Luận văn Xích Markob

. Chu'dngIII : MARTINGAL I/Trungblobcodi~uki~ndolvdiphanho~chvadolvdiham. 1.l/MOts6khiiini~mv~d~ihiQngngi1unhH~n. Chom<)tkhonggianXac sua't(Q,ffi,j.1) - Hamf xacdinhtrenQ va la'ygiatri trenR gQila d(;lilu'<jngng~u nhienne'uf dodu'<jc -M(f) == ffdJ.1:Trungblnhcuaf Q - M(I /R)= 1. fIdJI :Trungblnhcuaf vdi di€u ki~nR JI(R) R 1.2/Trungblob codi~uki~nd6ivOiphanho~ch Binhnghia1.2.1: Chot~pkhacr6nglaQ. HQcact~pconcuaQ la 91={Rj,R2,...,Rn,...GQila m<)tphanho(;lchcuaQ ,ne'u: (1) Ri n Rj =1>ne'u i :;t:j ( 2) u Rn=Q.n Binh nghia1.2.3:Cho2 phanho(;lchtrenQ Ia 91va 6l',ne'u: mQiph§ntU' cua91d€u la h<jpcuacacph§ntU'cua~ta n6i91ch((atrong67.Ky hi~u: 91c67. Ghi chu:Ne'u91c ~ta n6i91thohdn~, ngu'<jcl(;li~min hdn91. Binh nghia 1.2.3:Cho 2 phanho(;lchtrenQ la 91va ~. Giao cua91va 61'la hQcact~pxacdinh nht(sail: 91C8)r;? ={Ri /\ Sj /Ri E 91,Sj E r;? Ghi chu: +Giaocua2 phanho(;lchla 1phanho(;lch +C6th6mar<)ngiaochon phanho(;lch Trang19 Dinh nghia1.2.4:ChokhonggianXac su(t (0,91,/.1),f la d~ilu'9ngng~u nhien, ky hi~ubdiM(f/91),xacdinhnhu'sail: Vdi :OJ E Ri E 9i: M ( f 191)(0))=M(f I Ri) +Nh~nxet: - M ( f I 91)la h~ngtrenm6iphffntItcua91 - ffdfJ =fM(f /9i)dfJ vdiRi E91 RI Ri - Ne'u91={a} th'iM(f/~H)=M(f) M~nhd~1.2.5:trungblnhcodi€u ki~ntrenphanho~chcocacHnhch(t sail: 0) M(M(f/Q9)/91))=M(f/91)ne'u91c 67 (2)M(M(f/91)=M/f) (3)Ne'ug la h~ngtrenm6iphffntItcua91 vaf la d(;lilu'9ngng~unhien saochom(fg)t6nt(;lithl: +M(gl91)=g hcc +M(gf/91)=g M(f/91) CM: Ta chicffnchungminh0), cactru'ongh9Pkhacdaubchob(;lndQc. Gia sIt91c67. Do M(M(f/d7)191) va M(f/91)lag h~ngtrencacphffntIt cua 91,nenchi cffnchungminh: fM(M(f / QY)/9i)dfJ =fM(f /9i)dfJV Rj E 9i Rl Rj Ta co : fM(M(f /67)/9i)dj.l= fM(f /67)dfJ RJ Rj = L fM(f /67)dj.l,vdiSi E67(do91c d7) i S;cRj =L f fdj.l = ffdj.l = fM(f /9i)dj.l i s;cRj Rj Rj Trang20 1.3/Trungblobc6.di~uki~nd6ivoiham. Blnh nghia1.3.1: Cho g la d(;tilu9ngng~unhienco t~pgiatfi dSmdU9C:{Cl,C2,...,Ci , } D~tRi ={m/g(m)=cJ, '\Ii Ta co :91={Rl ,R2,...,Ri,...}la mQtphanho(;tchcuaQ va dU9CgQila phan ho(;tchsinhrabdig,ky hi~u91g. ~ Blnh nghia1.3.2: Cho g la d(;tilu9ngng~unhienco t~pgiatridSmdU9C. Trungblnhcuaf voi di€u ki~ng la: M(f/g)=M(f/91g) Blnh nghial.3.3: Chocacd(;tilu9ngng~unhienfo,fl,...,fn.fn+l.Trungblnh cuafn+lvoidi€u ki~nfo,fl,...,fnla: M(fn+/fO I\f11\...l\fn)=M(fn-l/9{fo (8)9{fi...