The McKean martingale for the homogeneous $R-D$-systems

An analog of the McKean martingale is constructed for branching processes with a continuous phase space.

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Datum:1993
Hauptverfasser: Safaryan, R. G., Сафарян, Р. Г.
Format: Artikel
Sprache:Russisch
Englisch
Veröffentlicht: Institute of Mathematics, NAS of Ukraine 1993
Online Zugang:https://umj.imath.kiev.ua/index.php/umj/article/view/5936
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Назва журналу:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
_version_ 1860512160979877888
author Safaryan, R. G.
Сафарян, Р. Г.
Сафарян, Р. Г.
author_facet Safaryan, R. G.
Сафарян, Р. Г.
Сафарян, Р. Г.
author_sort Safaryan, R. G.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2020-03-19T09:21:13Z
description An analog of the McKean martingale is constructed for branching processes with a continuous phase space.
first_indexed 2026-03-24T03:24:23Z
format Article
fulltext Y,UK519.21 P. r. Ca<t,apJ1H, KaHn .. q>H3. -MaT. HayK (EpeBaH. nea. HH-T) MAPTUHr AJI MAKKIIHA, CBH3AHHLIII C O,UHOPO,UHhIMU R-D-CUCTEMAMU T he analog of the McKean martingale is constructed for branching processes with a continuous phase space. nooynoBaHO am1.11or MaPTHHra.na MaKKiHa llJUI ri.nJI.RCTHX npmieciB 3 HenepepBHHM q>a30BHM npoc­ TOpoM. 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R-D-CHCTeMy (1) 6yn;eM Ha3bIBaTb OAHOPOAHOA (TO"!Hee, O1vmpon;1rnA no npocTpaHCTBy), ecJIH f(x, u) He 3<1.BHCHT OT XE £, f(x, 11) = /(u), (5) a x, - npou;ecc C He3aBHCHMblMH npHpa [J.lemp!MH co 3Ha"leHHJIMH B JIHHeAHOM npocTpancTee E. 3AeCb Mb! paCCMOTPHM CJiy"lait, KOl"A8 £ = R1 - BeUleCTBeHIIUJI np51M8JI, e = fl - 6opeJieBCKaJI cr-anre6pa, © P. r . CA<I>APRH, 1993 ISSN 0041-6053. YKp. ~,am . JKJP" ·· /993. 111. 45, N" 9 1315 1316 P. r. CA<l>APRH Lu(x) = au'(x) + bu"(x) + J [u(x + y)- u(x) - yu'(x)] IT (dy). (6) b ~ O. IT(dy) ~ O. IT{O} = 0, J y2IT(dy) < 00 Eo11ee Toro. OTHOCHTeJibHO Mepbl IT npe,[1,nOJIO)l(HM BbinOJIHeHHbIM ABYCTOpOI-IHee yCJIOBHe KpaMepa: Hatt,[l,yTC.ll 'IHCJia s- ::;; 0 s si-. s- < s+. nurne. 'ITO 00 J (1 A l)esy ITtfy) < 00 (7) npH s- < s < s+. B Ka'leCTBe MHO)l(eCTBa f:J paCCMOTpHM MHO)l(eCTBO ,[l,Ba)l(,[l,bl HenpepbIBHO ,[l,Hq:>cpepeHQHpyeMhIX q:>yHKQHtt, HC'le3a!OLQHX Ha ± 00 BMeCTe co CBOHMH nepBblMH ABYM.ll npOH3B0,[l,HblMH (CM. no noMY llOBOAY 3aMe'laHHe 2 B [ 1 ]). ITpH 3TOM TPHBHaJibHO BbllIOJIHeHO yCJIOBHe (9) H3 [ 1 ]. KaK noKa3aHO B [2]. pewenHe ypamfeHH.ll (1), (2) c f BH,[l,a (3) H OS q S 1 npeACTaBHMO B BH,[l,e V(I) ui{:r) = lEx TI g(xk(t)), k=l rAe lEx-YCJIOBHoe cpeAHee npH yc110BHH X(O) = x; X1(t), X2(t), ... -ne3aBHCHMbie KOllHH npoQecca X(t); V(t) - 