WikiDer > Stoxastik jarayon

A-ni kompyuter tomonidan taqlid qilish Wiener yoki Braun harakati shar sirtidagi jarayon. Wiener jarayoni ehtimollar nazariyasida eng ko'p o'rganilgan va markaziy stoxastik jarayon hisoblanadi.^[1][2][3]

Yilda ehtimollik nazariyasi va tegishli sohalar, a stoxastik yoki tasodifiy jarayon a matematik ob'ekt odatda a sifatida belgilanadi oila ning tasodifiy o'zgaruvchilar. Ko'plab stoxastik jarayonlar vaqt qatorlari bilan ifodalanishi mumkin. Biroq, stoxastik jarayon tabiatan uzluksiz, vaqt qatori esa butun sonlar bilan indekslangan kuzatuvlar to'plamidir. Stoxastik jarayon bir nechta tasodifiy o'zgaruvchilarni o'z ichiga olishi mumkin.

Umumiy misollarga a ning o'sishi kiradi bakterial aholi, an elektr toki tufayli o'zgaruvchan termal shovqinyoki a harakati gaz molekula.^[1][4][5][6] Sifatida stoxastik jarayonlardan keng foydalaniladi matematik modellar tasodifiy ravishda o'zgarib turadigan ko'rinadigan tizim va hodisalar. Kabi ko'plab fanlarda qo'llanmalar mavjud biologiya,^[7] kimyo,^[8] ekologiya,^[9] nevrologiya,^[10] fizika,^[11] tasvirni qayta ishlash, signallarni qayta ishlash,^[12] boshqaruv nazariyasi, ^[13] axborot nazariyasi,^[14] Kompyuter fanlari,^[15] kriptografiya^[16] va telekommunikatsiya.^[17] Bundan tashqari, tasodifiy ko'rinadigan o'zgarishlar moliyaviy bozorlar stoxastik jarayonlardan keng foydalanishga turtki bergan Moliya.^[18][19][20]

Ilovalar va hodisalarni o'rganish o'z navbatida yangi stoxastik jarayonlar taklifiga ilhom berdi. Bunday stoxastik jarayonlarning misollariga quyidagilar kiradi Wiener jarayoni yoki Braun harakati jarayoni,^[a] tomonidan ishlatilgan Louis Bachelier bo'yicha narx o'zgarishini o'rganish Parij birjasi,^[23] va Poisson jarayonitomonidan ishlatilgan A. K. Erlang ma'lum bir vaqt ichida sodir bo'lgan telefon qo'ng'iroqlari sonini o'rganish.^[24] Ushbu ikkita stoxastik jarayon stoxastik jarayonlar nazariyasida eng muhim va markaziy hisoblanadi,^[1][4][25] Bachelier va Erlangdan oldin ham, keyin ham turli xil sharoitlarda va mamlakatlarda bir necha bor va mustaqil ravishda topilgan.^[23][26]

Atama tasodifiy funktsiya stoxastik yoki tasodifiy jarayonga murojaat qilish uchun ham ishlatiladi,^[27][28] chunki stoxastik jarayon a da tasodifiy element sifatida talqin qilinishi mumkin funktsiya maydoni.^[29][30] Shartlar stoxastik jarayon va tasodifiy jarayon bir-birining o'rnida ishlatiladi, ko'pincha o'ziga xos xususiyatlarsiz matematik makon tasodifiy o'zgaruvchilarni indekslaydigan to'plam uchun.^[29][31] Ammo ko'pincha bu ikkita atama tasodifiy o'zgaruvchilar tomonidan indekslanganida ishlatiladi butun sonlar yoki an oraliq ning haqiqiy chiziq.^[5][31] Agar tasodifiy o'zgaruvchilar Dekart tekisligi yoki undan yuqori o'lchovli Evklid fazosi, keyin tasodifiy o'zgaruvchilar yig'indisi odatda a deb nomlanadi tasodifiy maydon o'rniga.^[5][32] Stoxastik jarayonning qiymatlari har doim ham raqamlar emas va ular vektor yoki boshqa matematik ob'ektlar bo'lishi mumkin.^[5][30]

Matematik xususiyatlaridan kelib chiqib, stoxastik jarayonlarni turli toifalarga birlashtirish mumkin, ular tarkibiga kiradi tasodifiy yurish,^[33] martingalalar,^[34] Markov jarayonlari,^[35] Levi jarayonlari,^[36] Gauss jarayonlari,^[37] tasodifiy maydonlar,^[38] yangilanish jarayonlariva dallanish jarayonlari.^[39] Stoxastik jarayonlarni o'rganish uchun matematik bilim va metodlardan foydalaniladi ehtimollik, hisob-kitob, chiziqli algebra, to'plam nazariyasiva topologiya^[40][41][42] filiallari kabi matematik tahlil kabi haqiqiy tahlil, o'lchov nazariyasi, Furye tahliliva funktsional tahlil.^[43][44][45] Stoxastik jarayonlar nazariyasi matematikaga muhim hissa bo'lib hisoblanadi^[46] va u nazariy sabablarga ko'ra ham, qo'llanilish uchun ham faol tadqiqot mavzusi bo'lib qolmoqda.^[47][48][49]

Kirish

Stoxastik yoki tasodifiy jarayonni ba'zi bir matematik to'plamlar tomonidan indekslangan tasodifiy o'zgaruvchilar to'plami deb ta'riflash mumkin, ya'ni stoxastik jarayonning har bir tasodifiy o'zgaruvchisi to'plamdagi element bilan o'ziga xos tarzda bog'liqdir.^[4][5] Tasodifiy o'zgaruvchilarni indekslash uchun ishlatiladigan to'plamga deyiladi indeks o'rnatilgan. Tarixiy jihatdan, indekslar to'plami biroz edi kichik to'plam ning haqiqiy chiziqkabi natural sonlar, indeksni berish vaqt talqini.^[1] To'plamdagi har bir tasodifiy o'zgaruvchi bir xil qiymatlarni qabul qiladi matematik makon nomi bilan tanilgan davlat maydoni. Ushbu holat maydoni, masalan, butun sonlar, haqiqiy chiziq yoki bo'lishi mumkin ${ displaystyle n}$- o'lchovli Evklid fazosi.^[1][5] An o'sish stoxastik jarayon ikki indeks qiymatlari orasida o'zgarib turadigan, ko'pincha vaqtning ikki nuqtasi sifatida talqin qilinadigan miqdor.^[50][51] Stoxastik jarayon ko'p narsalarga ega bo'lishi mumkin natijalar, tasodifiyligi tufayli va stoxastik jarayonning yagona natijasi, boshqa nomlar qatorida a namuna funktsiyasi yoki amalga oshirish.^[30][52]

Bitta kompyuter simulyatsiyasi namuna funktsiyasi yoki amalga oshirish, boshqa terminlar qatorida, 0 ≤ t time vaqt davomida uch o'lchovli Wiener yoki Braun harakati jarayonining 2. Ushbu stoxastik jarayonning indeks to'plami manfiy bo'lmagan sonlar, uning holat maydoni esa uch o'lchovli Evklid fazosidir.

Tasnifi

Stoxastik jarayonni har xil yo'llar bilan, masalan, holat maydoni, indekslar to'plami yoki tasodifiy o'zgaruvchilar o'rtasidagi bog'liqlik bo'yicha tasniflash mumkin. Tasniflashning keng tarqalgan usullaridan biri kardinallik indekslar to'plami va holat maydoni.^[53][54][55]

Vaqt deb talqin qilinadigan bo'lsa, agar stoxastik jarayonning indekslar to'plamida cheklangan sonlar to'plami, butun sonlar to'plami yoki natural sonlar kabi sonli yoki hisoblanadigan sonli elementlar bo'lsa, u holda stoxastik jarayon quyidagicha bo'ladi: diskret vaqt.^[56][57] Agar indekslar to'plami haqiqiy chiziqning biron bir oralig'i bo'lsa, u holda vaqt deyiladi davomiy. Ikki xil stoxastik jarayonlar navbati bilan ataladi diskret vaqt va uzluksiz stoxastik jarayonlar.^[50][58][59] Diskret vaqtli stoxastik jarayonlarni o'rganish osonroq, chunki uzluksiz vaqtli jarayonlar yanada rivojlangan matematik texnika va bilimlarni talab qiladi, ayniqsa indekslar to'plami hisoblanmaydi.^[60][61] Agar indekslar to'plami butun sonlar yoki ularning ba'zi bir to'plami bo'lsa, unda stoxastik jarayonni a deb ham atash mumkin tasodifiy ketma-ketlik.^[57]

Agar holat maydoni butun sonlar yoki natural sonlar bo'lsa, u holda stoxastik jarayon a deb ataladi diskret yoki butun son bilan baholanadigan stoxastik jarayon. Agar davlat maydoni haqiqiy chiziq bo'lsa, u holda stoxastik jarayon a deb nomlanadi real baholanadigan stoxastik jarayon yoki a uzluksiz holat maydoni bilan jarayon. Agar davlat maydoni bo'lsa ${ displaystyle n}$-o'lchovli Evklid fazosi, keyin stoxastik jarayon a deb ataladi ${ displaystyle n}$-o'lchovli vektor jarayoni yoki ${ displaystyle n}$-vektor jarayoni.^[53][54]