(8)9{fn) Blnh Nghia 1.3.4:Daycac d(;tilu9ng ng~unhien {fn}gQila dQcl~pnSu: P(fn+l E A / fa 1\...1\fJ =P(fn+lE A),VA,'\Ii Nh~nxet: NSu day{fn}dQcl~pthl : M(fn+l / fa 1\...1\fn) =M(fn+l)'V n Blnh Nghia 1.3.6:NSu P(fn EA) khongphl,lthuQcn, VA ta noi day {in} cophanph6id€u. 2.1CacthuQcHnhcuaMartingal. Blnh nghia2.1:ChokhonggianXac sua't(Q,91,~L)va khonggiancac tr(;tngthaiS la t~pdSmdlt9ccacs6thl,l'c,cho{in}la daycacd(;tilu9ng ng~unhienva {91n}ladaytangcuaphanho(;tchcuaQ l~p(fn,91n)gQila Martingal,nSu: (l)M(lfnl LOO,Vn (2) fnla h~ngtrencacph~ntitcua91n (3)M(fn+lI 91n)=fn,Vn. Trang21 Ghi chl1: -Di€u ki~n (3) co nghlala trungblnhcuafn+lvdi di€u ki~ntrenthl m6i gia tricuafnchinhb~ngtn. - NSu di€u ki~n(3)thaybdi :M(fn+l,I iRn)~fn , Vn Ta noi (fn,iRn)latrenMartingal. - NSu di€u ki~n(3) thaybdi : M(fn+lliRn~fn, Vn Ta noi (fn,iRn)ladu'diMartingal. - Cho daycacd(;lilu'<jngng~unhien{filL d~t:91n=91/0/\.../\ 91/0 .C~p (fn,iRn)laMartingalkhivachikhi : M(fn+llfo /\.../\fn)=fn,Vn Ta kyhi~uddngianla {fn} - Cho (fn,iRn)laMartingal, d~t ,iR*nla tru'ongbore1 sinh bdi phan ho(;lch,iRn Ta co : (fn"iR*n) la mQtquatrlnhng~unhien. Ta xet vai vi dl.lv€ Martingal Vi dl).l:Cho {Yn}la daycacd(;lilu'<jngng~unhiendQcl~pvdi t~pgia tri de-ill du'<jc. D~t: Sn=YO+ Yo + ...+Yn Ta co :M(sn+11So /\.. ./\sn)=M(sn+Yn+/sO /\.../\sn) =M(Yn+llYo /\.. ./\Yn) +Sn =M(Yn+l)+Sn Do do {sn}la martingalkhi va chikhi :M(Yn+l)=0 ,Vn Ne-uYns6 ti€n dat cu'<jcthudu'<jcua ngu'oidat cu'<jcsail van bai thli n. Sn:T6ngs6ti€n codu'<jcsailvanbai thlin. {Sn}la Martingalkhi va chi khi s6ti€n trungblnhthudu'<jcsailm6ivan baiphaib~ngO. Trang22 - NSu{sn}laMartingalthltac6: M(Sn+llSO/\../\Sn)=Sn,Vn. C6 nghlala t6ngs6ti~nmongd<;5ic6du'<;5cd m6i thaidiemvdi di~uki~n bitt du'<;5ct6ngs6 ti~nc6 du'<;5cd acthaidiemtru'dcd6 chinhla t6ngs6 ti~nthudu'<;5cd thaidiemngaytru'dcd6.Ta n6i trochdikhongthuathi~t. - NSu {Sn}la trenMartingalthl tan6itrochdic6 l<;5ichonhat6chilc. - NSu {Sn}la du'diMartingalthl ta n6i tro chdi c6 l<;5icho ngu'aid~t cu'<;5c. - Vi d1,12: Cho cha'tdi~mchuyendQngtrendu'angth~ng, sailm6ibu'dc