'IHCJIO 'laCTHQ B BeTB.llLQeMC>I npoQecce, onpeAe­ m1eMOM q:>yHKQHett f(u) BHAa (3) (CM., HanpHMep, [3]) H He :k'lBHCRLQeM OT nocJieAO­ BaTeJibHOCTH Xit), k ~ 1. EcJIH )Ket HMeeT BRA (4), TO perneHHe ypaBHeHH>I (1), (2) c g ~ 0 npeACTaBHMO B BHAe (1) (8) rAe M ,(dy} - TaK Ha3blBaeMbltt BeTBHLQHitCH npoQecc Hp)l(HHbl C MHO)l(eCTBOM THilOB E. PeweHHe (7) eAHHCTBeHHO B HeKOTOpoM KJiacce K . ll0Apo6uoe OnHcanHe KOTOporo COAep)l(HTCH s [ 1 ]. MaKKHH B [ 4] OTMeTHJI ,llJIR Cny'laR ,no npH nI06oM s npo1~ecc Lu = lu" 2 • f(u) = u2 - u, v(1) 2 ':t _ -1 ~ -sX,(1)-s 1/2 <11 1 - e L, e k=I (9) 06pa3yeT MapTHHraJI, H HCnOJib30BaJI 3TO o6cTO>!TeJibCTBO AJIH AOKa3aTeJibCTBa TeO­ peM nma KonMoropoBa - ITeTpoBCKoro - ITHcKyttosa. Ero KOHCTPYKUH.ll TPHBH­ aJibHO nepeHOCHTC>I c CHTyaQHH (9) Ha (3), (6). (7), a HMeHHO, MapTHiffaJIOM >IBJIHeT­ cg cny'la8Hbltt npouecc V(I) ":t --0.1 ~ sX,(1)-L(s)I <11 1 = e L, e , ·k=l ISSN 0041 -6053 . YKp. Mam. xypn., 1993, m. 45, N" 9 MAPTHHrAJlMAKKHHA ... 1317 - L(s) =as+b.?+ J (e' 11 -l-sy)TI(dy), a= c [ L,~=O kPt - 1] (npeanonaraeTC.ll , 'ITO a< oo ). UeJib HaCTOJl(Uett CTaTbH - ITOCTpDHTb ananorH'IHbitt MapTHHraJI t).JIJI R-D-CHC­ TeMbl (1), (2), (4). HanoMHHM, 'ITO M i(dy) - npou;ecc lfp)KHHbl H3 (7), a - napa­ MeTp H3 (4). Teope1>1a 1. B yc11oeuJ1.X (4) - (7) npoijecc - ~1 = e--0.1-L(s')I J Mr(dy)/11 npu s- < s < s+ 06pa3Jem M.apmuHW/1. aoKa3Dllle/lbClll60. TIOKa)KeM CHalfaJia, 'ITO (10) (11) tJ.JI.ll Bcex s E (s-, s+). C 3TOtt u;enb!O o6<>3Ha'IHM qepe3 T1 no.nyrpynny onepaTopos B 6aHaxOBOM npocTpaHCTBe .B, OObl'IHblM o6pa30M DOp0)KtJ.eHHYIO MapKOBCKHM npo­ u;eCCOM xi' T. e. T/g(x) = lExg(Xr), g E .B, XE R 1, H ITOKa)KeM, 'ITO (12) KaK y)Ke OTMelfaJIOCb, B (1] DOKa3aHO, 'ITO npaBaJI lfaCTb (8) JIBJI.lleTCJI et).HHCTBeH­ HblM B K Jiacce K peweHHeM ypasneHHJI (1), (2). TaM )Ke noKa3aHo, 'ITO npaBa.ll lfaCTb (8) ytJ.OBJieTBOp.sieT HeJmnettHOMY HHTerpaJibHOMY ypaBHeHHIO I U1 = T1g + J T1 - slf(us)]ds , 0 (13) B KOTOpoM Tr _ slf(us)] cneayeT ITOHHMaTb KaK pe3yJibTaT aettCTBHJI onepaTopa Tl_ s Ha cpyHKQHIO /,(~) = /(u,(x)). He orpaHH'IHBaJI OOIUHOCTH, 6yaeM C'IKraTb cpyHKQHIO g H3 (12) Heo-rpuu;aTeJib­ HOtt. )lm1 'A.> 0 o6o3Ha'IHM H 3aMeTHM, 'ITO u~(.\:) = -logJExexp{-Al M1(dy)g(y)} - JEX J M1(dy)g(y) = lim {- u~(x). t...J.O /1, CornacHo (13) HMeeM I u";-(x) = AT1g(,x) + J T1 _slf(u~(x))](x)ds. 0 KpoMe Toro, s [l] ycTaHOBJieHO, 'ITO O :S Ur(,x) :S eaJTlg(x), H, CTaJIO 6b1Tb, ISSN 0041-6053. YKp. Mam. JKyp11. , 1993, m. 45, N' 9 (14) (15) 1318 P. r . CA<l>AP}IH (16) O6o3Ha'IHM JieBbie 'laCTH (12), (14) qepe3 v,(x). HepaseHCTBO (16), B 'IUCTHOCTH, nOKU3bIBaeT, 'ITO u~(x) ~ 0 paBHOMepHO no XE R1, 0 ~ s ~ t J].J151 scex t > 0. OTCIO.lla