MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 16 11/04/2013. Ito integral. Properties

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1 MASSACHUSES INSIUE OF ECHNOLOGY 6.65/15.7J Fall 13 Lecture 16 11/4/13 Ito itegral. Propertie Cotet. 1. Defiitio of Ito itegral. Propertie of Ito itegral 1 Ito itegral. Exitece We cotiue with the cotructio of Ito itegral. Combiig the reult of Propoitio 1-3 from the previou lecture we proved the followig reult. Propoitio 1. Give ay proce X L there exit a equece of imple procee X L uch that lim E[ (X (t) X(t)) dt] =. (1) Now, give a proce X L, we fix ay equece of imple procee X L which atifie (1) for a give. Recall, that we already have defied Ito itegral for imple procee I t (X ). Propoitio. Suppoe a equece of imple procee X atifie (1). here exit a proce Z t M,c atifyig lim E[(Z t I t (X )) ] = for all t. hi proce i uique a.. i the followig ee: if Xˆ t i aother proce atifyig (1) ad Ẑ i the correpodig limit, the P(Ẑ t = Z t, t [, ]) = 1. Proof. Fix >. Applyig liearity of I t (X) ad Ito iometry E[(I (X m ) I (X )) ] = E[I (X m X )] = E[ (X m (t) X (t)) dt] E[ (X(t) X m (t)) dt] + E[ (X(t) X (t)) dt]. 1

2 But ice the equece X atifie (1), it follow that the equece I (X ) i Cauchy i L ee. Recall ow from heorem.. previou lecture that each I t (X ) i a cotiuou quare itegrable martigale: I t (X ) M,c Applyig Propoitio, Lecture 1, which tate that M,c i a cloed pace, there exit a limit Z t, t [, ] i M,c atifyig E[(Z I (X )) ]. he ame applie to every t ice (Z t I t (X )) i a ubmartigale. It remai to how that uch a proce Z t i uique. If Ẑ t i a limit of ome equece Xˆ atifyig (1), the by ubmartigale iequality for every E > we have P(up t Z t Ẑ t E) E[(Z Ẑ ) ]/E. But E[(Z Ẑ ) ] 3E[(Z I (X )) ] + 3E[(I (X ) I (Xˆ )) ] + 3E[(I (Xˆ ) Ẑ ) ], ad the right-had ide coverge to zero. hu E[(Z Ẑ ) ] =. It follow that Z t = Ẑ t a.. o [, ]. Sice wa arbitrary we obtai a a.. uique limit o R +. Now we ca formally tate the defiitio of Ito itegral. Defiitio 1 (Ito itegral). Give a tochatic proce X t L ad >, it Ito itegral I t (X), t [, ] i defied to be the uique proce Z t cotructed i Propoitio. We have defied Ito itegral a a proce which i defied oly o a fiite iterval [, ]. With a little bit of extra work it ca be exteded to a proce I t (X) defied for all t, by takig ad takig appropriate limit. Detail ca be foud i [1] ad are omitted, a we will deal excluively with Ito itegral defied o a fiite iterval. Ito itegral. Propertie.1 Simple example Let u compute the Ito itegral for a pecial cae X t = B t. We will do thi directly from the defiitio. Later o we will develop calculu rule for computig the Ito itegral for may iteretig cae. We fix a equece of partitio Π : = t < < t = ad coider B t = B tj, t [t j, t j+1 ). Aume that lim Δ(Π ) =, where Δ(Π ) = max j t j+1 t j. We firt how that thi i ufficiet for havig lim E[ (B t Bt ) dt] =. ()

