Mô hình xác suất và một số ứng dụng trong kinh tế luận văn thạc sĩ

BO GIAO DUC VA DAO TAO TR£fC‘ING DAY HOC KINH TG TP HO CHI MINH CflUYEN N6ANfl : DIEU KfllEN fl§C KINH TE MA S0 : 5.02.20 LUANANTIENSIKINMTE NGU#IHU0NGD MNOAHOC: -GS.TSTR %NGV TP HO CHI MINH - 2001 KHANG LII CAM DOAN Tfii xin cam doan day la cfing trinh nghien cilu cua rieng toi C:ie so lieu, ket qua néu luan :in la trung thitc Nh’itng ket luan cua lua( n :in chita tu’’ng ditclc cfi ng bet bat ky cong trinh nao kh:ic Ngitdi cam dean NGUYEN VAN SI UjCLTjC MU DAU i CHCI€ING XICH MARKOV VA CAC TRANG THAI 1.1 Tinh markov va xich Markov 1.1.1 C:ic dJnh nghia cci ban 1.1.2 Ma tr(an xac suat chuyén 1.1.3 Phan phoi hftu han chiéu 1.1.4 Phan phoi ban dau 1.2 Phan ld’p c:ic trang th:ii 1.2.1 Cac trang thiii lien thfi ng va s phan lfi’p 1.2.1.1 Cac trang thiii lien thong 1.2.1.2 Chu ky cua trang th:ii 1.2.2 Trang th:ii hoi quy va trang th:ii khong hoi quy 9 10 10 1.2.2.1 C:i c dinh nghia i0 1.2.2.2 Tieu chuan hoi quy va khong hoi quy 13 1.3 Xii c suat gidi han 13 1.3.1 Dinh ly Ergodic 13 1.3.2 Thcli gian la) p trung binh 16 1.4 Xich Markov http thu 16 1.5 h Markov vfii them gian lien tuc Xic 20 1.5.1 h nghia va cac khéi niem cd ban Din 20 1.5.2 i trlnh sinh va huy Qu: 22 1.5.3 ‘i han cua xac suat Gid 23 1.5.4 :i trinh Poisson va qua trinh dem Qu 25 1.5.5 Qu:i trinh Poisson co phan loai 29 1.5.6 Qu:i trinh doi mcii 29 1.5.6.1 DJnh nghia 29 1.5.6.2 Phfin phoi va trung binh cua N (t) 30 CHLI€ING 32 CAC MO HINH XICH MARK0V 32 2.1 Mo hinh kiém ke 32 2.2 Mo hinh phuc vu d:inn dong 35 2.3 Xich Markov chay lien tiép 2.3.1 Dang ma tran xii c suat chuyen 2.3.2 Ciic vi du 2.× Mo hinh pha ri hoach thJ phan 38 Mo hinh trang th:ii hap thy 40 2.6 II’ng dung cua qu:i trinh Poisson 41 2.7 Mo hinh tang trifcl ng tuyén tinh vfii s nhap cif 42 2.8 I.I’ng dung cua qu:i trinh dot mdi 44 CLUNG 49 NG DUNG CUA XICH MARK0V 49 3.1 I.I’ng dung mo hinh phan hoach thJ phan cho h5ng hang khong ci Viet Nam 3.1.1 him ma tran xac suat chuyén va kiem tra tinh Markov 49 50 3.1.1.1 Thuat giai tim ma tran x:I c suat chuyén 50 3.1.1.2 Kiém tra tinh Markov 52 3.1.2 Litu kiém tra tinh Markov va tim ma tr(an chuyen bai toan phan hoach thJ phan cho hai h5ng hang khfing 54 3.1.3 Chitctng trlnh phan mém kiem tra tinh Markov va tim ma tran chuyen bai tod n phan hoach th[ phan cua hat hang hang khong 55 3.1.4 Phfin hoach thi| phan cho hat h5ng hang khong vfii so lieu th¿c te 62 3.1.4.1 Kiem tra tinh Markov va ti’m ma tran x:ie suat chuyen 62 3.1.4.2 Phan hoach thi phan 63 3.2 Phan tich s bien dong cua gié vang tar TP Ho Chi Minh 65 3.2 1.him ma tran xiic su$t chuyén va kiém tra tinh Markov 67 3.2 1.1 Phifdng phép tim mak(an xiic su5t chuyén 67 3.2 1.2 Kiém tra tinh Markov 69 3.2.2 Litu kiem tra tinh Markov va la( p ma tra( n chuyén de dii