dung l~i t~icac di~mc6 tQadQ nguyen.saum6i bu'dccha'tdi~m chuy~ndQngdSnvi tri k = 2,-1,0,1,2,...vdi xac sua'tPk,trongd6 Pk :;t:0 vdihfi'uh~nk va LPk =1 k D~t:Yn:Vi tri cha'tdi~md bu'dcthil n so vdi vi tri cuacha'tdi~md bu'dc ngaytru'dc. Tac6 :M(Yn) =Lk.Pk k Xn=Yo+Yl +"'+Yn:La vi tri cha'tdiemsailn bu'dcsovdi vi triband~u. jet)=LPktk hamsinh. k Ta tha'y:f(l)=1,f'(l) =M(Yn),limf(t)=limf(t)=00 t--70 t--7oo - NSu phanph6i {Pn}du'<;5ccho saocho:f'(1)=0thl t = 1 la nghi~mduy nha'tcuaphu'dngtrlnhf(t)=1khi d6{Xli}la Martingal. Th~tv~y: {Xli}la Martingal. Q M(Yn)=0 Q f'(1) =0 - NSu phanph6i { Pk}dlt0 ho~cf'(1)<0 thlc6 them1nghi~mt=r <1khi d6rxnla Martingal. Th~tv~y: {rxn}la Martingal. Q M(rxn+llxn =k)=l Q M(l+yn+llxn=k )=l Trang23 l M(ryn+llxn=k)=l. :Li.rl = 1 f ( r) = 1 Trongvi dV1ngu'oitamu6nHmrah~th6ngquyt~cchdib~lisaochoco IQinha'trongtru'onghQp{So}la Martingalsaildaytasexetcacdinhly chungminhr~ngh~th6ngquyt~cdola vonghla,tag9icacdinhly nay lacacdinhly h~th6ng. 3/DinhIy h~thffngnha-t. B6 d~3.1:cho91nla dayphanho~chtangva {fn}ladaycacd~illfQng ng~unhien.£)~t91*nla tru'ongBorel sinhbdi91n Voi Rn=91n,taco : (1) Neu (fn,91n)la Martingalthl : f fn+ldJl = f J:dJl Rn Rn (2) Neu (fn,91n)la tren- Martingal ffn+ldJl ::;ffndJl Rn Rn (3)Ne'u(fn,91n)la dlfoi- Martingalthl: ffn+ldJl ~ fJ:dJl Rn Rn CM: (1):voi Rn E 91*n thl Rn Ia hQpde'mdu'<jcac phgntlt cua 91n. Do do tachi cgnchungminhvoi Rn E 91n. , 1 Taco: J:= (R).ffn+ldJl (do (fn,91n)Jl n Rn =>ffn+ldJl=ffndJl ( fnla h~ngtren 91n) Rn Rn laMartingal). (2),(3)tlfdngt\f Trang24 Nh~nxet: - Neu(fn,91n)la - Martingalthl: M(fn+l)=M(fn) ,'\!n - Neu (fn,91n)la tren- Martingalthl: M(fn+l) ~ M(fn) ,'\!n - Ne'u(fn,91n)la du'oi- Martingalthl: M(fn+l) ;:::M(fn) ,'\!n Dinh ly 3.2: Cho (fn,~Hn)la du'oi- Martingal va enla dC;liu'<;1ng~u nhien Iffy2 gicitri 0 &1 do du'<;1cd6i voi 91n*la tru'ongBorel sinhboi 91n.f)~t: 1\ fn = fa +eo(f1 - fa)+e1(/z- jJ+...+en(fn - fn-J Taco: (L9\" }a dUmMartingalva:M(J, ) S M (fJ CM : Ta chungminh: M(f:+ll~n)~Jn Ta co: M(i+l ~n) =M(Jn+eJfn+l- fJI ~n ) 1\ =M( fn I~n )+M(en(fn Ifn+l - fn)l~n) 1\ 1\ =fn +enM(fn+l - fn liRn)(dofn,en la h~ngtren91n) "' =f+ eJM(jn+l I~J- fJ(do fn lah~ngtrencua~n) !