s n nporn oundr funconl of h procss Ths rndom vrl pl n nporn rol n solwng of mos prol prolms rsng n conrol of rndom lvls of socs n wrhous whch s funconng ccordng o h procss For hs rson, h consdron wh dld of rndom vrl sms vr nrsng from scnfc nd prccl pon of vw THE AACE TRANSFORMATION OF In hs scon, l us clcul h plc rnsformon of rndom vrl, s h plc rnsformon s h mos mporn chrcrsc of rndom vrl B h ol prol formul, w cn wr, mn ds s s W dno, G Thn h quon hs h followng form, d s G ds G s whr,,

5 Th plc Trnsform Of A Boundr Funconl Of Th Sm-Mrovn Rndom Wl rocss Wh Two Dlng Brrrs ~ us dno h plc rnsformon of h funcon G G Thus, w cn wr I s ovousl ~ G G d d, whr E xp In smlr w, h quon cn wrn n h followng form G G d G For h rrr dsrud rndom vrl, h quon cn solvd wh mhod conscuv pproch Bu such dcson s no usful for pplcons Thrfor w shll solv, for xmpl, f hs h plc dsruon x x, x x, x whr nd In hs cs h quon cn wr s G ~ ~ G ~ G 7

6 SMADEN, B G SHAMIOVA 8 G ~ Now, w shll followng rplcmn ~ G Thn h quon hs h followng form B consdrng hs ngron quon, w cn wr h followng dffrnl quon on :, 5 whch hs h soluon с с, 6 whr,, u h roos of h chrcrscs quon of h dffrnl quon 6, h s, From h ngrl quon w fnd h followng oundr condons:,

7 Th plc Trnsform Of A Boundr Funconl Of Th Sm-Mrovn Rndom Wl rocss Wh Two Dlng Brrrs 9 Whnc w hv h followng ssm of h lnr lgrc non-homognous quons c, c с c Afr som clculons w g с nd с Susung nd c c n 6 w fnd h Applng h ol prol formul, w hv d

8 SMADEN, B G SHAMIOVA 5 d d Now, w shll fnd E nd Vr B h dfnons of xpcd vlu nd vrnc nd h propr of h plc rnsformon, w nown h E nd Vr Thrfor, w cn wr for h xpcon of h rndom vrl E E Also w g h followng xprsson for h vrnc of : Vr

9 Th plc Trnsform Of A Boundr Funconl Of Th Sm-Mrovn Rndom Wl rocss Wh Two Dlng Brrrs 5 Vr и

10 SMADEN, B G SHAMIOVA 5 E In hs xprssons f, lmng s w g E E nd E Vr Vr REFERENCES AA Borovov, On h rndom wl n h srp wh wo dlng rrrs, Mh Zm 7,, , 975 AA Borovov, Sochsc rocsss n Quung Thor, Sprngr-Vrlg, Nw Yor, 976 Fllr, W,An Inroducon o rol Thor nd Is Applcons, Vol I, Wl, Nw Yor 968 Khnv, TA, Dsruon of sm-mrov Wl wh wo dl scrns, Som quson of h hor of sochsc procsss, Collc sc Wors, Kv, 6-98 Khnv, TA, Ünvr, I, Th su of h lvl ro crossng m of sm-mrovn rndom wl wh dlng scrn, Tursh J of Mhmcs, Khnv, T A, Ödmr, H, nd Mdn, S, Clculng h prol chrcrscs of oundr funconls of sm-connuous rndom procss wh rflcng nd dlng scrns, Appl Sochsc Modls D Anl,, 7-998

11 Th plc Trnsform Of A Boundr Funconl Of Th Sm-Mrovn Rndom Wl rocss Wh Two Dlng Brrrs oov, VI Oh h smpoc of dsruons n wo-sdd oundr prolms for rndom wls dfnd on Mrov chn, S Adv Mh, o, Mdn, S, On h sm-mrovn rndom wl procss wh rflcng nd dlng rrrs, Tursh Journl of Mhmcs 5, 6-7 Nsrov, ТH rocsss of Sm-Mrov Rndom Wl, EM, Bu, 98 Khnv, T A, Unvr, I nd Mdn, S, On h sm-mrovn rndom wl procss wh wo rflcng rrrs, Sochsc Anlss nd Applcons, 95, rhu, NU, Sochsc Sorg rocsss, Nw Yor, Hdlrg: Sprngr-Vrlg, 98 Spr, F, rncpls of Rndom Wl, Vn Nosrnd, rncon, 969 Ünvr, I, On h oundr funconl of h sm-mrovn rndom wl wh ngv drf nd posv jumps nd wo dlng scrns, Jour of Ins of Mh & Comp Sc Vol, No

CMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk

CMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk CMPS 2200 Fll 2014 Grps Crol Wnk Sls ourtsy o Crls Lsrson wt ns n tons y Crol Wnk 10/14/14 CMPS 2200 Intro. to Alortms 1 Grps Dnton. A rt rp (rp) G = (V, E) s n orr pr onsstn o st V o vrts (snulr: vrtx),