doiin st tang giam cua gi:i vang 72 3.2.3 Chifdng trinh ph:i n mém kiem tra tinh Markov va lap ma tr3an chuyén de dy doan st tang giam cua gia vang 3.2.4 Ket n,ua vfii so lieu th c te 73 92 3.2.4.1 Ma tran x:ic suat chuyen va tinh Markov 82 3.2.4.2 Ap dung mo hinh xich Markov http thu 84 3.2.5 Xac suat gifii han 86 3.2.6 Kiem tra trang th:ii hoi qui 88 3.2.6.1 Thua(t giai 88 3.2.6.2 Litu kiem tra trang thai hoi qui 90 3.2.6.3 Chu’cJng trinh ph$n mém de kiem tra trang thiii hoi qui 90 KET LUAN 95 1/ Nh’ifng két qua da hoan 95 2/ Nhu”ng ddng gop mcii cila luan :in 96 3/ Nhifng van dé co the :ip dung tiép 97 DANH MQC TAI L@U THAM KSO 99 Trong viec nghien ci?u nham nang cao hi(eu qua cua mo( t he› thong kinh te viec t’if trang th:ii hien tai dp bao trang th:ii tiidng lai cua he thong la dieu het si?c cap thiet Bdi lC, nhcl c:ic nha kinh te, c:ie nha doanh nghiep hoach d)nh c:ie chien lu’dc kinh doanh, lpa chon quyet dJnh toi itu cii c rang bupc, dira c:ic bien ph:i p t:ie dpng de loi kéo kh:ich hang Khi tien hanh c:ic nghien clit nay, chiing ta luon dung vfii c:ic nhan‘ to ngau nhien, chiing t:ic dpng mpi ncli, mpi fire Nhat la giai doan kinh te thi tru’fing hien nay, c:i c yéu to ngau nhién t:ie dpng cang du’ dpi cang da dang hctn bao gif het Vi vay cii c mo hinh xac suat trci c:ic cong cu quan trpng khong the thieu cac nghien cit’u Mo hinh Xich Markov ditclc nha bac hpc Nga la A.A Markov (1856-1922) difa tiI dau the ky 20 da ditclc nhieu nha toé n hpc phat trien them va sit dung vao nhieu linh vie nghien ctiti khoa hpc : sinh hpc, vat ly hpc, x5 hpi hpc, van hpc, kinh té hpc Trong nh’itng nam gan day, fi c:ic nu’fic kinh té phiit trien Xich Markov diidc sit dung nhieu kinh tfi : van de phan hoach th) phan, van de phuc vu d:inn dong, van de dp trC hang hfi a, van de doi mdi cfc thiet bJ D niffic ta viec nghien ciiu quit trinh Markov ciing da ducic quan tain tit lau : c:ie gi:to sit Nguyen B:ie Van, Nguyen Duy Tien, Da) ng Hiing Thang ( Dai hpc Tong hdp Ha Npi ), Nguyen Ho Quynh ( Dai hpc Birch khoa Ha Npi ), Nguyen Van Thu ( Vien To:in hpc Ha Noi ) v .v d5 co nhieu cong trlnh c6 gia tri nghien ci’tu ve qu:i trinh Markov Tuy nhien nhiing cong trinh dfi la CaC cong trinh thu:in tii y ly thuyét va mtii nhpn la ciic dJnh ly gifii han Trong cong trinh chiing tfii mong muon st dung cac két qua cua ly thuyet Xich Markov vfii so lieu thpc té de tra left nhtfng cau hoi ct the kinh doanh Hai ncli chiing toi lifa chpn :ip dung la : hang khfi ng dan dung va kinh doanh vang ba c da quy, nhting linh vie ma chiing tot xem la khé mui nhpn nén kinh th quoc gia hien Cu the la chil ng tfii giai quyfit bai toan - Phan hoach thJ phan giffa hat hang hang khong tuyén bay TP Ho Chi Minh — Ha Npi - DQ doé n tang, giam gi:i vang tar TP Ho Chi Minh Theo nhan xét ciia Phfi Giiio sit Nguyen Bac Van — mpt c:ie chuyen gia hang dau ve ly thuyét xii c suat u n.itic ta tih viec cli9r iTlG I.'