\ ~f(do(fn,~n) la du'oiMartingal) Ta chUngminh M(f" - f"):2: 0 bangquyn~ptheon. Vdi n=0tac6 fo =J 0 =>M(fo- ;, ) =0 GiaSIT:M( f, - f") :2:O. 1\ 1\ Taco : In+l=In +en(1:+1- 1:) 1\ 1\ =>In+1- I n+1=In+1- 1- en(In+1- In) =(1-eJ.(ln+1- IJ+ 1:- In ~ M(IM1- L)= !(fMl-L}fl ~f(1-en).(In- 1:)df.l= f( In+1 - In)df.l~0 Q {w/en(W)=O} Ohi chil : -Neu(fn,91n)la trenMartingalthl(In,9\n)la trenMartingalva M(Jn) ~M(tJ -Neu(fn,91n)laMartingalthl (In,9\n)la trenMartingalva M(J. ) =M(JJ -Neufnla Sntrongvi dv 1thl fn+1- fn=Yo:la s6 ti~nthudu'<jcsan khichdivanthlln+1 -Neu en+1=1thld vanthlln+l co thalligiachdi -Neu en+1=0thld vanthlln +1khongco thalligiachdi -Neu (fn,91n)la Martingal thl: M(Jn)=M(tJ: tilcla bit ky h~ th6ngcacquyt~cchdinaGclingcos5ti~nmongd<jinhu'thalligia vaG tit cacacvanbai. 4/DinhIf h{)i t1;1Martingal : Dinh nghia4.1:Cho (fn)la d(;lilu'<jngng~unhienxacdinhtrenkhong giancacday(Q,~,J.l)va var,sla2s6saGchor <r .tanoir~ngtrenquy d(;lo0) co1du'angditenquado(;ln[r,s]tUthaidi~mn-kdenn,ntu: (1) fn-k(ro):::;r (2) r < fn-k+m(ro)< S ,vdi 0< m<k fn (3) fn ~ S S r Fn-k B6 d~4.2:Neu (fn,inn)va (gn,inn)la du'diMartingalthl sup(fn,gn),inn) la du'diMartingal . CM: Taco: Mfj suP(ln,gJ I)~M(sup(1In I,IgnO) = I I In I dJ! + II gn I dJ! <00 IM<:lgnl Ifnl<lgnl Do day fnva gn la h~ngtreninnliensup(fn, gn) la h~ngtreninn Taco: M(sup(ln,gJ I9iJ ~M(ln I9iJ ~In M(sup(ln,gn)I9iJ ~M(gnI9iJ ~gn =>M(sup(ln,gn)19in)~suP(ln,gJ Ohi chli : - Neu (fn,inn)va (fn,gn)la trenMartingal thl (in f (fn,gn),inn)la tren Martingal B6 d~4.3:(B6 d~du'ongdi nen) I Cho (fn,inn)la du'diMartingal . OQib(ro)la s6du'angdi leutrenro qua[r,s]tli'thaidiem0 d€n n.Ta co : M(b):::;M~ In - r I) :::;Me!In I)+r s-r s-r CM:Tru'octieDtachungminh chotru'angh<jpfn~Ova r=O. 1\ Bi;lt :1: =10+eo(h - 10)+... +en-1 (1: - 1:-1) Voiemxacdinhnhu'sau: . N€u fm(co)=Othlem(co)=1khi fn+t(co)<svoi 1:S;t:s;k . em(co)=0 trongcaetru'angh<jpkhac fn s r=O fo 1\ Voi eachxay dlfngh~th6ng(en) nhu'tieDthl r5 rang In chi de> tangtren caedltangdi leDtli'thaidiem0 de'nn. 1\ Do d6 :In ?:.b.s 1\ -=>b:::;1: s => M(b)S; M(J") S;M(fJ s s Voi fnba'tky va do~n[r,s]ba'tky X6t (fn-rt =sup(fn-r,O). Trang28 Dofnladu'aLMartingal.