Etimologiya

So'z stoxastik yilda Ingliz tili dastlab "gumonga tegishli" ta'rifi bilan sifat sifatida ishlatilgan va a dan kelib chiqqan Yunoncha "belgini nishonga solish, taxmin qilish" va Oksford ingliz lug'ati 1662 yilni eng dastlabki voqea sifatida beradi.^[62] Ehtimollik bo'yicha ishlarida Ars Conjectandi, dastlab 1713 yilda lotin tilida nashr etilgan, Yakob Bernulli "taxmin qilish yoki stoxastika san'ati" ga tarjima qilingan "Ars Conjectandi sive Stochastice" iborasini ishlatgan.^[63] Ushbu ibora Bernulli tomonidan ishlatilgan Ladislaus Bortkievich^[64] bu so'zni 1917 yilda nemis tilida yozgan stoxastik tasodifiy ma'noga ega. Atama stoxastik jarayon birinchi bo'lib 1934 yilda chop etilgan maqolada ingliz tilida paydo bo'ldi Jozef Dub.^[62] Ushbu atama va ma'lum bir matematik ta'rif uchun Doob yana 1934 yilgi maqolani keltirdi, bu erda bu atama stochastischer Prozeß tomonidan nemis tilida ishlatilgan Aleksandr Xinchin,^[65][66] garchi nemischa atama ilgari ishlatilgan bo'lsa ham, masalan, Andrey Kolmogorov tomonidan 1931 yilda.^[67]

Oksford ingliz lug'atiga ko'ra, so'zning erta paydo bo'lishi tasodifiy tasodif yoki omad bilan bog'liq bo'lgan hozirgi ma'no bilan ingliz tilida XVI asrga to'g'ri keladi, ilgari qayd etilganlar XIV asrda "dadillik, katta tezlik, kuch yoki zo'ravonlik (minishda, chopishda, hayratda qoldiradigan va h.k.) ". Bu so'zning o'zi o'rta tezlikdagi frantsuzcha "tezlik, shoshilish" so'zidan kelib chiqqan bo'lib, ehtimol frantsuzcha "chopish" yoki "chopish" ma'nosidagi fe'ldan kelib chiqqan. Terminning birinchi yozma ko'rinishi tasodifiy jarayon oldindan sanalar stoxastik jarayon, Oksford Ingliz Lug'ati ham sinonim sifatida keltirilgan va tomonidan maqolada ishlatilgan Frensis Edgevort 1888 yilda nashr etilgan.^[68]

Terminologiya

Stoxastik jarayonning ta'rifi turlicha,^[69] ammo stoxastik jarayon an'anaviy ravishda ba'zi to'plamlar tomonidan indekslangan tasodifiy o'zgaruvchilar to'plami sifatida tavsiflanadi.^[70][71] Shartlar tasodifiy jarayon va stoxastik jarayon sinonimlar deb qaraladi va indekslar to'plami aniq ko'rsatilmagan holda bir-birining o'rnida ishlatiladi.^{[29][31][32][72][73][74]} Ikkala "to'plam",^[30][72] yoki "oila" ishlatiladi^[4][75] "indekslar to'plami" o'rniga, ba'zida "parametrlar to'plami" atamalari^[30] yoki "parametr maydoni"^[32] ishlatiladi.

Atama tasodifiy funktsiya stoxastik yoki tasodifiy jarayonga murojaat qilish uchun ham ishlatiladi,^[5][76][77] garchi ba'zida u faqat stoxastik jarayon haqiqiy qiymatlarni qabul qilganda ishlatiladi.^[30][75] Ushbu atama indekslar to'plamlari haqiqiy chiziqdan tashqari matematik bo'shliqlar bo'lganida ham qo'llaniladi,^[5][78] shartlari esa stoxastik jarayon va tasodifiy jarayon odatda indekslar to'plami vaqt sifatida talqin qilinganda ishlatiladi,^[5][78][79] kabi boshqa atamalardan foydalaniladi tasodifiy maydon indeks to'plami bo'lganda ${ displaystyle n}$- o'lchovli Evklid fazosi ${ displaystyle mathbb {R} ^ {n}}$ yoki a ko'p qirrali.^[5][30][32]

Notation

Stoxastik jarayonni boshqa usullar qatori bilan ham belgilash mumkin ${ displaystyle {X (t) } _ {t in T}}$,^[58] ${ displaystyle {X_ {t} } _ {t in T}}$,^[71] ${ displaystyle {X_ {t} }}$^[80] ${ displaystyle {X (t) }}$ yoki shunchaki ${ displaystyle X}$ yoki ${ displaystyle X (t)}$, garchi ${ displaystyle X (t)}$ sifatida qaraladi funktsiya yozuvlarini suiiste'mol qilish.^[81] Masalan, ${ displaystyle X (t)}$ yoki ${ displaystyle X_ {t}}$ indeks bilan tasodifiy o'zgaruvchiga murojaat qilish uchun ishlatiladi ${ displaystyle t}$va butun stoxastik jarayon emas.^[80] Agar indeks to'plami bo'lsa ${ displaystyle T = [0, infty)}$, keyin yozish mumkin, masalan, ${ displaystyle (X_ {t}, t geq 0)}$ stoxastik jarayonni belgilash uchun.^[31]

Misollar

Bernulli jarayoni

Eng oddiy stoxastik jarayonlardan biri bu Bernulli jarayoni,^[82] bu ketma-ketligi mustaqil va bir xil taqsimlangan (iid) tasodifiy o'zgaruvchilar, bu erda har bir tasodifiy miqdor bir yoki nol qiymatini oladi, ehtimol ehtimoli bilan ${ displaystyle p}$ va ehtimollik bilan nol ${ displaystyle 1-p}$. Ushbu jarayon tanani qayta-qayta aylantirish bilan bog'liq bo'lishi mumkin, bu erda bosh olish ehtimoli mavjud ${ displaystyle p}$ va uning qiymati bitta, quyruqning qiymati esa nolga teng.^[83] Boshqacha qilib aytganda, Bernulli jarayoni - bu ketma-ketlik iid Bernulli tasodifiy o'zgaruvchilar,^[84] bu erda har bir tanga aylanasi a-ning misoli Bernulli sudi.^[85]

Tasodifiy yurish

Tasodifiy yurish odatda yig'indisi sifatida aniqlanadigan stoxastik jarayonlardir iid Evklid fazosidagi tasodifiy o'zgaruvchilar yoki tasodifiy vektorlar, shuning uchun ular diskret vaqt ichida o'zgaradigan jarayonlardir.^{[86][87][88][89][90]} Ammo ba'zilari bu atamani doimiy ravishda o'zgarib turadigan jarayonlarga nisbatan ishlatadilar,^[91] xususan moliya sohasida qo'llaniladigan Wiener jarayoni, bu biroz chalkashlikka olib keldi va natijada uning tanqidiga sabab bo'ldi.^[92] Tasodifiy yurishning boshqa har xil turlari mavjud, shuning uchun ularning holati bo'shliqlari boshqa matematik ob'ektlar, masalan, panjaralar va guruhlar bo'lishi mumkin va umuman olganda ular juda yaxshi o'rganilgan va turli fanlarda juda ko'p qo'llanmalarga ega.^[91][93]

Tasodifiy yurishning klassik namunasi sifatida tanilgan oddiy tasodifiy yurish, bu butun makon sifatida butun sonlar bilan diskret vaqtdagi stoxastik jarayon bo'lib, Bernulli jarayoniga asoslanadi, bu erda har bir Bernulli o'zgaruvchisi ijobiy yoki salbiy qiymatni oladi. Boshqacha qilib aytganda, oddiy tasodifiy yurish butun sonlarda sodir bo'ladi va uning qiymati ehtimollik bilan bittaga ko'payadi, aytaylik: ${ displaystyle p}$, yoki ehtimollik bilan biriga kamayadi ${ displaystyle 1-p}$, shuning uchun bu tasodifiy yurishning indeks to'plami tabiiy sonlar, uning holat maydoni esa butun sonlardir. Agar ${ displaystyle p = 0.5}$, bu tasodifiy yurish nosimmetrik tasodifiy yurish deb ataladi.^[94][95]

Wiener jarayoni

Wiener jarayoni statsionar va statsionar jarayondir mustaqil o'sish bu odatda taqsimlanadi o'sish hajmiga asoslanib.^[2][96] Wiener jarayoni nomlangan Norbert Viner, uning matematik mavjudligini isbotlagan, ammo bu jarayon tarix uchun bog'langanligi sababli Braun harakat jarayoni yoki shunchaki Braun harakati deb nomlanadi. Braun harakati suyuqliklarda.^[97][98][99]

Driver bilan Wiener jarayonlarini (yoki Braun harakat jarayonlarini) amalga oshirish (ko'k) va driftsiz (qizil).