•.nf rich Marr:ov de giai quyet ve van de dat d day la toi tin “ khong mpt phu’cIng phiip nao kh:1c co the dat tfii difclc “ Tat nhien qua hai van de giai quyét luan :in chung toi muon xay d ng mpt quy trlnh th c hien nhftng bai to:in tilfing th thupc mo hinh vfii phan mem vi tinh co the xem la cha) t ché Npi dung luan :in cua chiing toi "in 33 chilctng - ChiI‹Jng trinh bay nhftng yéu linh can ban can cho viec hiéu nhting mo hlnh chiing toi trinh bay ci chu’fing Cl chiming chilng tfii chi chting minh nhitng kit qua quan trpng Chi?ng minh nhu’’ng két qua kh:ic t dpc gia co the tim tai li u [7] cua Gi:t o sit Nguyen Duy Tien — mpt tai lieu trinh bay kha day du nhftng ket qua cap nh(at nhat ve qu:i trlnh Markov - Chifctng : Trinh bay ciic mo hinh ring dung Xich Markov vao kinh té Trong chiicing nay, ngoai mo hinh chung toi da sit dung cu the o chu’‹Jng 3, chiing tot trinh bay mpt “ mo hinh khac ma khufin kho luan :in va chita du so lieu, nen chiing toi chifa :ip dung nhifng chiing toi nhan thay co the ring dung tot kinh te niffic ta - Chitcing : Trinh bay :ip dung th c té d5 tién hanh co két qua Ciing theo lfii nhan xét cua Phfi Gi:to sit Nguyen B:ic Van thi “ day la luan :in cap tien st dau tien d Viet Nam ve ung dung mo hinh Markov vao thiic té “ XICH MARKOV VA CAC TR@NG Tl½é‹l 1.1 Tirih markov va xiclx Markov 1.1.1 C:ie di.nh nghia cd ban Gia sit chiing ta can nghien ct st tién trien theo thcii gian cua mpt he kinh te, vat ly hoac sinh thii i nao dfi Ky hieu X(t) la vJ tri cua he fi thcli diem t Tap help c:ie vi trf co the co cua he ditpc gpi la khong gian trang théi Gia st tru’fic thcli diem s he n trang th:ii nao dfi , tar thcii diem s he ct bãang thai i Tì cnn uiộ t tai thoi oiem I (t > S‘) titclng lai he cI trang thai j vfii x:ie suat la bao nhiéu Neu xii c suat chi phu thupc vao s, t, i, j thi dieu co nghia la st tien trien cua h e( titling lai chi phu thu pc vao hien tai va d pc lap vfii qua khi? Tinh chat gpi la tinh Markov co tinh chat gpi He( la quit trinh Markov Chang han, neu gpi X(t) la dan so ci tai thcli diem t (trong titling lai) thi co the xem X(t) chi phu thuo( c vao dan so hien tar va doe lap vfii qu:i khi’t Nfii chung c:i c he khfing co tri nhfi hoac sit’c y lh nhting he co tinh Markov Ta ky hieu E la tap gom cac giii tri cua X(t) va E gpi la khong gian trang th:ii cua X(t) Nfi u X(t) co tinh Markov va E d:inh so du’cic thi X(t) dupc gpi la xich Markov Tru’cing hcip, néu t = 0, 1, 2, thi ta co khai niem xich Markov vfii th’oi gian rcii rac Con neu t c [0,s) thl ta c6 kh:ii niem xich Markov vfii thdi gian lien tuc Tinh Markov dope dinh nghia theo ngon ngft to:in hpc nhif sau Dinh nghia Ta not rang X(t) co tinh Markov néu jcE