The Bk of Thld Fcl Isuos Polcy Group Que Models & Fcl Egeerg Tem Fcl Mhemcs Foudo Noe 8 STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION. ก Through he use of ordry d/or prl deres, ODE/PDE c rele oupu rle o oe or more pu rles, w.r.. whom he former s deermsc fuco s well s smuleously soluo o sd ODE/PDE. Cosder PDE wo pu rles, oe of whom specfclly deoes he pssge of me [,, whle he oher s some quy whose lue depeds deermsclly o me, whece self fuco of me x x. The soluo s of he form y y x,, [,, whose ol dere dfferel oo or ol dfferel s ge he ch rule y: y y y dx y y dx y y y x, dy dx d d d d x x d x d A me-dom sochsc process { X, } s fmly of me-dexed rdom rles, d s such c lso e hough of eher s re deermsc fuco of me d relso, X X, ω, [,, ω Ω, or s ure rdom fuco of me, X X, [,, u eer o e equed wh ure deermsc fuco of me. Dffcules rse whe oe res o formule PDE wo pu rles, oe of whom specfclly deoes he pssge of me [,, whle he oher s some quy whose lue depeds o me d rdom chces, hece self sochsc process { X, }. [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.78][Mr.55B.E.]
The soluo, for whom o ol dere my e defed, les o y wy of he rdol ch rule, s of he form: Y Y X, Y X, ω,, [,, ω Ω 3 STOCHASTIC DIFFERENTIAL oo for fesml chge sochsc process {, },.e. he sese of ryg o defe: X dx lm { X X } 4 BRONIAN INNOVATION TERM sochsc dfferel oo for sdrd Brow moo/eer process: d lm{ } 5 ~ Ν, d ~ Ν, d STOCHASTIC DIFFERENTIAL EQUATION SDE geerlsed dfferel equo h cos les oe sochsc process s pu rle, hece he presece somewhere of sochsc dfferel, prculr he Brow oo erm esselly wh mkes SDE much more h jus regulr PDE wh some rdom oses dded o. GEOMETRIC BRONIAN MOTION GBM sochsc process { S, } descrle y he followg SDE: ds S µ d d, d ~ Ν, d 6 σ ITO PROCESS sochsc process { X, } descrle y he followg SDE: s deermsc pr 64748 4 dx d sochsc pr 6444447 444448 µ d σ d, d ~ Ν, σ d 7 3 drf 443 dffuso { oo [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.79][Mr.55B.E.]
DRIFT TERM deermsc s pr of SDE o s ow forms ODE/PDE. DIFFUSION TERM sochsc d pr of SDE cog sochsc dfferels. GENERALISED IENER PROCESS sochsc process { X, } descrle y he followg SDE: s deermsc pr 6447 448 dx d sochsc pr 6444444 74444448 µ S, d σ S, d, d ~ Ν, σ d 8 443 drf 443 dffuso { oo I y ge modellg pplcos, such geerlsed eer process s o prode he uderlyg rdom dymcs w.r.. whom some quy of eres s deermsc fuco hereof; y ee, SDE s formuled erms of such ques d her deres. STOCHASTIC CALCULUS I ITO INTEGRAL As wh deermsc clculus, where he rue meg of dy y' dx s fc y dx dy ', sochsc clculus SDE s smlrly megful sofr s s hough of s smply shorhd oo for he correspodg egrl equo ; he rems o defe wh s me excly y egrl or performg egro oer sochsc dfferels d egrds! h fxed me erl [, ], sdrd Brow moo/eer process, d some rdom rgh-couous fuco of me, [, ], [, ] F : Sep Defe PARTITIONING h ddes [, ] up o > coguous, ooerlppg, lef-closed, equl-legh suerls: [, [,,, K, [,, K, [, [,, ],, K, 9 [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.8][Mr.55B.E.]
Sep Defe, w.r.., [, ], ogeher wh he suerls, he BRONIAN INCREMENTS { B,, K, }, whch s deed couous-lue, dscree-me sochsc process: B ~ Ν,,, K, Sep 3 Defe, w.r.. F, [, ], ogeher wh he suerls, pproxmg RANDOM STEP FUNCTION F, [, ], whch s deed dscree-lue, couous-me sochsc process: F < F,, K, F Noe how such pproxmo c e mde rrrly close sequece coergece erms of fuco orm: F F Sep 4 Sum-mulply he lues of he rdom sep fuco F, [, ] wh he correspodg Brow cremes { B,, K, }, herey defg hs rdom rle, wh relso y me, s he DISCRETE STOCHASTIC ITO INTEGRAL F : F F B F 3 Noe how he fuco rgume s F, s per Io s formulo, d delerely eher F, s per Mll s formulo, or F F, s per Srooch s formulo. [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.8][Mr.55B.E.]
The choce reflecs s moed y he ure of modelled pplco wherey F represes some kd of decso fuco h depeds o hg ll he lle formo ll ucery resoled s of me. Techclly, F s dped o he { }, {, K, } flro, u eher F or F F s; hese ler wo re cully dped o slghly lrger flro { }, {, K,, }. Icdelly, he Rem-egro logue of he dscree sochsc Io egrl s smply clled he Rem sum,.e. ohg lke dscree deermsc Rem egrl! Sep 5 I he lm s, le hs rdom rle sequece coerges me-squre o ye oher rdom rle, deed oe defed s he ITO INTEGRAL: F F B F d me squre s 44 44 44443 443 dscree sochsc Io egrl Io egrl 4 I essece: F d 443 Io egrl } Θ rdom rle lm Ε 44 443 Expeco [ F Θ ] F Θ dω Ω4 4444 3 Lesesgue egrl I geerl, he Io egrl s defed w.r.. he dffuso erm, hece sochsc egro oer sochsc egrd d sochsc dfferel: 5 σ Su, u d 4 43 { u 6 sochsc egrd sochsc dfferel I prculr, cosder he followg kow resuls: [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.8][Mr.55B.E.]
[Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.83][Mr.55B.E.] σ σ d dy d dy d d d 443 7 Here s sruce o go hrough deled sep 4 clculo w.r.. oe cse prculr, he Io egrl of where he egrd s he sdrd Brow moo/eer process self T d : For hs, he dscree sochsc Io egrl s ge y he followg rdom rle: 8 Nog he dey: 9 I he follows h:, f ω 3 Usg he followg lm resul: lm d d 3
[Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.84][Mr.55B.E.] From whch he sochsc Io egrl he follows: { Io correco T d lm lm lm 3 Noe he cors wh he Rem egrl, wherey: x dx 33