Ehtimollar nazariyasida markaziy rol o'ynagan Viner jarayoni ko'pincha boshqa stoxastik jarayonlar bilan bog'langan holda eng muhim va o'rganilgan stoxastik jarayon hisoblanadi.^{[1][2][3][100][101][102][103]} Uning indeks to'plami va holat maydoni mos ravishda manfiy bo'lmagan sonlar va haqiqiy sonlardir, shuning uchun ham doimiy indekslar to'plami, ham bo'shliq mavjud.^[104] Ammo jarayonni umuman ko'proq aniqlash mumkin, shuning uchun uning holat maydoni bo'lishi mumkin ${ displaystyle n}$- o'lchovli Evklid fazosi.^{[93][101][105]} Agar anglatadi har qanday o'sishning nolga tengligi, keyin paydo bo'lgan Wiener yoki Brownian harakatlanish jarayoni nolga siljish deb aytiladi. Vaqtning istalgan ikki nuqtasi uchun ortishning o'rtacha qiymati bir xil doimiyga ko'paytirilgan vaqt farqiga teng bo'lsa ${ displaystyle mu}$, bu haqiqiy son, keyin hosil bo'lgan stoxastik jarayonning o'zgarishi aytiladi ${ displaystyle mu}$.^{[106][107][108]}

Deyarli aniq, Wiener jarayonining namunaviy yo'li hamma joyda uzluksiz, ammo hech qaerda farqlash mumkin emas. Buni oddiy tasodifiy yurishning uzluksiz versiyasi deb hisoblash mumkin.^[51][107] Jarayon, boshqa tasodifiy yurishlar kabi boshqa stoxastik jarayonlarning matematik chegarasi bekor qilinganligi sababli paydo bo'ladi,^[109][110] qaysi mavzusi Donsker teoremasi yoki funktsional markaziy chegara teoremasi deb ham ataladigan invariantlik printsipi.^{[111][112][113]}

Wiener jarayoni stoxastik jarayonlarning ba'zi muhim oilalari, jumladan Markov jarayonlari, Levi va Gauss jarayonlari a'zosi.^[2][51] Jarayon, shuningdek, ko'plab dasturlarga ega va stoxastik hisoblashda ishlatiladigan asosiy stastik jarayondir.^[114][115] Bu miqdoriy moliyalashtirishda asosiy rol o'ynaydi,^[116][117] qaerda ishlatiladi, masalan, Blek-Skoulz-Merton modelida.^[118] Jarayon turli sohalarda, shu qatorda tabiatshunoslikning aksariyat qismida, shuningdek ijtimoiy fanlarning ayrim tarmoqlarida, turli xil tasodifiy hodisalar uchun matematik model sifatida ishlatiladi.^{[3][119][120]}

Poisson jarayoni

Puasson jarayoni - bu turli xil shakl va ta'riflarga ega bo'lgan stoxastik jarayon.^[121][122] Uni hisoblash jarayoni deb ta'riflash mumkin, bu stoxastik jarayon bo'lib, bir muncha vaqtgacha bo'lgan voqealar yoki hodisalarning tasodifiy sonini aks ettiradi. Jarayonning noldan ma'lum bir vaqtgacha bo'lgan oralig'ida joylashgan nuqtalari soni o'sha vaqtga va ba'zi parametrlarga bog'liq bo'lgan Puasson tasodifiy o'zgaruvchisidir. Bu jarayonda natural sonlar, uning holat maydoni va manfiy bo'lmagan sonlar, indekslar to'plami sifatida mavjud. Ushbu jarayonni Puassonni hisoblash jarayoni deb ham atashadi, chunki uni hisoblash jarayonining misoli sifatida talqin qilish mumkin.^[121]

Agar Puasson jarayoni bitta musbat konstantasi bilan aniqlansa, u holda jarayon bir hil Puasson jarayoni deyiladi.^[121][123] Bir hil Puasson jarayoni Markov jarayonlari va Leviy jarayonlari kabi stoxastik jarayonlarning muhim sinflari a'zosi hisoblanadi.^[51]

Bir hil Poisson jarayonini har xil usullar bilan aniqlash va umumlashtirish mumkin. Uning indekslari to'plami haqiqiy chiziq bo'lishi uchun shunday belgilanishi mumkin va bu stoxastik jarayon statsionar Puasson jarayoni deb ham ataladi.^[124][125] Agar Puasson jarayonining parametr konstantasi ning ba'zi manfiy bo'lmagan integral funktsiyalari bilan almashtirilsa ${ displaystyle t}$, natijada hosil bo'lgan jarayon bir hil bo'lmagan yoki bir hil bo'lmagan Puasson jarayoni deb ataladi, bu erda jarayon nuqtalarining o'rtacha zichligi endi o'zgarmaydi.^[126] Kuyruk nazariyasining asosiy jarayoni bo'lib xizmat qiladigan Puasson jarayoni matematik modellar uchun muhim jarayon bo'lib, u ma'lum vaqt oynalarida tasodifiy sodir bo'lgan hodisalar modellari uchun dasturlarni topadi.^[127][128]

Haqiqiy yo'nalishda aniqlangan Puasson jarayoni stoxastik jarayon sifatida talqin qilinishi mumkin,^[51][129] boshqa tasodifiy narsalar qatorida.^[130][131] Ammo keyin uni belgilash mumkin ${ displaystyle n}$- o'lchovli Evklid fazosi yoki boshqa matematik bo'shliqlar,^[132] bu erda ko'pincha stoxastik jarayon o'rniga tasodifiy to'plam yoki tasodifiy hisoblash o'lchovi sifatida talqin etiladi.^[130][131] Ushbu parametrda, Poisson jarayoni, shuningdek, Poisson nuqta jarayoni deb ham ataladi, ehtimollar nazariyasida ham amaliy, ham nazariy sabablarga ko'ra eng muhim narsalardan biri hisoblanadi.^[24][133] Ammo ta'kidlanishicha, Puasson jarayoni boshqa matematik maydonlarda emas, balki aksariyat hollarda faqat haqiqiy chiziqda ko'rib chiqilganligi sababli kerakli darajada e'tiborni tortmaydi.^[133][134]

Ta'riflar

Stoxastik jarayon

Stoxastik jarayon umumiyga aniqlangan tasodifiy o'zgaruvchilar to'plami sifatida tavsiflanadi ehtimollik maydoni ${ displaystyle ( Omega, { mathcal {F}}, P)}$, qayerda ${ displaystyle Omega}$ a namuna maydoni, ${ displaystyle { mathcal {F}}}$ a ${ displaystyle sigma}$-algebrava ${ displaystyle P}$ a ehtimollik o'lchovi; va tasodifiy o'zgaruvchilar, ba'zi bir to'plam tomonidan indekslangan ${ displaystyle T}$, barchasi bir xil matematik bo'shliqda qiymatlarni qabul qiladi ${ displaystyle S}$, bo'lishi kerak o'lchovli ba'zilariga nisbatan ${ displaystyle sigma}$-algebra ${ displaystyle Sigma}$.^[30]

Boshqacha qilib aytganda, berilgan ehtimollik maydoni uchun ${ displaystyle ( Omega, { mathcal {F}}, P)}$ va o'lchanadigan joy ${ displaystyle (S, Sigma)}$, stoxastik jarayon bu to'plamdir ${ displaystyle S}$- quyidagicha yozilishi mumkin bo'lgan tasodifiy o'zgaruvchilar.^[82]

${ displaystyle {X (t): t in T }.}$

Tarixiy jihatdan, tabiatshunoslikning ko'plab muammolarida bir nuqta ${ displaystyle t in T}$ vaqtning ma'nosiga ega edi, shuning uchun ${ displaystyle X (t)}$ vaqtida kuzatilgan qiymatni ifodalovchi tasodifiy o'zgaruvchidir ${ displaystyle t}$.^[135] Stoxastik jarayonni quyidagicha yozish mumkin ${ displaystyle {X (t, omega): t in T }}$ aslida bu ikkita o'zgaruvchining funktsiyasi ekanligini aks ettirish uchun, ${ displaystyle t in T}$ va ${ displaystyle omega in Omega}$.^[30][136]

Stoxastik jarayonni ko'rib chiqishning boshqa usullari mavjud, yuqoridagi ta'rif an'anaviy deb hisoblanadi.^[70][71] Masalan, stoxastik jarayonni a deb izohlash yoki aniqlash mumkin ${ displaystyle S ^ {T}}$-qiymatli tasodifiy o'zgaruvchi, qaerda ${ displaystyle S ^ {T}}$ barcha mumkin bo'lgan bo'shliq ${ displaystyle S}$- baholangan funktsiyalari ning ${ displaystyle t in T}$ bu xarita to'plamdan ${ displaystyle T}$ kosmosga ${ displaystyle S}$.^[29][70]

Indeks o'rnatilgan

To'plam ${ displaystyle T}$ deyiladi indeks o'rnatilgan^[4][53] yoki parametrlar to'plami^[30][137] stoxastik jarayon. Ko'pincha bu to'plam. Ning ba'zi bir kichik to'plamidir haqiqiy chiziqkabi natural sonlar yoki to'plamni beradigan interval ${ displaystyle T}$ vaqt talqini.^[1] Ushbu to'plamlarga qo'shimcha ravishda indekslar to'plami ${ displaystyle T}$ boshqa chiziqli tartiblangan to'plamlar yoki undan ko'p umumiy matematik to'plamlar bo'lishi mumkin,^[1][56] dekartiya tekisligi kabi ${ displaystyle R ^ {2}}$ yoki ${ displaystyle n}$- o'lchovli Evklid fazosi, bu erda element ${ displaystyle t in T}$ kosmosdagi nuqtani aks ettirishi mumkin.^[50][138] Umuman olganda indekslar to'plami buyurtma qilinganida stoxastik jarayonlar uchun ko'proq natijalar va teoremalar mumkin.^[139]