Ta xem ID hi(en tai vfi , la tifclng lai, ( , t i , t,.,) la qua khu Do biéu thiic tren chinh la tinh Markov cua X(t) Da)t p (s, i, t, j) = P{X(I) = j / X(s) = i} , (s < t) Do la x:ic suat co dieu kien de he tar thdi diem s d trang thai i, din thcli diem t he chuyen sang trang théi j Vi the ta gpi p(s, i, t, j) la xii c suat chuyen cua he Néu x:ie suat chuyen chi phu thuoc vao (t-s) to’c la p(s, i, t, j) = p (s + h, i, I + h, j) thi ta nfii he la thuan nhat theo thfii gian Trong !i:ar tire :i«y, I'm khong nfii gi them, chung ta chi xét xich Markov thuan nhat 1.1.2 Ma tran x:ic su3t chuj en Gia sit (X,; n = 0, 1, 2, ) la xich Markov rcli rac va thuan nhat Khi dfi co m9t khong gian x:ic suat (Cl, A, P), X, : Cl › E la cii c bien ngau nhien nh a(n gia tri tap dem difpc E, E la khong gian trang th:ii, cac phfin tit cua no du’cIc ky hieu la i, j, k, Tinh Markov va tinh thuan nhat cua (X,) co nghia la 86 G |p* B T Trong dfi G 0,2222 0,2417 0,2222 0,1574 0,2417 0,5782 B 0,1574 0,5782 T 0,6204 0,1801 = Q R O I Titclng th nhil tren, ta tim dif c ma tran dao W=(I-Q )"= G G 1,4544 B 0,5427 B 0,8334 2,6818 tit ket qua néu gi:i vang dang cI trang thai giam thi no giam trung blnh 1,4544 va giu’ nguyén trung binh 0,5427 rot mcli tang, tit’c la co trung binh 1,4544 + 0,5427 = 1,9971 tru’fic bat dau tang Neu gi:i vang dang fi trang th:ii khfing dfii thi no giam trung binh 0,8334 va gin nguyén trung binh 2,6818 rot mdi tang, ttic la co trung binh 0,8334 + 2,6818 = 3,5152 trirfic bat dau tang 3.2.5 X:i c suat gidi han Mot van de khiic , can quan tain la néu qu:i trinh ‹I trang th:1i i sau d6 rcli d ’i thi mat mot thfii gian trung binh la bao lau sé trfi lai trang th:ii i Theo phan xac su$t gidi han thi vec ter phan phoi ciic trang thd i co dinh la nghiem cila he› phifdng trinh 87 dfi IP' la ma tran chuyén vJ cua ma tran xii c suat chuyén d5 tim diiclc ci tren it ›>b • g la th’d i gian gia vang tang, giam , giC nguyén tii he phu’dng trinh tren ta ditclc ×= ( 0,3568; 0,2538 ; 0,3894 ) Theo cfing thtic ( 1.11 ) d chifdng thi thcli giang trung binh trot qua giita nhu’ng lan qu:i trinh fi trang th:ii i bang nghJch dao phan thii i cua véc td phan phoi xiic suat a Nhif va( y, ta co Trang th:ii T B G Thdi gian trung binh(ngay) 2,80 3,94 2,57 3.2.6 Kiem tra trang th:ii hid qui Mpt van de khiic nfta ma c:ic kinh doanh rat quan tain la linh vpc kinh doanh cua minh neu Hi cang nhieu cang tot va han ché Hi da mi?c d p thiet hai, nghia la - Néu da mua vang vao giii vang tang ngay horn sau giii bat dau giam, hp can biét gi:i co trcl lai khong ? - Ho)ac néu da bo l0 khong mua vang vao giii giam ngay horn sau gi:i bat dau tang, hp can biet gié co lilc giam trfi lai khong thi sé ddi giam roi mua hay cti phai mua ? Do dfi, néu c:ie nha kinh doanh biét du’clc cac van de thi hp sé quyet