=>(fn-rt la du'aiMartingal Hdn nuab(co)la s6du'ongdi leu trenco qua[r,s]cuafnclingchinhla s6 du'ongdi leucua(fn-rttrenco qua[O,s-r] Dodo: M(b)~ (in -r~r ~ M~ in -rs I) ~Mj in I+rs s-r s-r s- rs s s Dinh Iy 4.4: Ne'u(fn,91n)la du'aiMartingal vaM(lfnl)<k<oo,Vnthl : P ( hm fn(OJ)t6n t(;lihUll h';ln ) =1 n~<X) CM: Gia sll'tnli l(;lico t~pE vai /-l(E)=m>0saDcho : Ne'uroEE thl limineco)=00ho~climineco)khongt6nt(;li. -Ne'uroEE thllimfn(co)=oo: Theodinhly Fatou,Taco: Iliminf I in Idj.1~liminf II in Idj.1 E n n E ~ liminf II in Idj.1 n Q =liminf M~ in I)<k <(X) n Ma:lim inf(OJ)=(X) vdi OJ E E n =>Ilim inf Iin Idj.1= (X) (do j.1(E)>0 E n =>Voly. -Ne'uCO E E thllimfJw )khongt6nt~i n Dodovaim6iCO E E, co2 s6hi1utyr(co)var(co)saDcho: Co vo h(;lnn : fn(co)<r(co) Co vo h(;lnn : fn(co)> s(co) =>co vo h(;lndu'ongdi leutrenco qua do(;ln[r(co), r(co)] Do t~ps6hi1uty la de-mdu'<;1c,lien co th€ daubs6cacdo(;ln[r(co), r(co)] thanhcacdo(;lnqk=[rk,sk]de'mdu'<;1c. E>~t : Ak ={roleovahC;ln:fn(ro)~rk(ro)va eova hC;ln : fn(ro)~Sk(ro)} Ne'uro E Ak thl eo va hC;lndu'angdi len roquadoC;ln[rk,Sk] Taco:~p(Ak):~"p(YAk) =p(E) =m>0 ::::>eok 0saoeho:J.1(Ako»O,vai Ako ={ro/eovahC;ln:fn(ro)<r va eoeova hC;ln: fn(ro)>s} . E>~t,bn(ro)la s6du'angdi len trenroqua[r,s]titthaidi€m 0 de'nn b(ro)- la s6du'angdi lentrenroqua[r,s]trongtoanbQquatrinh Taeo:M(b)=fbdJ1~ fbdJ1=oo Q Aka TheoBE>3-4taeo: M(bJ~ M~In O+r~k+r =cs-r s-r Ma bnla daytanghQitl,lv€ b Do do theodinhly hQitl,lddndic$uTa eo: M(bn)~ M(b)=00va ly H~ qua 4.5:Ne'u(fn,91n)Ia trenMartingal khongam thilim fnt6nt(;li hil'u hC;lnhee . E>~ebic$tNe'u(fn,91n)la Martingal khongamthilim fnt6ntC;lihil'uhC;ln hee. CM : VI (fn,91n)la trenMartingal khongam ::::>(-fn,91n)la du'aLMartingal Ta eo :M( II-fill ) =M(fn)::; M(fo) ~ lim(- fJt6n t~ihUllh~nheen ~ lim(fn)t6nt~ihUllh~nheen Vi dt}.1: Xet ehuy€n dQngng~unhieneuaI ehfftdi€m trendlfangth~ngnhu'trong VD2 eua mve2. Trang30 Giasaphu'dngtrlnhf(t)=1eonghi~mt=r <1 Taeo: (rX,,)laMartingalkhongam. =>LimrXu t6nt(;lilien tvehee Ma Xnla'ygiatri nguyen,n€u eo2 khan[mgsail: (1)Vdiheeco :tlmdu'QeN(co)saoeho:n >N(co)=>Xn=XN (2)lim xn(co)=00hee Neueo(1): Ta eo(1)=n(xn=Xn+l=... =Xn+k) k Ma p(xn =xn+l=...