Davlat maydoni

The matematik makon ${ displaystyle S}$ stoxastik jarayonning o'zi deyiladi davlat maydoni. Ushbu matematik makon yordamida aniqlanishi mumkin butun sonlar, haqiqiy chiziqlar, ${ displaystyle n}$- o'lchovli Evklid bo'shliqlari, murakkab tekisliklar yoki ko'proq mavhum matematik bo'shliqlar. Vaziyat maydoni stoxastik jarayon qabul qilishi mumkin bo'lgan turli xil qiymatlarni aks ettiruvchi elementlar yordamida aniqlanadi.^{[1][5][30][53][58]}

Namuna funktsiyasi

A namuna funktsiyasi bitta natija stoxastik jarayonning, shuning uchun u stoxastik jarayonning har bir tasodifiy o'zgaruvchining bitta mumkin bo'lgan qiymatini olish yo'li bilan hosil bo'ladi.^[30][140] Aniqrog'i, agar ${ displaystyle {X (t, omega): t in T }}$ stoxastik jarayon, keyin har qanday nuqta uchun ${ displaystyle omega in Omega}$, xaritalash

${ displaystyle X ( cdot, omega): T rightarrow S,}$

namunaviy funktsiya deyiladi, a amalga oshirishyoki, ayniqsa, qachon ${ displaystyle T}$ vaqt deb talqin etiladi, a namuna yo'li stoxastik jarayon ${ displaystyle {X (t, omega): t in T }}$.^[52] Bu shuni anglatadiki, sobit bo'lganlar uchun ${ displaystyle omega in Omega}$, indekslar to'plamini xaritalaydigan namunaviy funktsiya mavjud ${ displaystyle T}$ davlat makoniga ${ displaystyle S}$.^[30] Stoxastik jarayonning namunaviy funktsiyasining boshqa nomlariga quyidagilar kiradi traektoriya, yo'l funktsiyasi^[141] yoki yo'l.^[142]

O'sish

An o'sish stoxastik jarayon - bu bir xil stoxastik jarayonning ikkita tasodifiy o'zgaruvchisi orasidagi farq. Vaqt deb talqin qilinishi mumkin bo'lgan indekslar to'plami bo'lgan stoxastik jarayon uchun o'sish stoxastik jarayonning ma'lum vaqt oralig'ida qanchalik o'zgarishini anglatadi. Masalan, agar ${ displaystyle {X (t): t in T }}$ davlat makoniga ega stoxastik jarayon ${ displaystyle S}$ va indekslar to'plami ${ displaystyle T = [0, infty)}$, keyin istalgan ikkita manfiy bo'lmagan son uchun ${ displaystyle t_ {1} in [0, infty)}$ va ${ displaystyle t_ {2} in [0, infty)}$ shu kabi ${ displaystyle t_ {1} leq t_ {2}}$, farqi ${ displaystyle X_ {t_ {2}} - X_ {t_ {1}}}$ a ${ displaystyle S}$- o'sish deb nomlanuvchi tasodifiy o'zgaruvchi.^[50][51] Qo'shimchalar qiziqtirganda, ko'pincha davlat maydoni ${ displaystyle S}$ haqiqiy chiziq yoki tabiiy sonlar, ammo shunday bo'lishi mumkin ${ displaystyle n}$- o'lchovli Evklid fazosi yoki shunga o'xshash mavhum bo'shliqlar Banach bo'shliqlari.^[51]

Boshqa ta'riflar

Qonun

Stoxastik jarayon uchun ${ displaystyle X colon Omega rightarrow S ^ {T}}$ ehtimollik maydonida aniqlangan ${ displaystyle ( Omega, { mathcal {F}}, P)}$, qonun stoxastik jarayon ${ displaystyle X}$ deb belgilanadi tasvir o'lchovi:

${ displaystyle mu = P circ X ^ {- 1},}$

qayerda ${ displaystyle P}$ ehtimollik o'lchovi, belgisidir ${ displaystyle circ}$ funktsiya tarkibini va ${ displaystyle X ^ {- 1}}$ - bu o'lchanadigan funktsiyani oldindan tasviri yoki shunga teng ravishda ${ displaystyle S ^ {T}}$-qiymatli tasodifiy miqdor ${ displaystyle X}$, qayerda ${ displaystyle S ^ {T}}$ barcha mumkin bo'lgan bo'shliq ${ displaystyle S}$-ning funktsiyalari ${ displaystyle t in T}$, shuning uchun stoxastik jarayon qonuni ehtimollik o'lchovidir.^{[29][70][143][144]}

O'lchanadigan kichik to'plam uchun ${ displaystyle B}$ ning ${ displaystyle S ^ {T}}$, ning oldingi tasviri ${ displaystyle X}$ beradi

${ displaystyle X ^ {- 1} (B) = { omega in Omega: X ( omega) in B },}$

shuning uchun a qonuni ${ displaystyle X}$ quyidagicha yozilishi mumkin:^[30]

${ displaystyle mu (B) = P ( { omega in Omega: X ( omega) in B }).}$

Stoxastik jarayon yoki tasodifiy o'zgaruvchining qonuni ham ehtimollik qonuni, ehtimollik taqsimotiyoki tarqatish.^{[135][143][145][146][147]}

Ehtimollarning chekli taqsimoti

Stoxastik jarayon uchun ${ displaystyle X}$ qonun bilan ${ displaystyle mu}$, uning chekli o'lchovli taqsimotlar quyidagicha aniqlanadi:

${ displaystyle mu _ {t_ {1}, dots, t_ {n}} = P circ (X ({t_ {1}}), dots, X ({t_ {n}})) ^ { -1},}$

qayerda ${ displaystyle n geq 1}$ bu hisoblash soni va har bir to'plam ${ displaystyle t_ {i}}$ indekslar to'plamining bo'sh bo'lmagan cheklangan kichik to'plamidir ${ displaystyle T}$, shuning uchun har biri ${ displaystyle t_ {i} subset T}$, bu shuni anglatadiki ${ displaystyle t_ {1}, dots, t_ {n}}$ indeks to'plamining har qanday cheklangan to'plamidir ${ displaystyle T}$.^[29][148]

Har qanday o'lchovli kichik to'plam uchun ${ displaystyle C}$ ning ${ displaystyle n}$- katlama Dekart kuchi ${ displaystyle S ^ {n} = S times dots times S}$, stoxastik jarayonning chekli o'lchovli taqsimotlari ${ displaystyle X}$ quyidagicha yozilishi mumkin:^[30]

${ displaystyle mu _ {t_ {1}, dots, t_ {n}} (C) = P { Big (} { big {} omega in Omega: { big (} X_ {) t_ {1}} ( omega), dots, X_ {t_ {n}} ( omega) { big)} in C { big }} { Big)}.}$

Stoxastik jarayonning cheklangan o'lchovli taqsimotlari izchillik shartlari deb nomlanuvchi ikkita matematik shartni qondiradi.^[59]

Statsionarlik

Statsionarlik stoxastik jarayonning barcha tasodifiy o'zgaruvchilari bir xil taqsimlanganda stoxastik jarayonga ega bo'lgan matematik xususiyatdir. Boshqacha qilib aytganda, agar ${ displaystyle X}$ bu statsionar stoxastik jarayon, keyin har qanday narsa uchun ${ displaystyle t in T}$ tasodifiy o'zgaruvchi ${ displaystyle X_ {t}}$ bir xil taqsimotga ega, ya'ni har qanday to'plam uchun ${ displaystyle n}$ indeks o'rnatilgan qiymatlar ${ displaystyle t_ {1}, dots, t_ {n}}$, mos keladigan ${ displaystyle n}$ tasodifiy o'zgaruvchilar

${ displaystyle X_ {t_ {1}}, nuqta X_ {t_ {n}},}$

barchasi bir xil ehtimollik taqsimoti. Statsionar stoxastik jarayonning indekslar to'plami odatda vaqt deb talqin etiladi, shuning uchun u butun sonlar yoki haqiqiy chiziq bo'lishi mumkin.^[149][150] Ammo statsionarlik tushunchasi nuqta jarayonlari va tasodifiy maydonlar uchun ham mavjud bo'lib, unda indekslar to'plami vaqt sifatida talqin qilinmaydi.^{[149][151][152]}

