Introduction to Ito's Lemma

1 Introduction to Ito s LemmaWenyu ZhangCornell UniversityDepartment of Statistical SciencesMay 6, 2015 Wenyu Zhang (Cornell)Ito s LemmaMay 6, 20151 / 21 Overview1 Background2 Ito Processes3 Ito s LemmaWenyu Zhang (Cornell)Ito s LemmaMay 6, 20152 / 21 BackgroundProved by Kiyoshi Ito(not Ito s theorem on group theory by NoboruIto)Used in Ito s calculus , which extends the methods of calculus tostochastic processesApplications in mathematical finance derivation of theBlack-Scholes equation for option valuesWenyu Zhang (Cornell)Ito s LemmaMay 6, 20153 / 21 Ito ProcessesQuestionWant to model the dynamics of processX(t) driven by Brownian motionW(t).

2 Wenyu Zhang (Cornell)Ito s LemmaMay 6, 20154 / 21 Ito Processes: Discrete-time ConstructionPartition time interval [0,T] into N periods, each of length t=TN;tn=n tXtn+1=Xtn+ tn t+ tn WtnIdrift Ivolatility Ifluctuations Wtn=Wtn+1 Wtn N(0, t)Wenyu Zhang (Cornell)Ito s LemmaMay 6, 20155 / 21 Ito Processes: Discrete-time ConstructionSumming the increments,XT=X0+N 1 n=0 tn t+N 1 n=0 tn WtnContinuous-time analogue asN ,XT? X0+ T0 tdt+ T0 tdWtWenyu Zhang (Cornell)Ito s LemmaMay 6, 20156 / 21 Ito Processes: Discrete-time ConstructionRegularity conditions for tand tadapted toFWtcontinuous in tintegrability conditions ( t ( T0 2tdt)< )Riemann-Stieltjes integralN 1 n=0 tn t T0 tdtIto integralN 1 n=0 tn WtnL2 T0 tdWtWenyu Zhang (Cornell)Ito s LemmaMay 6, 20157 / 21 Ito IntegralsQuestionWhat is T0 tdWt?

3 Wenyu Zhang (Cornell)Ito s LemmaMay 6, 20158 / 21 Ito IntegralsTheorem (Existence and Uniqueness of Ito Integral)Suppose that vt M2satisfies the following: For all t 0,A1) vtis continuousA2) vtis adapted to FWtThen, for any T>0, the Ito integral IT(v) = T0vtdWtexists and isunique for proof1 Construct a sequence of adapted stochastic processesvnsuch that v vn M2= E( T0|vn(t) v(t)|2dt) 02 Show that IT(vn) IT(v) L2 03 Show the uniqueness of the limitIT(v)Wenyu Zhang (Cornell)Ito s LemmaMay 6, 20159 / 21 Ito Integrals: ExampleExample (Ito Integral) T0 WtdWtwith approximating sumsN 1 n=0 Wtn WtnN 1 n=0 Wtn(Wtn+1 Wtn) =N 1 n=0[12(W2tn+1 W2tn) 12(Wtn+1 Wtn)2]=12W2T 12N 1 n=0(Wtn+1 Wtn)2L2 12W2T 12 TasN Wenyu Zhang (Cornell)Ito s LemmaMay 6, 201510 / 21 Ito Integrals: ExampleExample (Riemann-Stieltjes Integral) T0 GtdGtwithG C1,G(0) = 0 T0 GtdGt= T0 GtG tdt=12G2 TWenyu Zhang (Cornell)Ito s LemmaMay 6, 201511 / 21 Ito Integral.

