Mathematil Engineering. Stochastic differential equations.

Resuming scheme of the formulas for the exam

Other

STOCHASTICPROCESSESUSEFULFORMULAS -- Equi :saune finitedindistributionCash(y)=e_×-1e'sinlx)=e-×-e"*--22Modificati :(f.+edt)HtXÉ e- Xe a.a *lndistiugui-sha.be : lfixed w)(w' Xtcwl-X.fm)htt)=I5in2a = 2. sina.coaa-varlX-YI-varlXI-varlYI-2covCX.it )BROWNIANMOTION - DEF1DEFZ-- gaussianoVECTORS o 13101=0 o Blasco -° ttoes.atBe - Botta ° (Ben ,_... Ben)- Nleir)Xun(µ ×, -2×7 → Y = AX + b ° Be - BsNNIO ,t-s)° EIBE - B.) = tra Y~NCAM-ib.AE > A " ) -o- [ Budbu e- BI - ft :#[cosce .- B . » . e-È "¥ 'tiae' PELÉ) ' E[ sin (Bt - Bs)) =0#[etx] =@tu +' ÷ : ÷ :* ! :c : " : - iii. ÷ :( iii. ÷ .it : > CHARACTERIZATIONOFBMINTERMSOFMARTINGALE' Xp isa BMiff → X. =O→e iii. ×> t'171'tis ao Xegaussiana a)1ITL=max(ti-ti-a)(2.act)2(Bee -a)i° BMCBIetf-→+toDONIINATED• Bene → D= convergere° ITOPROCESSXe - X. + {ÈDB ,-1 # sds THEOREM•Eexitfrom C-a.b)(×! = (tosrdsE[Xe]= Xo =-a RCBz =- a) +b.PICB ,=b) STOCHASTICINTEGRALSCHANGEOFPROBABILITIES -- [ Xsdbs=L;ÌÈXI 'Bei "- Bei)7- = DIE'IXJ = E ' [Z - X]dp IXEMTI.EE/abXtdBeI7aJ=OEOIXI = EP-LZXIDI.tk/aXedBt)47aI--EffIXIdt/7a )EXPONENTIALMARTINGALE /¨"[Z'd'° EI[ Xudbufotyudbu) = #f ![%da ] /¨ exp{/! 'sdbs - ftp.ids}dzt = ZeteOIBTIct) = ! sdbsisa squareinmartingale o iflffe{È#s'%) ,a ÷ :È:÷÷: ÷ : : : " : :/ " ÷: ÷ .GIRSANOVTHEOREMXe = X. + [ F.da+ { " GsDBS - Bd -dinBM , tiC-Mia{X = {te;da'X.Xr! = { " Galilea")daAssumethattheprocesso ° d'X.th - Xdtt) = DX . Xz-iyn.dyz.dey.ir,Zt ' exp {fattoDBS - £( +14 . l'ds } ITOFORMULAisa martingaleandlettQbethe - F : RlxIo + a) → RFEC 'in×, e ' .atprobabilitagivenbyZta)Yt = ECX e. t)isan Itaproces , da = ZedpThentheprocessob)dye = jffixeihdttffllattd # ' {IÌ 'XHHGÉDTWe = Be - { " tsdsU i Rlnx[a. + a) →17UEE '"isae'- BMondaad.Be = dwttpedtdye.az?dttEffflXitfiI#JxjdIindXI NELLKNOWNPROCESSESGagadt- MARTINGatesGBM i←= bxlttdttoltdbt -×_. ItFeo