Completed notes of the course

Page 1

SDE INTRODUCTIONStochastic= probabilistic, notdeterministicevolution ODE= /×' = b'× I x.it)=d XCO)= XoER1" dt b:131 " →R" ×: [0, too)→B" +It) FNhmmwhwwM' t XIt)+noise { ×it)=blxlt))toCXLHI. ICH o: A"→ Rn×m Xlol=Xo Ic.in vector = whitenoise ateverytimetwe don'tknowtherealvaluexit)butit'sdestination thesolutionis xlt)in → R" ①STOCHASTICPROCESS functiondefinedmn spaceandtime X:hx [0, too)→ R" s.t.tt to XLHis a windowvariable ②NOISE XLH+ noise= trajectoryECTIis" thedifferentialofthenoise " (gaussiandestitutionwithmeow 0) BrownianMotionBtNN10,t) ContinuoustrajectoryEct)→13¥) dt DXIt)=blXlHIdt+OCXIH)dBt STOCHASTICDIFFERENTIAL{XLOI= xo EQUATION 㱺xltt-xoxf.tblxlsllds-f.toCXCSIIDBCS) - integrationina brownianmotion ③STOCHASTICINTEGRAL ④STOCHASTICCALCULUS ⑤STOCHASTICDIFFERENTIALEQUATION ⑥APPLICATIONS• PLH= nowherepriceofa stock ° evolvesas a stochasticdifferentialequation 4¥=µdtxodBt1 I drift volatility ELEMENTSOFPROBABILITY DEFsAmeasurablespaceis a painIE,E)where • Eis a set ° Eis a o- algebraof subsetsofE 4A G-algebrais a familyofsubsetsofE set. 1) EEE 2) AEE㱺A' = E1AEEa 3) Az. . . AnEE㱺 UAnC-E.k=h ExamplesofG-algebra: - 9=101, EItriviala-algebra- o- algebrageneratedbyAtE : \o/,A.ATE} >Ifn=R" , BC1R")= boreliauo- algebrawhichisgeneratedbytheopensets, or closed sets, or C- oo,a] at . >A probabilityspaceisa tripleLE, I, RI SI ° (E, Itisa measure space ° Pisa positivemeasure s.at. RLEI=L Ri E→[0,23 IfAz, . . . . Anis a countablesequenceofelementsofE, pairwisedisjoint(AnnAutohtm) RIIIAut.IEPLAN > ⊥, 7.B)probabilityspace, IE,E)measure space, a randomvariable isa measurablefunction X: I → E sitVAET X- YA1E7 X- ' (A)=/wer IXCWIEA} > IfE=Rtherandomvariableis called RealRandomVariableE=B(B) LetXbea RV. withvalueintheueeasutue spaceLE,E) thefunctiondefinedon E µ×(A)=PIX- 'LAIIis a…