dinh thiii dp kinh doanh ciia minh cho co lcli nhat De giai quyet van de chiing tfii d5 ring dung tinh chat cua trang th:ii hoi qui nhit chilling I d5 trinh bay va in:iy tinh la cfing cu dac l¿c giiip ta tim két qua mpt ciich nhanh chfing va chinh x:i c 3.2.6.1 Thuat giai dfi p;';" la phan til nam tren dong i, cot i en a ma tran P 89 Chiing toi sit dung tieu chuan hoi m Cauchy giai tich de giai quyét van de n$y nhif sau : (1) Vfii P la ma tra( n x:i c suat chuyén Tinh P' ( vfii n>1) (2) Co N cho vfii mpi n :• N • < thi ket luan trang th:ii i khong hot qui 90 3.2.6.2 Lifu dli kiém tra trang thiii hoi qui Begin nhap data trang th:ii i hoi qui kang th:ii i khong hoi qui End 3.2.6.3 Chitdng trinh phan niem dé kiém tra trang th:ii hoi qui if ! thisform.test wait ' Ch—a nhap ma {trn ! ' wind nowa return f endif form xemmatran * Tao mang P 91 sele P dime P(in.sc,m.sc) P=0 for i= l to in.sc fld="p.P"+a1lt(str(j,3)) P(i,j)=&f1d endfor endfor thisform.ghikq(1,”Ma trn P") for i= l to in.sc in.kq="" for j=1 to in.sc in.kq=m.kq+tran(P(i,j),'999.99999999') endfor thisform.ghikq(2,m.kq) endfor * Ma tran tich dime tich(in.sc,m.sc) dime ketqua(in.sc,m.sc) dime test(in.sc) test=0 * Gan : test=1 la khong hoi quy ; tets=2 la hoi quy *Pi for i= l to in.sc for j=1 to in.sc tich(i,j)=P(i,j) endfor endfor * Tinh P2 -> Pn *define wind zzzz from 2,2 to 32, 60 nozoom nof loat *acti wind zzzz 92 k=1 whil t k—k+1 thisform.tichmatran(k) thisform.kiemtrahoiquy(k) in.exit=.t for i=1 to in.sc if test(i)=0 in.exit=.f endif endfor if (k to 1000 = 0) or in.exit clear ?k for h1 = to in.sc for h2 = to in.sc ?? tran(tich(h1, h2)*(1/k), "999.99999999") next next endif if in.exit exit endif if inkey()=27 exit endif if 1astkey()=32 inkey(0) endif enddo *rele wind zzzz thisform.ghikq(2,"") t}rn P"+allt(str(in.k,6))) thisform.ghikq(1,"Ma for i=l to in.sc in.kq="" for j=1 to in.sc in.kq=m.kq+tran(tich(i,j),'999 99999999') endfor thisform.ghikq(2,m.kq) endfor thisform.ghikq(2,"") thisform.ghikq(1,"Kiem tra : ") in.kq="" for i= l to in.sc in.kq=m.kq+tran(test(i),'9')+space(8) endfor thisform.ghikq(2,m.kg) xemmatran.release sele kq go top report form i_kqhq preview Két qua kiém tra Vfii so lieu th c té d5 xét mo hinh va chilng tot da dung chilling trinh phan niem de kiem tra trang thlii hot qui difclc ket qua la cii c trang thai T, B, G la c:ic trang th:ii hoi qui Trong tinh hinh bien dong cua so lieu khac, chilfing trinh phan mem cua chéng toi sC giiip nha kinh doanh ph:It hien difcic mot trang thai nao dfi co hoi qui hay khfing Tom lai Vfii so lieu th¿c te de dp doén st tang, giam cua gi:I vang tar Thanh Ho Chi Minh nhu’ d5 trinh bay d tren nham dita den giai quyét bai to:in d¿ dodn ve gid co phiéu tu’fing lai d Viet Nam Ma hien thi trifd ng co phiéu tren 94 the gicli rat sfii dong Liic can dy