=Xk+n)=POk =>P((1»=0. =>(1)khongthSxftyra =>eo(2),tue: limxn(co)=00 hee Vi dV2: ehof la hamddndi~ut6nt(;litren[0,1] D~tiRnla phanho(;lehdo(;ln[0,1], nhu'sau: [0,2-n],[2-n,2.2-n],...,[(2n-1) .2-n,1] fnladayhamxaedinhnhltsail: XE[j.2-n,Q+1).2-n],d~t: fn(x)= f( (j+1).2-D) - f(j.2-D) 2-D Ta tha'y: lim fn = f' , n€u lim fn t6n t(;liTa d~tha'yr~ng (fn, iRn)la Martingal khongam. =>limfn t6n t(;lihUll h(;lnhee =>f't6n t(;lihUll h(;lnhee V~y ham f ddn di~utangthl d(;loham f' ehi khong t6n t(;li tren t~pd9 do o. Trang31 5/Dinhly h~th6ngthO'2: 5.1/Thaidi~mdungng~unhien: Cho(Q,~,J.1)lakhonggianeaeday (Qn,~n,J.1n)khonggianeaedayeoehi€u dain+1. Dinh Nghla 5.1.1:Thai di6mng~unhienla d~ilu<jngng~unhien t(O) thoa: (1) t(O)l~ygia tri la s6nguyenkhongamho~e+00 (2){0)/t(0)=n}E ~n,Vn Nh~nxet: -Chodayd~iht<jngng~unhien(fn)vathaidiemng~unhient(O),d~t: ft(O)=fn(O), ne'ut(O) =n ft(0) khong xae dinh ,ne'ut(0)=00 OC! Ta eo : {wi/t(w)<c}=U{wlt(QJ)=n!\/n(w)<c} n=1 =>ft(0) dodu<jedO'ivdi ~ Dinh nghla 5.1.2:Thai diem ng~unhien t(O) gQi la thai diem dung, ne'u t(0) huu h(;lnhee. Nh~nxet: -N6u t(OJ)Iii thai diSm dungthl: .u(O{w/t(w)2': n})=0 D inhnghla5.13:Cho(fn)laquatrlnhng~unhien, ne'u: fn(0) lah~ngsO'vdinkhaIOnhee Tanoifnlaquatrlnhng~unhiendung Dinh nghla5.1.4:Chofnla quatrlnhng~unhien va t(O)la thaidiem 1\ ng~unhien,d~t:fn(m)=fmin(n,t(OJ)/m) Ta noi: (in) laquatrlnhbidungeua(in>dthaidiemng~unhient(O). 5/DjnhIy h~th6ngthu2: 5.1/Thaidi~mdungngfiunhien: Cho(Q,~,J.1)lakh6nggianeaeday (Qn,~n,J.1n)kh6nggianeaedayeoehiSudain+1. Djnh Nghia 5.1.1:Thai di~mng~unhienla d<;lihi<;fngng~unhien t(ro) thoa: (1) t(ro)la'ygiatri la s6nguyenkh6ngamho~e+00 (2){ro/t(ro)=n}E ~n,Vn Nh~nxet: -Cho dayd<;lilu'<;fng~unhien(fn)va thaidi~mng~unhient(ro),d~t: ft(ro)=fn(ro), ne'ut(ro)=n ft(ro)kh6ng xae dinh ,ne'ut(ro)=00 00 Ta eo: {wift(w)<c}=U{wlt(aJ)=nAfn(w)<c n=1 =>ft(ro)do du'<;fed6i vdi ~ Djnhnghia5.1.2:Thaidi~mng~unhient(ro)gQila thaidiemdung,ne'u t(ro)hUll h<;lnhee. Nh~nxet: -Ntu l(oo)Iii thai diSm di'lngthi: Jl(E! {OJ/ t(OJ)2 n})=0 D jnhnghia5.13:Cho(fn)laquatrlnhng~unhien, ne'u:fn(ro)lah~ngs6 vdinkhaIOnhee.tanoifnlaquatrlnhng~unhiendung. Dlnh nghia5.1.4:Chofnla quatrlnhng~unhien va t(ro)la thaidi~m 1\ ng~unhien, d~t: fn(OJ) =fmill(n,t(OJ)/OJ) Ta noi: (in) laquatrlnhbidungeua(f,J dthaidi~mng~unhient(ro). Trang34 Nh~nxet: in(m)= { fn (00) , n~ut(oo):::n ft (m),neu t(m)<n Do done'utern)la thaidiSmdungthl J n laquatrinhng~unhiendung. 5.2/Bioh ly h~th6ogthO'2: ~ B6 d~5.2.1:Neu (fn,91n)la Martingalva tern)la thaidiSmng~unhien saocho:fJ;dfl t6nt(;liTaco: Q fitdfl = f indfl + fttdfl (2 {t,;n) {t>n) n n CM Ta co: fitdfl =L fitdfl + fitdfl =L fikdfl +fitdfl (2 k=O(t=k\ {t>n} k=O{t=k\ t>n n =L findfl + fitdfl (do(t=k)E 91nva (fn, 91n)la k=O{t=k} t>n Martingal ) =fi"dfl + fitdfl {t,;n\ It>,,} Bioh ly 5.2.2:Neu(fn,91n)laMartingalvatern)la thaidiSmdungthoa: (1) M(lftl)<oo (2) li~ findfl =0 (t2n\ Thl :M[ft]=M[fo] Ghi chu: - Neu (fn,91n)la tren- Martingal thl:M[ftJ ~M[fo] Neu(fn,91n)ladu'di- Martingal thl:M[ft] ~M[fo] CM: Taco: fi"dfl = findfl + fitd;.,=f fndfl- ffndfl + fJ;dfl (2 (t,;n\ (t>,,\ OJ {t:>n} {t>n} Trang35 Ma lim ffndJ.l=0 (giathie't) n~oo{t>n} hm fJ;dJ.l =0 (dot(OJ)Hi thaidi€m dung) t>n => fftdJ.l =ffodJ.l Q Q H~qua:5.2.3:Cho(fn,91n)la Martinga1xacdjnhtrenkhonggianQ co dQdohfi'uh~nvat(0) 1athaidiemng~unhien.NSu: Ifni~k, Vn thl M[ft]=M[fo] CM: Do Ifni~k va Q codQhfi'uh~n ~M(lftl) <(fJ Taco: I findJ1l::; f IinIdJ1::; fkdJ1-+O,khi-+00 (VIt(O)la thaidiemdung) {c>nj {c>nj {c>nj Do do theo Dinh ly 5.2.2,tadu'Qcke'tqua Nh~nxet: -NSufn1at6ngsoKti€ncodu'Qcsanvanchdithancuangu'aid~tcu'Qcthl M[ft]=M[fo]co nghla la dil chQnh~th6ngthai diem dung nao cling khonglamthayd6i t6ngs6ti€n mongdQithudu'Qcne'utrochdi1avo h~i 1a(Martingal) M~nhd~5.2.4:Cho (fn,91n)la Martingalva t(O) la thaidiemng~u 1\ nhienva in (m)=fimin(n,C(aJ));(m) 1\ Khi do:(in' 91n) laMartingal. CM: 1\ n 1\ Taco :1in I::;L 1J; 1=>1in I khiitich ;=0 1\ D€ tha'yf n la h~ngtn~n91n Trang36 Vdi eoERE91nTa eo : [ /\ J I /\ M I n+1linn (m)= JL(R)I I n+1dJL =Jl(R) ( ff:+ldJl + ff:+ldJl ]Rn(tn = peR)( ffndJL+ ff.. n+1dJL )Rn(t:5,n) R (. n I>n) = (lR)( f!ndJl + ffndJl ] ( do Rn(t>n) E 91n) Jl Rn(tn) 1 ( f\ f\ J=R ffndJl + ffndJlJl( ) Rn(tn) I /\ /\ /\ "- =JL(R) I J:dJL =In(doInhangtreninJ H~qua5.2.5:Cho(fn,91n)HiMartingalIffygiatri Ia 86nguyen,teo)Ia /\ thai diem ng~unhien .