Indeks o'rnatilganda ${ displaystyle T}$ vaqt deb talqin qilish mumkin, agar uning cheklangan o'lchovli taqsimotlari vaqt tarjimalari ostida o'zgarmas bo'lsa, stoxastik jarayon statsionar deb aytiladi. Stoxastik jarayonning bu turidan barqaror holatdagi, ammo baribir tasodifiy tebranishlarni boshdan kechiradigan jismoniy tizimni tavsiflash uchun foydalanish mumkin.^[149] Statsionarlikning sezgi shundaki, vaqt o'tgan sayin statsionar stoxastik jarayon taqsimoti bir xil bo'lib qoladi.^[153] Tasodifiy o'zgaruvchilar ketma-ketligi faqat tasodifiy o'zgaruvchilar bir xil taqsimlangan taqdirdagina statsionar stoxastik jarayonni hosil qiladi.^[149]

Yuqorida keltirilgan statsionarlik ta'rifi bilan stoxastik jarayon ba'zan qat'iy statsionar deyiladi, ammo statsionarlikning boshqa shakllari ham mavjud. Masalan, diskret vaqt yoki uzluksiz stoxastik jarayon ${ displaystyle X}$ keng ma'noda statsionar deyiladi, keyin jarayon ${ displaystyle X}$ hamma uchun cheklangan ikkinchi moment bor ${ displaystyle t in T}$ va ikkita tasodifiy o'zgaruvchining kovaryansiyasi ${ displaystyle X_ {t}}$ va ${ displaystyle X_ {t + h}}$ faqat songa bog'liq ${ displaystyle h}$ Barcha uchun ${ displaystyle t in T}$.^[153][154] Xinchin ga tegishli tushunchani taqdim etdi keng ma'noda statsionarlik, shu jumladan boshqa nomlarga ega kovaryans statsionarligi yoki keng ma'noda statsionarlik.^[154][155]

Filtrlash

A filtrlash sigma-algebralarning ketma-ketligi bo'lib, ba'zi ehtimollik fazosiga nisbatan belgilanadi va indekslar to'plamiga ega umumiy buyurtma munosabatlar, masalan, haqiqiy sonlarning ba'zi bir to'plami bo'lgan indekslar to'plamida. Rasmiy ravishda, agar stoxastik jarayonda umumiy tartib bilan indeks o'rnatilgan bo'lsa, u holda filtratsiya qilinadi ${ displaystyle {{ mathcal {F}} _ {t} } _ {t in T}}$, ehtimollik oralig'ida ${ displaystyle ( Omega, { mathcal {F}}, P)}$ sigma-algebralar oilasidir ${ displaystyle { mathcal {F}} _ {s} subseteq { mathcal {F}} _ {t} subseteq { mathcal {F}}}$ Barcha uchun ${ displaystyle s leq t}$, qayerda ${ displaystyle t, s in T}$ va ${ displaystyle leq}$ indekslar to'plamining umumiy tartibini bildiradi ${ displaystyle T}$.^[53] Filtrlash tushunchasi bilan stoxastik jarayonda mavjud bo'lgan ma'lumot miqdorini o'rganish mumkin ${ displaystyle X_ {t}}$ da ${ displaystyle t in T}$, bu vaqt sifatida talqin qilinishi mumkin ${ displaystyle t}$.^[53][156] Filtrlashning sezgi ${ displaystyle { mathcal {F}} _ {t}}$ vaqt kabi ${ displaystyle t}$ o'tadi, tobora ko'proq ma'lumot ${ displaystyle X_ {t}}$ ushlanib qolgan ma'lum yoki mavjud ${ displaystyle { mathcal {F}} _ {t}}$, natijada qismlarning yanada nozik va nozik qismlari ${ displaystyle Omega}$.^[157][158]

O'zgartirish

A o'zgartirish stoxastik jarayon - bu boshqa stoxastik jarayon bo'lib, u asl stoxastik jarayon bilan chambarchas bog'liqdir. Aniqrog'i, stoxastik jarayon ${ displaystyle X}$ bir xil indeksga ega ${ displaystyle T}$, bo'sh joyni o'rnating ${ displaystyle S}$va ehtimollik maydoni ${ displaystyle ( Omega, { cal {F}}, P)}$ boshqa stoxastik jarayon sifatida ${ displaystyle Y}$ ning modifikatsiyasi deb aytilgan ${ displaystyle Y}$ agar hamma uchun bo'lsa ${ displaystyle t in T}$ quyidagi

${ displaystyle P (X_ {t} = Y_ {t}) = 1,}$

ushlab turadi. Bir-birining modifikatsiyasi bo'lgan ikkita stoxastik jarayon bir xil cheklangan o'lchov qonuniga ega^[159] va ular aytilgan stoxastik jihatdan teng yoki teng.^[160]

O'zgartirish o'rniga atama versiyasi shuningdek ishlatiladi,^{[151][161][162][163]} Ammo ba'zi bir mualliflar ikkita stoxastik jarayon bir xil sonli o'lchovli taqsimotga ega bo'lganda, bu atama versiyasidan foydalanadilar, ammo ular har xil ehtimollik oralig'ida aniqlanishi mumkin, shuning uchun bir-birining modifikatsiyasi bo'lgan ikkita jarayon ham ikkinchi ma'noda bir-birining versiyasidir , lekin aksincha emas.^[164][143]

Agar doimiy ravishda real vaqtda baholanadigan stoxastik jarayon uning o'sishida ma'lum bir moment shartlariga javob bersa, u holda Kolmogorov uzluksizligi teoremasi Ushbu jarayonning ehtimolligi bitta bo'lgan doimiy namuna yo'llariga ega bo'lgan modifikatsiyasi mavjud, shuning uchun stoxastik jarayon doimiy modifikatsiyasiga yoki versiyasiga ega.^{[162][163][165]} Teorema tasodifiy maydonlarda ham umumlashtirilishi mumkin, shuning uchun indekslar to'plami ${ displaystyle n}$- o'lchovli Evklid fazosi^[166] shuningdek, stoxastik jarayonlarga metrik bo'shliqlar ularning davlat makonlari sifatida.^[167]

Ajratib bo'lmaydi

Ikki stoxastik jarayon ${ displaystyle X}$ va ${ displaystyle Y}$ bir xil ehtimollik maydonida aniqlangan ${ displaystyle ( Omega, { mathcal {F}}, P)}$ bir xil indeks bilan ${ displaystyle T}$ va bo'sh joyni o'rnating ${ displaystyle S}$ deb aytiladi ajratib bo'lmaydigan agar quyidagilar bo'lsa

${ displaystyle P (X_ {t} = Y_ {t} { text {for all}} t in T) = 1,}$

ushlab turadi.^[143][159] Ikki bo'lsa ${ displaystyle X}$ va ${ displaystyle Y}$ bir-birlarining modifikatsiyalari va deyarli uzluksiz, keyin ${ displaystyle X}$ va ${ displaystyle Y}$ bir-biridan farq qilmaydi.^[168]

Ajratish

Ajratish ehtimollik o'lchoviga nisbatan o'rnatilgan indeksga asoslangan stoxastik jarayonning xususiyati. Xususiyat stoxastik jarayonlarning funktsiyalari yoki hisoblanmaydigan indekslar to'plamiga ega bo'lgan tasodifiy maydonlar tasodifiy o'zgaruvchilar hosil qilishi uchun qabul qilinadi. Stoxastik jarayon ajralib turishi uchun, boshqa shartlardan tashqari, uning indeks to'plami a bo'lishi kerak ajratiladigan joy,^[b] demak, indekslar to'plami zich hisoblanadigan kichik to'plamga ega.^[151][169]

Aniqrog'i, real qiymatga ega bo'lgan doimiy stoxastik jarayon ${ displaystyle X}$ ehtimollik maydoni bilan ${ displaystyle ( Omega, { cal {F}}, P)}$ agar indeks o'rnatilgan bo'lsa, ajratish mumkin ${ displaystyle T}$ zich hisoblanadigan kichik to'plamga ega ${ displaystyle U subset T}$ va to'plam bor ${ displaystyle Omega _ {0} subset Omega}$ ehtimollik nolga teng, shuning uchun ${ displaystyle P ( Omega _ {0}) = 0}$, har bir ochiq to'plam uchun ${ displaystyle G subset T}$ va har bir yopiq to'plam ${ displaystyle F subset textstyle R = (- infty, infty)}$, ikkita voqea ${ displaystyle {X_ {t} in F { text {for all}} t in G cap U }}$ va ${ displaystyle {X_ {t} in F { text {for all}} t in G }}$ pastki qismida bir-biridan maksimal darajada farq qiladi ${ displaystyle Omega _ {0}}$.^{[170][171][172]}Ajralish ta'rifi^[c] boshqa indekslar to'plamlari va holatlar uchun ham ko'rsatilishi mumkin,^[175] masalan, tasodifiy maydonlarda bo'lgani kabi, bu erda ham indeks o'rnatiladi, balki holat maydoni ham bo'lishi mumkin ${ displaystyle n}$- o'lchovli Evklid fazosi.^[32][151]