3 Ideed We have 1 j+1 (B t B t ) dt = (B t B tj ) dt. j= t j j+1 j+1 E[ (B t B tj ) dt = E[(B t B tj ) ]dt t j t j t j+1 = (t t j )dt t j (t j+1 t j ) =, implyig E[ (B t B ) dt] = (t j+1 t j ) Δ(Π ) (t j+1 t j ) = Δ(Π ), t j= j= a. hu () hold. hu we eed to compute the L limit of I (B ) = B tj (B tj+1 B tj ) a. We ue the idetity implyig j Bt B = (B tj+1 B tj ) j+1 t j + B tj (B tj+1 B tj ), B ( ) B () = B B t j+1 t = (B j t j+1 B tj ) + B tj (B tj+1 B tj ), j= j= j= But recall the quadratic variatio property of the Browia motio: 1 lim (B tj+1 B tj ) = j= i L (recall that the oly requiremet for thi covergece wa that Δ(Π ) ). herefore, alo i L We coclude 1 Btj (B t B j+1 t ) 1 j B ( ). j= 3

4 Propoitio 3. he followig idetity hold I (B) = B t db t = 1 B ( ). Further, recall that ice B t M,c the it admit a uique Doob-Meyer decompoitio Bt = t + M t, where t = (B t ) i the quadratic variatio of B t ad M t i a cotiuou martigale. hu we recogize M t to be I t (B).. Propertie We already kow that I t (X) M,c, i particular it a i cotiuou martigale. Let u etablih additioal propertie, ome of which are geeralizatio of heorem. from the previou lecture. Propoitio 4. he followig propertie hold for I t (X): I t (αx + βy ) = αi t (X) + βi t (Y ), α, β, (3) E[(I X t (X) I (X)) F ] = E[ du F ], < t. (4) Furthermore, the quadratic variatio of I t (X) o [, ] i u X dt. t Proof. he proof of (3) i traightforward ad i kipped. We ow prove (4). Fix ay et A F. We eed to how that E[(I t (X) I (X)) I(A)] = E[I(A) Xu du]. Fix a equece X L atifyig (1). he E[(I t (X) I (X)) I(A)] = E[(I t (X) I t (X )) I(A)] + E[(I t (X ) I (X )) I(A)] + E[(I (X ) I (X)) I(A)] + E(I t (X) I t (X ))(I t (X ) I (X ))I(A) + E(I t (X) I t (X ))(I (X ) I (X))I(A) + E(I t (X ) I (X ))(I (X ) I (X))I(A) But E[(I t (X) I t (X )) I(A)] ice E[(I t (X) I t (X )) ] (defiitio of Ito itegral). Similarly E[(I (X) I (X )) I(A)]. Applyig Cauchy-Schwartz iequality E(I t (X) I t (X ))(I t (X ) I (X ))I(A) (E[(I t (X) I t (X )) ]) 1/ (E[(I t (X ) I (X )) ]) 1/ 4

5 from the defiitio of I t (X). Similarly we how that all the other term with factor i frot coverge to zero. By property (.6) heorem. previou lecture, we have E[(I t (X ) I (X )) I(A)] = E[I(A) (Xu ) du] Now E[I(A) (Xu ) du] E[I(A) Xudu] = E[I(A) (Xu X u )(Xu + X u )du] E[ (Xu X u )(Xu + X u ) du] E 1 [ (X u X u ) du]e 1 [ (X u + X u ) du] where Cauchy-Schwartz iequality wa ued i the lat tep. Now the firt term i the product coverge to zero by the aumptio (1) ad the ecod i uiformly bouded i (exercie). he aertio the follow. Now we prove the lat part. Applyig Propoitio 3 from Lecture 1, it uft fice to how that I t (X) X d i a martigale, ice the by uiquee of the Doob-Meyer decompoitio we mut have that (I X t (X)) = d. But ote that (4) i equivalet to E[It (X) I (X) F ] = E[It (X) F ] I (X) = E[ Xu du F ] = E[ Xu du F ] E[ Xu du F ] = E[ X du F ] X du. Namely, u E[It (X) F ] E[ Xu du F ] = I (X) Xu du, t t amely, I (X) X d i ideed a martigale. u 3 Additioal readig material Karatza ad Shreve [1]. Økedal [], Chapter III. 5

6 Referece [1] I. Karatza ad S. E. Shreve, Browia motio ad tochatic calculu, Spriger, [] B. Økedal, Stochatic differetial equatio, Spriger,

7 MI OpeCoureWare J / 6.65J Advaced Stochatic Procee Fall 13 For iformatio about citig thee material or our erm of Ue, viit: