Mathematil Engineering. Stochastic differential equations.

Completed notes of the course

Complete course

SDE INTRODUCTION Stochastic = probabilistic , notdeterministicevolutionODE = / ×'=b'× Ix.it)=dXCO)=XoER1 " dtb :131"→ R "×: [ 0,too ) → B "+It) FNhmmwhwwM 'tXIt)+noise{ × it)=blxlt))toCXLHI . ICHo: A "→ Rn×mXlol=XoIc.invector = whitenoiseateverytimet we don'tknowtherealvaluexit)butit'sdestinationthesolution isxlt) in→ R " ①STOCHASTICPROCESSfunctiondefined mn spaceandtimeX:hx[ 0,too)→ R " s.t.tttoXLH isa windowvariable ②NOISEXLH + noise = trajectoryECTI is" thedifferentialofthenoise" (gaussiandestitutionwithmeow0)BrownianMotionBtNN10 , t)ContinuoustrajectoryEct) → 13¥)dtDXIt)=blXlHIdt+OCXIH)dBtSTOCHASTICDIFFERENTIAL { XLOI = xoEQUATION IfE=Rtherandomvariable is calledRealRandomVariableE=B(B)LetXbe a RV . withvalueintheueeasutuespaceLE , E)thefunctiondefined on Eµ×(A)=PIX -' LAIIisa probability measureon E" = µ×=loverofmonotonevariableofX~µ× PR0PLetX:(1,7 , B) → CE , E)be a RV . withlawµ , thenthemeasurablefunctionf- : (E.E) → IR1 , BIRIII Is µ - integrableifandonlyifFIXI is P - integrableand "¥ fcxIµ× (DX) ={ FCXCWIIPCDWI -LINREMifX isa realRV , LE ,21=1113, BIRD , hlXI=x , thenX is R - integrableifandonlyifX →xis µ - integrableEIXJ = µ Rldw )= "¥ xµ×Cdx )ifXNNfu ,o' )fix)=. expI - } Ya >0 ELX = # ×' µ×ldx!Ifa-2andEIX ' ] andµ,coincideon 9themµ ,andµ ,coincideon EProofX=IX ,,... ,Xn) vV(V: lawofX)µ = product measure I = fowleyofsubsetsoftheformA , xAz× ...× Amwithti=1....~, AIEEIIisstableunderfiniteintersectionsandgenerates& , bydefinitionBythedefinitionofimdipaudenoe , µandUcoinoe.de on IVIA ,×A,×× Ant = µ , LA1 PR0PLinearaffinetransformationsmapnormallawintonormallawThatmeans: ifXNN1A,M)Y=A X+ b ~ NIA-a+b, ATAT)REMt - ifXandYreal r. i r.andIX ,Y) isa gaussianvector In}thenInst . dlxs,B ' ) iftset.seMEN, Since4 is continuousdCXs , 13971that t THREverysubmartingale(Xnln can bedecomposeduniquelyintothe sum of a martingale(Mnhand an increasingpredictableprocess(An)n t-a Mtg to )Misa.s.constanton ( t Mxt -←M>s=D)Its,tJI't- EH12 it-s)@iii.×>+ I 1×1't=e= eicx.is >+ I 'X''s= ys×Efei 't,#I+ IH141FS] =@i←t.is>+ I 1×1'sEfeit."-×s> frf , ] =e- flat 'it-s) Ta k i n gtheexpectationEfei #it-×.> ] =e- I '""t-s)(*,Xt-Xs~NC0,t-A) ToprovethatXt-Xsis independentofAsweuse again1*1PR0PIfforeverytepidEL e'''"×> ID3 =#[ eit"" ] tNQ, ttItDefineZt= dttdptttThen :(i)(It)tisaP- www.mgabeIii)(¥)tisaQ- martingale (iii)(XtItisaQ- uuantingale c0/¨ I @n'% '' ] aca• AT Eto,TJ t=-D PR0PCorollary7. 2IfM,Nare continuouslocalmartingalethenIa uniqueprocessAwithfinitevariationsit . It =Mt- Nt -Atisa continuouslocalmartingaleREMAt= I CaMTN>t-'M-N>t )At =EM, N7EcovaniationofMandNIfM=NEM,MI=< MEIfMt - Ntisa martingale =qandforevery x. yEDte [v.T31BIix.t)- bicy ,E)ItLlx-y)I oi cx.tl - oily ,E)IE1-Ix- yl IfZii=1,2aretheexittimeof XifromDThenEz AT= Ez nTa.s. PCX ,it)=XzCt)httuetee ,nT)=LPR0P(ExittimefromBr)letb,o bemeasurableandI bcx.tl/EMlt-1×1) } µ×eR" fte-u.TT/olX,t)IEMlt +1×1)letBandIzEL'll ,7in,P)begiven(etXbethesolutionofSDE {dXt=bCXt,t)dt +CCXEit)dBtXu=qLetR>0andI,z=1nF It >os-t.IXTIIR}theexittimefromtheballofradiusR(itisa stoppingtime)ThenPl=pET)e CCM,T , a)C1-1EI14123) -R'II P(supIXTIIR)tE[up] ASSUMPTIONCA ' )1)b,oaveleeeasuiuablefunctioninCX,t)2)IM>0 [ lbCX.tl/EMl1tlx1)t×eR" te-u.TTIOCX,t)IEM1It1×1)3)tN so0set§ ,lbcx.tl- biy.tl/ELnIxtYlf×, y1×1IN V-ttlu.TTlotx.t)- ocy.tl/eLn1x-yIIY1E N THEOREMUnderassumption(A'), letz c- L2Cd.An,P)ThereexistaprocessXsolutionofSDE{dXt= bcxt.tldt+ olXt,t)dBtXu=qMoreover ,EfsupIXTI' ] tooandpathwiseuniquenessholds te[u,T] ProofEXISTENCEAssumeUE0ForNS04NEEILR1" )stOE 4nsI1×1IN4 ,ex)=' ( tthePTteLu,T 3 0IX12NTI=1I →IF-NN' NTLetbnlx.tl =bcx,t)Ionlx)Gn(×,t)= ocx.tlOnCx) bn ,onsatisfyassumptionA(they areLip)BytheI!theoremunderassumptionAI!XNsit. Efsup1×+121N }bylocalizationtheoremforSDEeeCN)=EzlN)a.s.andX,= Xzooiadea.s. { E,en)>T }9ha.s.