4 PropertiesLinear in the integrandTime-additiveMartingaleProofFor t<T, increase the partition by an extra pointtk= [IT(vn) It(vn)|Ft] =E[N 1 n=k tn Wtn|Ft]=E[N 1 n=kE( tn Wtn|Ftn)|Ft]=E[N 1 n=k tnE( Wtn|Ftn)|Ft]= 0IT(vn) is a martingale. Martingales are preserved Zhang (Cornell)Ito s LemmaMay 6, 201512 / 21 Ito ProcessesXt X0= t0 sds+ t0 sdWsSDE notation:dXt= tdt+ tdWtWenyu Zhang (Cornell)Ito s LemmaMay 6, 201513 / 21 Ito s LemmaTheorem (Ito s Lemma )Suppose that f C2. Then with probability one, for all t 0,df(Xt) = f x(Xt)dXt+12 2f x2(Xt)(dXt)2f(Xt) f(X0) = t0f (Xs)dXs+12 t0f (Xs)dsExplicit statement:df(Xt) =( t f x(Xt) +12 2t 2f x2(Xt))dt+ t f x(Xt)dWtWenyu Zhang (Cornell)Ito s LemmaMay 6, 201514 / 21 Ito s Lemma : IdeaCan be obtained heuristically by second order Taylor expansion offaboutXt(dXt)2= ( tdt+ tdWt)2term cannot be droppedI(dWt)2=dtdrop terms dt T0(dt)p= limN N 1 n=0( t)p= limN N(TN)p= 0asN ifp>1 Wenyu Zhang (Cornell)Ito s LemmaMay 6, 201515 / 21 Ito s Lemma .

5 Idea(dWt)2=dtsince T0(dWt)2= limN N 1 n=0( Wtn)2L2=T= T0dtE[N 1 n=0( Wtn)2]=N 1 n=0E( Wtn)2=N 1 n=0 t=TE[N 1 n=0( Wtn)2 T]2=Var[N 1 n=0( Wtn)2]=N 1 n=0 Var( Wtn)2=N 1 n=0(E( W)4 [E( W)2]2) =N 1 n=0(3( t)2 ( t)2)=2T2N 0asN Wenyu Zhang (Cornell)Ito s LemmaMay 6, 201516 / 21 Ito s Lemma : More detailsMore rigorously, for bounded continuousg,N 1 n=0g(Wtn)( Wtn)2L2 T0g(Wt)dtasN ProofSincet7 g(Wt) is continuous, N 1n=0g(Wtn) t T0g(Wt) :IN=N 1 n=0g(Wtn)[( Wtn)2 t]L2 0 Wenyu Zhang (Cornell)Ito s LemmaMay 6, 201517 / 21 Ito s Lemma : More (I2N) =E{ N 1n=0(g(Wtn))2[( Wtn)2 t]2}E{g(Wtn)g(Wtm)[( Wtn)2 t][( Wtm)2 t]}=E{E(g(Wtn)g(Wtm)[( Wtn)2 t][( Wtm)2 t]|Ftm)}=E{g(Wtn)g(Wtm)[( Wtn)2 t]E[( Wtm)2 t|Ftm]}= 0since{W2t t}is (I2N) CT2NE(I2N) g 2 N 1 n=0E[( Wtn)2 t]2sincegis bounded= g 2 N 1 n=0( t)2E(W21 1)2since Wtn t W1=CN 1 n=0(TN)2=CT2 NWenyu Zhang (Cornell)Ito s LemmaMay 6, 201518 / 21 Ito s Lemma .

6 ExampleExample (Ito s Lemma )Use Ito s Lemma , writeZt=W2tas a sum of drift and diffusion (Xt)with t= 0, t= 1,X0= 0,f(x) =x2dZt=df(Xt)=f (Xt)dXt+12f (Xt)(dXt)2= 2 WtdWt+122(dWt)2= 2 WtdWt+dtWenyu Zhang (Cornell)Ito s LemmaMay 6, 201519 / 21 Ito s Lemma : Higher dimensionsIto s LemmaIfXtandYtare Ito processes andf:R27 Ris sufficiently smooth, thendf(Xt,Yt) = f xdXt+ f ydYt+12 2f x2(dXt)2+12 2f y2(dYt)2+ 2f x y(dXt)(dYt)Wenyu Zhang (Cornell)Ito s LemmaMay 6, 201520 / 21 ReferencesRene L. Schilling/Lothar PartzschBrownian Motion - An Introduction to stochastic Processes (2012)CUHK course notes (2013)Chapter 6: Ito s stochastic CalculusKarl SigmanColumbia course notes (2007) Introduction to stochastic IntegrationWenyu Zhang (Cornell)Ito s LemmaMay 6, 201521 / 21