H H3 (4) BbITeKaeT, 'ITO f(u~(x)) I A~ av/x) TaK)Ke paBHOMepHO no x e R 1, 0 ~ s ~ t .llJ151 scex t > 0. ITocne tJ.eJieHHa o6eHx 'IUCTett (14) Ha A H nepexo.lla K npe.lle.rcy npH A J, 0 HaXO.llHM I v 1(x) = T1g(x) + a J T, _,v,(x)ds. (17) 0 Ho ypasHeHHe (17) HMeeT e.llHHCTBeHHOe paBHOMepno orpaHH'le1moe no XE R 1 H JIOKaJihHO orpaHH'lenuoe no t ~ 0 pewenHe v,(x) = e01T,g(x). PasencTBO (12) JJ.0KU3aHO. 11cnOJib3ya npetJ.eJibHbltt nepeXO.ll no MOHOTOHHbIM nocJieJJ,OBaTeJibHOCTaM cpyu- KQHtt H3 13 , no.rcyqaeM, 'ITO (12) cnpaBeJJ,JIHBO JJ,JHI scex 1-1eOTpm.1;aTeJibHbIX .l3-H3Me­ pHMblX q>yHKQHtt (npH 3TOM, pa3yMeeTC}I, .llJ151 o6eHx 'IUCTett (12) .llOITyCKaeTCR 31-Ia- 'leHHe + 00 ). Tenepb, 'ITOObl y6etJ.HTbC51 B cnpaBeJJ;JlHBOCTH (11 ), .llOCTUTO'IHO npose­ pHTb KOHe'IHOCTb cpetJ.nero IExexp{sX,} npH_ s- < s < s+. Ho, KaK CJie.llyeT H3 q>op­ MYJibI JlesH-XHH'IHHa H npu1-1u;Hna ananHTH'leCK0ro npo.ll()Jl)KeHH51, IEx exp{sX,} = = e5xetL(I) .llJ151 TeX s, .llJ151 KOTOpbIX onpe.lleJieH CHMBOJI L(s) onepaTopa L BHJ).a (6), T. e . .llJlH s- < s < s+. 3THM .llOKa3aHO He T0JibKO COOTHOUie1me (11), HO H YCTaH0BJieHO paseHCTBO 00 JEX J M,(dy) esy = esxerA+L(s)I_ (18) Tenepb H3 (18) JierKO BbITeKaeT YTBep)KtJ.eHHe TeOpeMbI. ,UettCTBHTeJibHO, nyCTb noTOK cr-anre6p ~, nopo)KtJ.eH npou;eccoM Hp)KHHbI M,(dy). TaK KaK npou;ecc M,(dy) - 0.llHOpo.llHbltt MapKOBCKHtt H BeTBal..l\HtlCH, TO J).J151 BC ex X E RI H Tex HeOTpHu;aTeJibHbIX .l3-H3MepHMbIX q>yHKQHit g, JJ.J151 KOTOpbIX 00 JEX f M,(dy)g(y) < oo, V x e R1, ~ 0. OTCIO.lla c y<leTOM (10), (18) npH r < s < s+ HMeeM IE(~ I~)= e-a(t+u)-L(s)(t+u)IE [ooJ M (dy)esy 1~] = s t+u I X t+u I ~ 'ITO H TpeOOBaJIOCb .llOKa3aTb. ISSN 0041-6053. YKp ~WIil. )/(Y/J"·· 1993, 111. 45, N9 9 M , ?THHrAJl MAKKHHA 1319 B 3aKJ11Ol{e1-me ycTaHOBHM ow-mpor.lHOCTb npouecca Hp)KHHbI M1(.) no npoc­ TpaHcrny AJUl CHCTeM BHAa (1 ), (2), ( 4). (6). TeopeMa 2. B ycAoouRx (4) , (5), (6) ycAO011oe pacnpeoe11em1e Mepbt M 1(dy) 11p11 ycAoouu X0 = x coonaoaem c yc11O0HblM pacnpeoe11e1me.M Mepbt M(dy •- x) npu yc11oouu X0 = 0. /(OKll3ll l1le.AbC11l60. )locTaTOl{HO nOKa3aTb, l{TO (19) I AJUI BCex x E R • g E J.9, g ~ 0. 3aq:rnKCHpyeM x0 E R 1 H o6o3ttal{HM 111 (x) = ui(x + x0), g (x) = g(x + x0). rAe g E J.9 , g ~ 0, u1(t) onpeAeJ1etto no (8). KaK noKa3a.t lO B [l] , cpyHKUIUI ll1(t) yn,oB.JiernopjjeT ypaBHe11111O (1), (2), a H3 (5) cnen,yeT, l{TO If cpyHKUHR iii (.r) yn,OBJJeTBOpjjeT ypaB11e1-1111O (1) C Hal{aJlbHbIM ycJ1O1meM 110 (x) = g (x). Cne11.0BaTeJ1b110 [I) , no11arM 3D,eCb X = o. nOJTyl{ aeM ( 19) C Xo BMeCTO X. TeopeMa AOKa3atta. B Kal{eCTBe 1-1enocp etKTBe1 111o r o CJieD,CTBlrn TeopeMbl 2 OTMeTHM CJ1en,y1oll(ee YTBep)K t.ler-m e. TeopeMa 3. n _vcrllb ll1(X) - pe11te1-1L1e 3GOG'lll (1 ). (2) (3 )'C/IO0ll JIX (4) - (7) C g(x) = e5X, r < s < s+. T02oa 2oe U1(t - Cf), c = aJs + L(s)ls. /(oKa3ame.;1bC11wo. Cor11ac110 (8) , ( 19). (2 l) HMeeM 11 1(r+ cl) = - log lEx + ctexp{- l M1(cly)c''Y} = l{TO u /~OK,l3bmaer (20). (20) (21) l-h1e 11110 3TO y ruep>Kn,e1111 e (nep11ee. ero AHCKpen11,1H a 11a.J 101·) u c no.111,3ona.11 MaKKHII B 14J. I. C,u/){IJ'fllt P . r . R- D CIIC I e>IIJ II lll'Tl\l!ll Ull'Cl! 11po11ccc 1,1 llp}K Hlll,I // YKp. "", . }K y p11 . - I ')'II . - 43, '<0 2. - C. 162 - 167. 2. C{J(pap11// P. r . Be I n >11 11m·c>1 11 11, ) ><1>Y3H011111,1 e 11prn1cn: 1,1 H c 1-1 c 1crn,1 1tml><l>c pc111u1a.r 11,111-,1 x y pa n- 11rn11f.l pcaKlllUl-/llHl">< l>YJll>I // Mar. c6. - 1987. - 134. l\04. - C. 530 - 545. J. Ce,i<ir1111,11//oo E . A . Be rBH1111 1ccl! 11po 1tt·cc 1,1 . - M .: I l ayK a , 197 1. - -B6 c. 4 . M ,K ca,1 I I . P. i\pplicalion n f Brown ian Mo tion lo the Equati on of Ko lmognrnv- 1'-:trovskii - l'i sku ­ nov // Commun, Pun: and i\ppl. ll'IJ tlt . - 1'17.'i . - 28. - l' .323 - :n I. ISSN 0041 -6053 . .Yhp. ·""111 . A-.\ '/!i t . • /911.i . 111 . ./5 . !\''' 'i 0129 0130 0131 0132 0133
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spelling umjimathkievua-article-59362020-03-19T09:21:13Z The McKean martingale for the homogeneous $R-D$-systems Мартингал Маккина, связанный с однородными $R-D$-системами Safaryan, R. G. Сафарян, Р. Г. Сафарян, Р. Г. An analog of the McKean martingale is constructed for branching processes with a continuous phase space. Побудовано аналог мартингала Маккіна для гіллястих процесів з неперервним фазовим прос­тором. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5936 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1315–1319 Український математичний журнал; Том 45 № 9 (1993); 1315–1319 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5936/8555 https://umj.imath.kiev.ua/index.php/umj/article/view/5936/8556 Copyright (c) 1993 Safaryan R. G.
spellingShingle Safaryan, R. G.
Сафарян, Р. Г.
Сафарян, Р. Г.
The McKean martingale for the homogeneous $R-D$-systems
title The McKean martingale for the homogeneous $R-D$-systems
title_alt Мартингал Маккина, связанный с однородными $R-D$-системами
title_full The McKean martingale for the homogeneous $R-D$-systems
title_fullStr The McKean martingale for the homogeneous $R-D$-systems
title_full_unstemmed The McKean martingale for the homogeneous $R-D$-systems
title_short The McKean martingale for the homogeneous $R-D$-systems
title_sort mckean martingale for the homogeneous $r-d$-systems
url https://umj.imath.kiev.ua/index.php/umj/article/view/5936
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