£)~t: In(m)=Imin(n,I(W)/m) /\ /\ Ne'u In bieh~ntrenvadu'dithl In Iaquatrlnhdung. CM: /\ -Ne'ufn~0 thl(In ,91n)IaMartingalkhongam. Dodo: hm t6nt(;tilien t\lehee /\ Ma In Iffygiatringuyen. f\ =?in Ia h~ngb~tdftutunkhaIOnhee 1\ =?fn Iaquatrlnhdung /\ -Ne'uIn ~e Trang37 /\ =:.>In -C~0 /\ Ta apd\lngk€t quatrencho in -c " /\ - Neuin ~d /\ =:.>- in ~ - d /\ =:.>d- in ~ 0 /\ Taapd\lngk€t quatrenchod-in Vi d\l : Cho(fn)la quatrlnhng~unhiend9Cl~pva cophanph5i d~u, la'ygiatri-1,0,1.Va M(fn)=0 D~t:Sn =Yo+ Yl +...+Yn GQit(ro)la thaidi€m ng~unhien masn(ro)=Mho~csn(ro)=-Nvdi M,N :la 2s5nguyendu'dngc5dinhnaodo X6t quatrlnh: M , n€u sn(ro)=M S n (w) = Smin(n,t(w))(w) -N, n€u sn(ro)=-N Sn,n€u- N< Sn(ro)<M /\ =:.> s bi chann . . /\ =:.> Sn laquatrlnhdung N€u Ynla s5ti~nthudu'<,jcsailvanthiln /\ S n (aJ)=Mkhi ngu'oit6chilclamphasanngu'oid~tcu'Qc /\. S n (w)=- N khi ngu'oit6chilcphasan /\ . Sn laquatrlnhdung =:.> p( quatrlnhdung)=1 =:.>P(Ngu'oit6chilclamphasanngu'oid~tcu'<,jc) +P(Ngu'oit6chilcbi phasan)=l Trang38 Vdi P=P(Ngu'oit6chuclamphasanngu'oid~tcu'<jc) q =p(Ngu'oit6chucbi phasan) Ma :0=M(So)=M(S+) =P.M+q.(-N)=P.M+(l-P).(-N) N M =>p=M+N,q= M+N 6/Lu(itsalon: Blnb ly 6.1:Cho(Yn) la daycacd(;lilu'<jngngfiunhiendQcl~pcophan s ph6id~uvdiM(Yn)=ahuuh(;ln.D~t:Sn=Yo+Yl+...+Yn va s*n=-E...Ta n co:P{limSn=a)=l~~oo CM: Ta chungminhtrongtru'ongh<jpYnIffyhliuh(;lnvdigia trjXac sufft khacO. Ta co : P(Yn=j)=Pj>0 vdihliuh(;lnj va L~.=1 . J D~t lfJ(t)=L Pj"tJ J Ta co :ljJ(l) =1,lfJ'(I)=a=M(yJ l D~t:f(k,n)= [lfJ(t)]nvdit>Osexacdinhsan Ta sechungminhding:f(Sn,n)laMartingal Taco: tk+J tk . M(Sn+l,n+1)lsn=k)=~[lfJ(t)]n+l'~ = [tp(t)Y+1'Z;Pj"tJ tk =[lfJ(t)]n=f(k,n) =f(sn,n) Trang39 =>f(sn,n)laMartingalkhongam. Theadinhly hQih,IMartingal,taco: [ . ] n tS" tS" f(Sn,n) = [cp(t)l= cp(t) hQih,Ihfi'uh(;lnhee tb Vdi 8>0, di;it:b=a+8vag(t)= qJ~) Ta co :g(l)=a , g'(1)=b-a>0. =>g(to)> 1khi tokhag§n 1 N€u S' >b th'n- 1: S' to" tb cp(to)~ cp(to)=g(to)>1 [ . ] n ( S" =>f(sn,n) = ;((0) >[g(to)Y~ cokhi n ~ co [ S' ] n Ma f(s.,n)= ~&:') h9i tIi hiluh~nhee * =>P(sn(ro));:::b)=0 =>S*n(OJ) ~b =a +E hee =>lim supSon(OJ~a +E hee ) Tllong tll voi E >0,d~tb =a- E , tadll<Je: =>liminf S*n(OJ)~a - Eheen =>a - E ~lim inf S*n(OJ)~ lim sup S*n(OJ)~ a +E heen =>P(lim S*n=a) ==1n ,V >0E: Trang 40