Stoxastik jarayonning ajralib turishi tushunchasi tomonidan kiritilgan Jozef Dub,^[169]. Ajratishning asosiy g'oyasi indekslar to'plamining hisoblanadigan to'plamlari stoxastik jarayonning xususiyatlarini aniqlashdir.^[173] Hisoblanadigan indekslar to'plami bo'lgan har qanday stoxastik jarayon allaqachon ajratish shartlariga javob beradi, shuning uchun diskret vaqt stoxastik jarayonlar har doim ajralib turadi.^[176] Doob teoremasi, ba'zida Doobning ajralish teoremasi deb ham ataladi, har qanday real qiymatli doimiy va doimiy stoxastik jarayonning ajraladigan modifikatsiyasi mavjud.^{[169][171][177]} Ushbu teoremaning versiyalari, shuningdek, haqiqiy chiziqdan tashqari indekslar to'plamlari va holat bo'shliqlari bo'lgan umumiy stoxastik jarayonlar uchun mavjud.^[137]

Mustaqillik

Ikki stoxastik jarayon ${ displaystyle X}$ va ${ displaystyle Y}$ bir xil ehtimollik maydonida aniqlangan ${ displaystyle ( Omega, { mathcal {F}}, P)}$ bir xil indeks bilan ${ displaystyle T}$ deb aytiladi mustaqil agar hamma uchun bo'lsa ${ displaystyle n in mathbb {N}}$ va davrlarning har bir tanlovi uchun ${ displaystyle t_ {1}, ldots, t_ {n} in T}$, tasodifiy vektorlar ${ displaystyle chap (X (t_ {1}), ldots, X (t_ {n}) o'ng)}$ va ${ displaystyle chap (Y (t_ {1}), ldots, Y (t_ {n}) o'ng)}$ mustaqil.^{[178]:p. 515}

Aloqasizlik

Ikki stoxastik jarayon ${ displaystyle left {X_ {t} right }}$ va ${ displaystyle left {Y_ {t} right }}$ deyiladi aloqasiz agar ularning o'zaro bog'liqligi ${ displaystyle operator nomi {K} _ { mathbf {X} mathbf {Y}} (t_ {1}, t_ {2}) = operator nomi {E} chap [ chap (X (t_ {1}) ) - mu _ {X} (t_ {1}) o'ng) chap (Y (t_ {2}) - mu _ {Y} (t_ {2}) o'ng) o'ng]}$ hamma vaqt uchun nolga teng.^{[179]:p. 142} Rasmiy ravishda:

${ displaystyle chap {X_ {t} o'ng }, chap {Y_ {t} o'ng } { matn {o'zaro bog'liq emas}} quad iff quad operator nomi {K} _ { mathbf {X} mathbf {Y}} (t_ {1}, t_ {2}) = 0 quad forall t_ {1}, t_ {2}}$.

Mustaqillik o'zaro bog'liq bo'lmaganlikni anglatadi

Agar ikkita stoxastik jarayon bo'lsa ${ displaystyle X}$ va ${ displaystyle Y}$ mustaqil, keyin ular ham o'zaro bog'liq emas.^{[179]:p. 151}

Ortogonallik

Ikki stoxastik jarayon ${ displaystyle left {X_ {t} right }}$ va ${ displaystyle left {Y_ {t} right }}$ deyiladi ortogonal agar ularning o'zaro bog'liqligi bo'lsa ${ displaystyle operator nomi {R} _ { mathbf {X} mathbf {Y}} (t_ {1}, t_ {2}) = operator nomi {E} [X (t_ {1}) { overline { Y (t_ {2})}}]}$ hamma vaqt uchun nolga teng.^{[179]:p. 142} Rasmiy ravishda:

${ displaystyle left {X_ {t} right }, left {Y_ {t} right } { text {ortogonal}} quad iff quad operator nomi {R} _ { mathbf {X} mathbf {Y}} (t_ {1}, t_ {2}) = 0 quad forall t_ {1}, t_ {2}}$.

Skoroxod maydoni

A Skoroxod maydoni, shuningdek yozilgan Skorohod maydoni, chap chiziqlar bilan o'ng uzluksiz barcha funktsiyalarning matematik maydoni bo'lib, masalan, haqiqiy chiziqning ba'zi bir oralig'ida aniqlangan. ${ displaystyle [0,1]}$ yoki ${ displaystyle [0, infty)}$, va haqiqiy chiziqda yoki ba'zi bir metrik bo'shliqlarda qiymatlarni oling.^{[180][181][182]} Bunday funktsiyalar frantsuzcha iboraning qisqartirilishiga asoslanib, cdlàg yoki cadlag funktsiyalari sifatida tanilgan continue à droite, limite à gauche, funktsiyalar chap chegaralar bilan o'ng uzluksiz bo'lishi sababli.^[180][183] Skorokhod funktsional maydoni, tomonidan kiritilgan Anatoliy Skoroxod,^[182] ko'pincha harf bilan belgilanadi ${ displaystyle D}$,^{[180][181][182][183]} shuning uchun funktsiya maydoni bo'shliq deb ham ataladi ${ displaystyle D}$.^{[180][184][185]} Ushbu funktsiya makonining yozuvi barcha cdlàg funktsiyalari aniqlangan intervalni ham o'z ichiga olishi mumkin, shuning uchun, masalan ${ displaystyle D [0,1]}$ -da aniqlangan cdlàg funktsiyalar maydonini bildiradi birlik oralig'i ${ displaystyle [0,1]}$.^{[183][185][186]}

Skoroxod funktsiyalari bo'shliqlari stoxastik jarayonlar nazariyasida tez-tez ishlatiladi, chunki u doimo doimiy stoxastik jarayonlarning namunaviy funktsiyalari Skoroxod fazosiga tegishli deb taxmin qilgan.^[182][184] Bunday bo'shliqlar doimiy ravishda funktsiyalarni o'z ichiga oladi, ular Wiener jarayonining namunaviy funktsiyalariga mos keladi. Ammo kosmosda uzilishlarga ega funktsiyalar ham mavjud, demak, sakrashli stoxastik jarayonlarning namunaviy funktsiyalari, masalan, Puasson jarayoni (haqiqiy chiziqda) ham shu fazoning a'zolari.^[185][187]

Muntazamlik

Stoxastik jarayonlarning matematik qurilishi sharoitida atama muntazamlik mumkin bo'lgan qurilish masalalarini hal qilish uchun stoxastik jarayon uchun muayyan shartlarni muhokama qilish va taxmin qilishda foydalaniladi.^[188][189] Masalan, stoxastik jarayonlarni hisoblab bo'lmaydigan ko'rsatkichlar to'plami bilan o'rganish uchun stoxastik jarayon namunaviy funktsiyalar uzluksiz bo'lishi kabi ba'zi bir muntazamlik shartlariga rioya qiladi deb taxmin qilinadi.^[190][191]

Boshqa misollar

Markov jarayonlari va zanjirlari

Markov jarayonlari an'anaviy ravishda stoxastik jarayonlardir diskret yoki uzluksiz vaqt, Markov xususiyatiga ega bo'lgan, ya'ni Markov jarayonining keyingi qiymati joriy qiymatga bog'liq, ammo u stoxastik jarayonning oldingi qiymatlaridan shartli ravishda mustaqil. Boshqacha qilib aytganda, jarayonning kelajakdagi xatti-harakatlari stoxastik ravishda o'tmishdagi xatti-harakatlaridan mustaqil bo'lib, jarayonning hozirgi holatini hisobga olgan holda.^[192][193]

Braun harakat jarayoni va Puasson jarayoni (bir o'lchovda) ikkalasi ham Markov jarayonlariga misoldir^[194] doimiy vaqt ichida tasodifiy yurish butun sonlarda va qimorbozning xarobasi muammo diskret vaqtdagi Markov jarayonlariga misoldir.^[195][196]

Markov zanjiri - bu diskret bo'lgan Markov jarayonining bir turi davlat maydoni yoki diskret indekslar to'plami (ko'pincha vaqtni aks ettiradi), ammo Markov zanjirining aniq ta'rifi turlicha.^[197] Masalan, Markov zanjirini ikkalasida ham Markov jarayoni deb ta'riflash odatiy holdir diskret yoki uzluksiz vaqt hisoblash mumkin bo'lgan bo'shliq bilan (shuning uchun vaqtning tabiatidan qat'i nazar),^{[198][199][200][201]} Markov zanjirini hisoblash yoki uzluksiz holat fazosida diskret vaqtga ega deb belgilash odatiy holga aylandi (shuning uchun holat makonidan qat'iy nazar).^[197] Markov zanjirining birinchi ta'rifi, unda diskret vaqt bor, endi ikkinchi ta'rif kabi tadqiqotchilar tomonidan ishlatilganiga qaramay, foydalanishga moyil ekanligi ta'kidlangan. Jozef Dub va Kay Lay Chung.^[202]

Markov jarayonlari muhim stoxastik jarayonlar sinfini tashkil etadi va ko'plab sohalarda qo'llanilishiga ega.^[41][203] Masalan, ular ma'lum bo'lgan umumiy stoxastik simulyatsiya usuli uchun asosdir Monte Karlo Markov zanjiri, bu aniq ehtimollik taqsimotiga ega bo'lgan tasodifiy ob'ektlarni simulyatsiya qilish uchun ishlatiladi va dasturni topdi Bayes statistikasi.^[204][205]

Markov xususiyati kontseptsiyasi dastlab doimiy va diskret vaqtdagi stoxastik jarayonlar uchun mo'ljallangan edi, ammo bu xususiyat boshqa indekslar to'plamlariga moslashtirildi, masalan. ${ displaystyle n}$Markov tasodifiy maydonlari deb nomlanuvchi tasodifiy o'zgaruvchilar to'plamlariga olib keladigan o'lchovli Evklid fazosi.^{[206][207][208]}

Martingeyl

Martingale - bu har bir lahzada, joriy qiymat va jarayonning barcha o'tgan qiymatlarini hisobga olgan holda, har bir kelajakdagi qiymatning shartli kutilishi joriy qiymatga teng bo'lgan xususiyatga ega bo'lgan diskret-vaqt yoki uzluksiz stoxastik jarayon. Diskret vaqt ichida, agar bu xususiyat keyingi qiymatga ega bo'lsa, unda u kelajakdagi barcha qiymatlarga tegishli bo'ladi. Martingalening aniq matematik ta'rifi, filtrlashning matematik kontseptsiyasi bilan birlashtirilgan yana ikkita shartni talab qiladi, bu vaqt o'tgan sayin mavjud ma'lumotlarning sezgi bilan bog'liq. Martingalalar odatda haqiqiy qiymatga ega,^{[209][210][156]} lekin ular ham murakkab baholanishi mumkin^[211] yoki undan ham umumiy.^[212]

Nosimmetrik tasodifiy yurish va Wiener jarayoni (nolga siljish bilan) har ikkisi ham navbati bilan diskret va uzluksiz vaqt ichida martingalalarga misol bo'la oladi.^[209][210] Uchun ketma-ketlik ning mustaqil va bir xil taqsimlangan tasodifiy o'zgaruvchilar ${ displaystyle X_ {1}, X_ {2}, X_ {3}, dots}$ nolinchi o'rtacha bilan stoxastik jarayon ketma-ket qisman yig'indilardan hosil bo'ldi ${displaystyle X_{1},X_{1}+X_{2},X_{1}+X_{2}+X_{3},dots }$ is a discrete-time martingale.^[213] In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.^[214]

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.^[210] Martingales can also be built from other martingales.^[213] For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.^[209][215]

Martingales mathematically formalize the idea of a fair game,^[216] and they were originally developed to show that it is not possible to win a fair game.^[217] But now they are used in many areas of probability, which is one of the main reasons for studying them.^{[156][217][218]} Many problems in probability have been solved by finding a martingale in the problem and studying it.^[219] Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.^{[214][220][221]}

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.^[222] They have found applications in areas in probability theory such as queueing theory and Palm calculus^[223] and other fields such as economics^[224] va moliya.^[19]

Levi jarayoni

Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.^[51][225] These processes have many applications in fields such as finance, fluid mechanics, physics and biology.^[226][227] The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process ${ displaystyle X}$ is a Lévy process if for ${ displaystyle n}$ non-negatives numbers, ${displaystyle 0leq t_{1}leq dots leq t_{n}}$, mos keladigan ${ displaystyle n-1}$ o'sish

${displaystyle X_{t_{2}}-X_{t_{1}},dots ,X_{t_{n-1}}-X_{t_{n}},}$

are all independent of each other, and the distribution of each increment only depends on the difference in time.^[51]

A Lévy process can be defined such that its state space is some abstract mathematical space, such as a Banach maydoni, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so ${displaystyle I=[0,infty )}$, which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and subordinators are all Lévy processes.^[51][225]

Tasodifiy maydon

A random field is a collection of random variables indexed by a ${ displaystyle n}$-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line.^[32] But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions.^[5][30][228] If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.^[229]

Nuqta jarayoni

A point process is a collection of points randomly located on some mathematical space such as the real line, ${ displaystyle n}$-dimensional Euclidean space, or more abstract spaces. Ba'zan atama point process is not preferred, as historically the word jarayon denoted an evolution of some system in time, so a point process is also called a random point field.^[230] There are different interpretations of a point process, such a random counting measure or a random set.^[231][232] Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process,^[233][234] though it has been remarked that the difference between point processes and stochastic processes is not clear.^[234]

Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space^[d] on which it is defined, such as the real line or ${ displaystyle n}$- o'lchovli Evklid fazosi.^[237][238] Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.^[239][240]

Tarix

Early probability theory

Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,^[241][242] but very little analysis on them was done in terms of probability.^[241][243] The year 1654 is often considered the birth of probability theory when French mathematicians Pierre Fermat va Blez Paskal had a written correspondence on probability, motivated by a qimor muammosi.^{[241][244][245]} But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae tomonidan Gerolamo Kardano, written in the 16th century but posthumously published later in 1663.^[241][246]

After Cardano, Yakob Bernulli^[e] yozgan Ars Conjectandi, which is considered a significant event in the history of probability theory.^[241] Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability.^{[241][248][249]} But despite some renowned mathematicians contributing to probability theory, such as Per-Simon Laplas, Avraam de Moivre, Karl Gauss, Simyon Poisson va Pafnutiy Chebyshev,^[250][251] most of the mathematical community^[f] did not consider probability theory to be part of mathematics until the 20th century.^{[250][252][253][254]}

Statistik mexanika

In the physical sciences, scientists developed in the 19th century the discipline of statistik mexanika, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Klauziy, most of the work had little or no randomness.^[255][256]This changed in 1859 when Jeyms Klerk Maksvell contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.^[257][258] The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Lyudvig Boltsman va Josiah Gibbs, keyinchalik bu ta'sir ko'rsatishi mumkin Albert Eynshteyn's mathematical model for Braun harakati.^[259]

Measure theory and probability theory

Da Xalqaro matematiklar kongressi yilda Parij 1900 yilda, Devid Xilbert presented a list of matematik muammolar, where his sixth problem asked for a mathematical treatment of physics and probability involving aksiomalar.^[251] Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, Anri Lebesgue va Emil Borel. In 1925 another French mathematician Pol Levi published the first probability book that used ideas from measure theory.^[251]

In 1920s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergey Bernshteyn, Aleksandr Xinchin,^[g] va Andrey Kolmogorov.^[254] Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.^[260] In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky va Nikolai Smirnov,^[261] and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.^[65][262][h]

Birth of modern probability theory

In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,^[men] where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.^[251][254]

After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as Jozef Dub, Uilyam Feller, Moris Frechet, Pol Levi, Volfgang Doeblinva Xarald Kramer.^[251][254]Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".^[254] Ikkinchi jahon urushi greatly interrupted the development of probability theory, causing, for example, the migration of Feller from Shvetsiya uchun Amerika Qo'shma Shtatlari^[254] and the death of Doeblin, considered now a pioneer in stochastic processes.^[264]

Matematik Jozef Dub did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales.^[265][263] Uning kitobi Stochastic Processes is considered highly influential in the field of probability theory.^[266]

Stochastic processes after World War II

After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.^[254][267] 1940-yillardan boshlab, Kiyosi Itô published papers developing the field of stoxastik hisob, which involves stochastic integrallar and stochastic differentsial tenglamalar based on the Wiener or Brownian motion process.^[268]

Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of potentsial nazariyasi, with early ideas by Shizuo Kakutani and then later work by Joseph Doob.^[267] Further work, considered pioneering, was done by Gilbert Xant in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.^{[23][269][270]}

In 1953 Doob published his book Stoxastik jarayonlar, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.^[267][266] Doob also chiefly developed the theory of martingales, with later substantial contributions by Pol-Andre Meyer. Earlier work had been carried out by Sergey Bernshteyn, Pol Levi va Jan Vill, the latter adopting the term martingale for the stochastic process.^[271][272] Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.^[267]

Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations.^[267] The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s fundamental work was done by Alexander Wentzell in the Soviet Union and Monroe D. Donsker va Srinivasa Varadhan in the United States of America,^[273] which would later result in Varadhan winning the 2007 Abel Prize.^[274] In the 1990s and 2000s the theories of Schramm – Loewner evolyutsiyasi^[275] va rough paths^[143] were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Maydonlar medallari being awarded to Vendelin Verner^[276] 2008 yilda va Martin Xayrer 2014 yilda.^[277]

The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.^[47][226]

Discoveries of specific stochastic processes

Although Khinchin gave mathematical definitions of stochastic processes in the 1930s,^[65][262] specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process.^[23][26] Some families of stochastic processes such as point processes or renewal processes have long and complex histories, stretching back centuries.^[278]

Bernulli jarayoni

The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied.^[83] The process is a sequence of independent Bernoulli trials,^[84] which are named after Jackob Bernoulli who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens.^[279] Bernoulli's work, including the Bernoulli process, were published in his book Ars Conjectandi 1713 yilda.^[280]

Tasodifiy yurish

1905 yilda Karl Pirson atamani o'ylab topdi tasodifiy yurish while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks.^[91][280] For example, the problem known as the Gambler's ruin is based on a simple random walk,^[196][281] and is an example of a random walk with absorbing barriers.^[244][282] Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods,^[283] and then more detailed solutions were presented by Jakob Bernoulli and Avraam de Moivre.^[284]

For random walks in ${ displaystyle n}$-dimensional integer panjaralar, Jorj Polya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.^[285][286]