doiin gi:i co phiéu thi mo hinh sé giiip ta giai quyet van de ma khfi ng can tién i i them 95 1/ Nhifng két qua da hoan Vé ma)t ly luan - Trong rat nhieu mo hlnh xac suat co sit dung qua trinh Markov, chung toi d5 ph$n tich, lya chpn nhifng mo hinh kha di ap dung ditclc vao kinh te Viet Nam cd so lieu th c té Chung tot trinh bay cac mo hinh d chitdng Dong their chiing toi chpn hai mo hinh ap dung vao thyc te diidc trinh bay d chitdng - De chi ro cd sci ly lu(an can thiet cho viec ung dung cac mo hinh d childng 2, chting toi da nghien cif’u va trinh bay ket qua can thiet nhat ve ly thuyet xich Makov va trinh bay d childng Vé ap dqng th c té — Da ap dung vdi cac so lieu th c té co ket qua cho hat bai toan a/ Phan hoach thi phan gii’fa hai hang hang khfing Cl bai toan cac tinh toan dJnh hu’clng ch(at ché va chinh xac da tra Hi cac cau hoi : ty ie khéch hang ma hang hang khong gianh difcic hien tai va titling lai nhif the nao ? Tii de xuat cac y kién nham tri va tang cifdng so khéch hang cua cac hang b/ Phtln tich tEng giam gié vang tar TP Hit Chf Minh D bai toan nay, chiing Hi tra ldi cac can hoi ma cac doanh nghiep rat quan tain + Ty le nhung tang, giam gié vang kong hien tar va tifDng lai + Thcli gian trung binh qud trinh tang trcl lai tang va giam trd lai gidm 96 + Their gian trung binh qua trinh tang chuyén sang giam hay tit giam chuyen sang tang + Chi trang th:ii nao la hoi gut ( trang thai 1a)p ) va trang th:ii nao la nhat thfii sé khfi ng quay trd lai niia - Lap chitdng trinh phan mem giai quyet hat bai toan tren Trong tifdng lai co the sit dung mo hinh dy doan st tang giam gin co phieu cl Viet Nam ma khf›ng can cai tién them 2/ Nhifng dong gop mdi cua luan :in So vfii nhitng cfi ng trinh :ip dung xich Markov ‹I cac nu’fic phat trien ma chiing toi tham khao diIcIc, cfing trinh cua chung toi co nhting net mdi nhil sau : a/ Trong bai toii n phan hoach thi phan, cac tiic gia md’i chi giai quyet cho tritctng help - So kh:i ch hang di co d(nh Thcli doan chuyen trang th:ii co dinh Chilng tot giai quyét tru”fi ng help so 1‹:hach khong co dinh va thfii doin ciing khfi ng co d(nh nhit vay sé phil hdp du’clc vfii nhieu hoan canh thpc té h‹Jn va c:in quay lai triffing hcip so kh4 ch hang va th’d i doan co dinh la hoan toan de dang b/ Dieu noi b(at nhfit la chung tot kiem tra tinh Markov bang tieu chuan kiem dinh gia thiet thong ké cha) t ché sau d6 thay thoa man thi m0’i ép dung cilc két qua vé xich Markov de phan tich Trong cite cong trinh ma chung tot tham khao khfi ng tain viec ma th’ita nh a(n tinh Markov — mot gia thiet mau chot nh$t ciia mpi phan tich 97 c/ Vd’i thuat giai rot den lifclc khoi va sau dfi la chifdng trinh phan mem chiing tfii co gang xay ding mot khufi n mau ve viec :ip dung xich Markov cho c:ic bai to:in tifdng th 3/ Nhifng vfi n de cii thé :ip dqng tiép Hien tar, chiia co so li(u va han che ve khufin kho luan an, chung tf›i md’i chi :ip dung mo hinh la : mo hinh phan hoach thJ phan cho hang hang khfing va mfi hinh d do:in st tang, giam cua gi:i vang tar TP Ho Chi Minh “ mo hinh nhu’ : mo hinh tang tru'dng tuyen tinh vfii s nh(ap cu’, mo hinh doi mcli thiet bJ v v co the :ip dung va co nhieu hit’a hen [1] Ye viec tip dqng mot sit bat dang thif‘c ly thuyét x:ic suat van thfing ké — thang 04 nEm 1999 Tap chi Ph:it trien Kinh te, so 102 [2] D)nh ly Ergodic va mii hinh di dan t - Th:ing 10/2000 Tap chi Ph:it trien Kinh Ie, so 120 [3] Qu:i trinh Pois"son, Hang phqc vq vo han - Th:ing 11/2000 Tap chi Phil t trien Kinh té, so 121 DANH VUCT LIEUTKALIM O Da)ng Han (1996), Xic sua’t tho’ng ké , Nha XB Thong ké Le Van Hot ( 1997 ), Gmt I/cli Tocin hoc, D H Kinh te Tp HCM Nguyen Tan Lap (1981), Ccic phitong phép Toén Kinh te’, Dai hpc Kinh te TP Ho Chi Minh Trifdng Van Khang ( 1982), Gi‹zp trinh Mo hinh Tocin Kinh te’, Dar hpc [4] Kinh te TP Ho Chi Minh Tran Van Thang ( 1998), Gide trinh LT Tho’ng ké , NXB Thong ke Nguyen Duy Tien, Vu Viet Yen (2000) LJ thuyé‘t Xac sua’t, Nha XB Giao duc Nguyén Duy Tien ( 2000), Cfc mo /iin/i xd c suci’t vé ting dung [8] Phan I Xich Markov vé ting dung, NXB Dpi hoc Quo“c gia Her Noi Nguyen Duy Tien, Nguyen View Phil ( 1983), Cd sd I y thuye ’t Xac sué’t, [9] Nha XB Dar hpc va Trung hpc chuyén nghiep Ha Npi Dang Hung Thang (1998), flu diii ve I y thuye’t xéc suit vé céc u:rig ding, In lan thil Nha Xuat ban Giao duc - Ha Npi [10] Nguyen B:i c Van (1996), Xfic sum vfi Xit Iy so’lie u, NXB Gi:t o duc [11] Vii Tien Vi t, Nguyen Van Si (th:ing 12-1999), Véi ii:rig dung mo hinh xéc sua’t tho’ng ke quan I j kinh te’ ver trqt tij: xé hot B:t o cao tar Hoi nghi ting dung toén hpc toan quoc lan thil nhat, Ha Noi [121 A.N.Shiryaev (1996), Probabilit:y Second Edition, Springer - Verlag, New York [13] Kai Lai Chung (1967), Markov chains with stationary tragitions probabilities Springer - Verlag, New York [14] H.M Taylor (1984), An Introduction to stochanstic modeling, Academic Press, Inc-New York [15] S.M Ross (1993), Introduction to probability models 5" ed Academic Press, Inc-New York ... sinh hpc, vat ly hpc, x5 hpi hpc, van hpc, kinh té hpc Trong nh’itng nam gan day, fi c:ic nu’fic kinh té phiit trien Xich Markov diidc sit dung nhieu kinh tfi : van de phan hoach th) phan, van... lai cua he thong la dieu het si?c cap thiet Bdi lC, nhcl c:ic nha kinh te, c:ie nha doanh nghiep hoach d)nh c:ie chien lu’dc kinh doanh, lpa chon quyet dJnh toi itu cii c rang bupc, dira c:ic... Nhifng van dé co the :ip dung tiép 97 DANH MQC TAI L@U THAM KSO 99 Trong viec nghien ci?u nham nang cao hi(eu qua cua mo( t he› thong kinh te viec t’if trang th:ii hien tai dp bao trang th:ii tiidng