Resuming scheme of the formulas for the exam

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STOCHASTICPROCESSES USEFUL FORMULAS- - Equi : saune finitedin distribution Cash(y)= e_ × -1e' sinlx)= e- ×- e" * - - 2 2 Modificati: (f.+edt)HtXÉe-Xea.a * lndistiugui-sha.be: lfixedw) (w 'Xtcwl-X.fm)htt)= I 5in2a= 2. sina.coaa-varlX-YI-varlXI-varlYI-2covCX.it) BROWNIANMOTION- DEF1 DEFZ- - gaussianoVECTORSo 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 - [Budbue- BI- ft: #[cosce. - B.». e-È ⊥'ti ae'PELÉ)' E[sin(Bt- Bs))= 0 #[etx]= @tu+ ' ÷:÷:*!:c:" : - iii.÷ :( iii.÷.it:>CHARACTERIZATIONOFBMINTERMSOF MARTINGALE' Xpis a BMiff → X.= O → e iii.×>t'171' t isa o Xegaussiana㱺 XtEL' 71spero Temartingalehtt 㱺 Finite variations >LEVY'sCHARACTERIZATIONOFBn CONDITIONAL EXPECTATIONXeisa BMiff - Xisa martingale - - <×!= t /<Xi,Xt>= S.tt {×dp=/, zdpttDEDZ-EEXID.tt IYARTINGALESDEFo Meintegrabilett- > E[ELXIDI]= EIXI o ElMelas]Ms , e[×110, a}]= EIXIE[Melas]Ms super , ×Ducas㱺 E[XIDIe-X E[Melas]7ms sub ,×#p 㱺 E[XIDI- ELX? > YC-La㱺 Yt= EIYIft)isa martingale , a@ 㱺 E-[ELXI@Il I> EIXIDi > Me= e "Be- È" isa martingale Ht= <M'e → MI- Aeisa 7. mail.mg. > E[!sola/d)= [ETXS10.7da MaRTlNGalEREPRESENTATIONTHEORe#E[XIY7.= cov (X- E[XJ)var (Y) TheneveryrandomvariabileNEL' isofthe form M= m + [Hisdbs FREEZINGLEMMA.ie:÷÷÷÷÷÷÷÷÷: " " " / ÷::c:S:÷÷:!!!!:: i:&:STOPPINGTIMES- VARIATIONS xnxx.si DEItt trot}c-te - STOPPINGTHEOREM NÉf= pÈlflti i- fctiil •forBM: Yt= B-l.ie- Beis a BMIl te o Vabf<a f=vi- va Tiffany• forMart. : E[Me1701= Meno o feE' 㱺 Vabf<n Meatis a temortuale (Xii. = ¥,:[ I Bei- Bei"REFLECTIONPRINCIPE lsupbs>a)=P(Ta eti-2171Be>a) 1ITL= max (ti- ti- a) (2.act)2 (Bee- a) i ° BM CBIet f-→+to DONIINATED• Bene→ D=convergere ° ITOPROCESSXe-X.+{ÈDB,-1#sdsTHEOREM • E exitfromC- a.b) (×!= (tosrdsE[Xe]=…