Wiener jarayoni

The Wiener jarayoni or Brownian motion process has its origins in different fields including statistics, finance and physics.^[23] 1880 yilda, Torvald Thiele wrote a paper on the method of least squares, where he used the process to study the errors of a model in time-series analysis.^{[287][288][289]} The work is now considered as an early discovery of the statistical method known as Kalman filtering, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.^[289]

Norbert Viner gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of Torvald Thiele, Louis Bachelierva Albert Eynshteyn.^[23]

Frantsuz matematikasi Louis Bachelier used a Wiener process in his 1900 thesis^[290][291] in order to model price changes on the Parij birjasi, a Fond birjasi,^[292] without knowing the work of Thiele.^[23] It has been speculated that Bachelier drew ideas from the random walk model of Jyul Regna, but Bachelier did not cite him,^[293] and Bachelier's thesis is now considered pioneering in the field of financial mathematics.^[292][293]

It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the 1950s by the Leonard Savage, and then become more popular after Bachelier's thesis was translated into English in 1964. But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas,^[293] which was cited by mathematicians including Doob, Feller^[293] and Kolmogorov.^[23] The book continued to be cited, but then starting in the 1960s the original thesis by Bachelier began to be cited more than his book when economists started citing Bachelier's work.^[293]

1905 yilda Albert Eynshteyn published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the gazlarning kinetik nazariyasi. Einstein derived a differentsial tenglamadeb nomlanuvchi diffuziya tenglamasi, for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement, Marian Smoluchovskiy published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.^[294]

Einstein's work, as well as experimental results obtained by Jan Perrin, later inspired Norbert Wiener in the 1920s^[295] to use a type of measure theory, developed by Percy Daniell, and Fourier analysis to prove the existence of the Wiener process as a mathematical object.^[23]

Poisson jarayoni

The Poisson process is named after Simyon Poisson, due to its definition involving the Poissonning tarqalishi, but Poisson never studied the process.^[24][296] There are a number of claims for early uses or discoveries of the Poissonprocess.^[24][26]At the beginning of the 20th century the Poisson process would arise independently in different situations.^[24][26]In Sweden 1903, Filip Lundberg nashr etilgan tezis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.^[297][298]

Another discovery occurred in Daniya in 1909 when A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.^[24]

1910 yilda Ernest Rezerford va Xans Geyger published experimental results on counting alpha particles. Motivated by their work, Garri Beytmen studied the counting problem and derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process.^[24] After this time there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.^[24]

Markov processes

Markov processes and Markov chains are named after Andrey Markov who studied Markov chains in the early 20th century.^[299] Markov was interested in studying an extension of independent random sequences.^[299] In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a katta sonlarning kuchsiz qonuni without the independence assumption,^{[6][300][301][302]} which had been commonly regarded as a requirement for such mathematical laws to hold.^[302] Markov later used Markov chains to study the distribution of vowels in Evgeniy Onegin, tomonidan yozilgan Aleksandr Pushkin, and proved a markaziy chegara teoremasi for such chains.^[6][300]

In 1912 Poincaré studied Markov chains on cheklangan guruhlar with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Pol va Tatyana Erenfest in 1907, and a branching process, introduced by Frensis Galton va Henry William Watson in 1873, preceding the work of Markov.^[300][301] After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé.^[303] Starting in 1928, Moris Frechet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.^[300][304]

Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.^[254][260] Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Viner's work on Einstein's model of Brownian movement.^[260][305] He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.^[260][306] Independent of Kolmogorov's work, Sidney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.^[307] The differential equations are now called the Kolmogorov equations^[308] or the Kolmogorov–Chapman equations.^[309] Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.^[254]

Lévy processes

Lévy processes such as the Wiener process and the Poisson process (on the real line) are named after Paul Lévy who started studying them in the 1930s,^[226] but they have connections to cheksiz bo'linadigan taqsimotlar going back to the 1920s.^[225] In a 1932 paper Kolmogorov derived a xarakterli funktsiya for random variables associated with Lévy processes. This result was later derived under more general conditions by Lévy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937.^[254][310] In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by Bruno de Finetti va Kiyosi Itô.^[225]

Mathematical construction

In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically.^[59] There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.^[311]

Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem^[j] to prove a corresponding stochastic process exists.^[59][311] This theorem, which is an existence theorem for measures on infinite product spaces,^[315] agar har qanday cheklangan o'lchovlar ikkita shartni qondirsa, deyiladi izchillik shartlari, keyin cheklangan o'lchovli taqsimot bilan stoxastik jarayon mavjud.^[59]

Qurilish masalalari

Uzluksiz vaqtli stoxastik jarayonlarni qurishda diskret vaqtli jarayonlar bilan yuzaga kelmaydigan indekslar to'plami tufayli ma'lum matematik qiyinchiliklar paydo bo'ladi.^[60][61] Bitta muammo shundaki, bir xil sonli o'lchovli taqsimot bilan bir nechta stoxastik jarayon bo'lishi mumkin. Masalan, Puasson jarayonining chap uzluksiz modifikatsiyasi ham, o'ng uzluksiz modifikatsiyasi ham bir xil sonli o'lchovli taqsimotlarga ega.^[316] Bu shuni anglatadiki, stoxastik jarayonning taqsimlanishi, albatta, stoxastik jarayonning namunaviy funktsiyalarining xususiyatlarini aniq ko'rsatmaydi.^[311][317]

Yana bir muammo shundaki, indekslar to'plamining hisoblab bo'lmaydigan sonli nuqtalariga tayanadigan doimiy ish jarayonining funktsiyalari o'lchovli bo'lmasligi mumkin, shuning uchun ba'zi bir hodisalar ehtimolligi yaxshi aniqlanmagan bo'lishi mumkin.^[169] Masalan, stoxastik jarayon yoki tasodifiy maydon supremumi aniq belgilangan tasodifiy miqdor emas.^[32][61] Uzluksiz stoxastik jarayon uchun ${ displaystyle X}$, indeks to'plamining hisoblab bo'lmaydigan soniga bog'liq bo'lgan boshqa xususiyatlar ${ displaystyle T}$ quyidagilarni o'z ichiga oladi:^[169]

• stoxastik jarayonning namunaviy funktsiyasi ${ displaystyle X}$ a doimiy funktsiya ning ${ displaystyle t in T}$;
• stoxastik jarayonning namunaviy funktsiyasi ${ displaystyle X}$ a cheklangan funktsiya ning ${ displaystyle t in T}$; va
• stoxastik jarayonning namunaviy funktsiyasi ${ displaystyle X}$ bu ortib borayotgan funktsiya ning ${ displaystyle t in T}$.

Ushbu ikkita qiyinchilikni engish uchun har xil taxminlar va yondashuvlar mumkin.^[71]

Qurilish masalalarini hal qilish

Stoxastik jarayonlarning matematik konstruktsiyasi masalalaridan qochish uchun bitta yondashuv Jozef Dub, stoxastik jarayon ajralib turadigan deb taxmin qilishdir.^[318] Ajratuvchanlik cheksiz o'lchovli taqsimotlarning namunaviy funktsiyalarning xususiyatlarini indekslar to'plamidagi zich hisoblanadigan nuqtalar to'plamidagi namunaviy funktsiyalarning mohiyati bo'yicha aniqlanishini talab qilish orqali ta'minlaydi.^[319] Bundan tashqari, agar stoxastik jarayon ajralib turadigan bo'lsa, u holda indekslar to'plamining hisoblab bo'lmaydigan sonli nuqtalarining funktsional imkoniyatlari o'lchanadi va ularning ehtimolliklarini o'rganish mumkin.^[169][319]

Dastlab ishlab chiqilgan yana bir yondashuv mumkin Anatoliy Skoroxod va Andrey Kolmogorov,^[320] uzluksiz vaqtdagi stoxastik jarayon uchun har qanday metrik faza o'zining holat fazosi sifatida. Bunday stoxastik jarayonni qurish uchun stoxastik jarayonning namunaviy funktsiyalari ba'zi mos funktsiyalar maydoniga tegishli deb taxmin qilinadi, bu odatda chap chegaralari bo'lgan barcha o'ng uzluksiz funktsiyalardan iborat bo'lgan Skoroxod fazosidir. Ushbu yondashuv hozirda ajralib chiqish taxminidan ko'ra ko'proq foydalanilmoqda,^[71][265] ammo ushbu yondashuvga asoslangan bunday stoxastik jarayon avtomatik ravishda ajralib turadi.^[321]

Kamroq ishlatilgan bo'lsada, bo'linish taxminlari umumiyroq hisoblanadi, chunki har bir stoxastik jarayonning ajratiladigan versiyasi mavjud.^[265] Shuningdek, u Skoroxod makonida stoxastik jarayonni qurish imkoniyati bo'lmagan hollarda ham qo'llaniladi.^[174] Masalan, tasodifiy maydonlarni qurishda va o'rganishda bo'linish qabul qilinadi, bu erda tasodifiy o'zgaruvchilar yig'indisi endi haqiqiy chiziqdan tashqari to'plamlar bilan indekslanadi. ${ displaystyle n}$- o'lchovli Evklid fazosi.^[32][322]

Shuningdek qarang

Izohlar

Adabiyotlar