MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 15 10/30/2013. Ito integral for simple processes

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.7J Fall 13 Lecure 15 1/3/13 I inegral fr simple prcesses Cnen. 1. Simple prcesses. I ismery. Firs 3 seps in cnsrucing I inegral fr general prcesses 1 I inegral fr simple prcesses. I ismery Cnsider a Brwnian min B adped sme filrain F such ha (B, F ) is a srng Markv prcess. As an example we can ake filrain generaed by he Brwnian min iself. Our gal is give meaning expressins f he frm X db = X (ω)db (ω), where X is sme schasic prcess which is adaped he same filrain as B. We will primarily deal wih he case X L, alhugh i is pssible exend definiins mre general prcesses using he nin f lcal maringales. As in he case f usual inegrain, he idea is define X (ω)db (ω) as sme kind f a limi f (randm) sums j X j (ω)(b j+1 (ω) B j (ω)) and shw ha he limi exiss in sme apprpriae sense. As X we can ake all kinds f prcesses, including B iself. Fr ex- T ample we will shw ha B db makes sense and equals (1/)BT (1/)T. Definiin 1. A prcess X L = L (Ω, F, (F ), P) is called simple if here exiss a cunable pariin Π : = < < n < wih lim n n = such ha X (ω) = X j (ω) fr all [ j, j+1 ), j =, 1,,... fr all ω Ω. The subspace f simple prcesses is dened by L We assume ha pariin is such ha j as j. I is impran ne ha we assume ha he pariin Π des n depend n ω. Thus n every piece-wise cnsan prcess is a simple prcess. Give an example f a piecewise cnsan prcess which is n simple. Ne ha since X F we have X j F j fr each j. As an example f simple prcess, fix any pariin Π and a prcess X L and cnsider he prcess Xˆ(ω) defined by Xˆ(ω) = 1

2 X j (ω), where j is defined by [ j, j+1 ). In he definiin i is impran ha Xˆ = X j and n X j+1. Observe ha he laer is n necessarily adped (F ) R+. Given a simple prcess X and, define is inegral by I (X(ω)) = j n 1 X j (ω)(b j+1 (ω) B j (ω)) + X n (ω)(b (ω) B n (ω)), where n = max{j : j }. Observe ha I (X) is an a.s. cninuus funcin (as B is a.s. cninuus). Therem 1. The fllwing prperies hld fr I (X) I (αx + βy ) = αi (X) + βi (Y ). (1) E[I (X)] = E[ Xs ds] [I ismery], () I (X) M,c, (3) E[(I (X) I s (X)) F s ] = E[ X du], s < T. (4) u s Nice ha (4) is a generalizain f I ismery. We nly prve I ismery, he prf f (4) fllws alng he same lines. Prf. Define n = fr cnvenience. We begin wih (1). Le { 1 j} and {j } be pariins crrespnding simple prcesses X and Y. Cnsider a pariin { j } bained as a unin f hese w pariins. Fr each j which belngs he secnd pariin bu n he firs define X j = X 1, where 1 i is he larges i pin n exceeding j. D a similar hing fr Y. Observe ha nw X = X j fr [ j, j+1 ). The lineariy f I inegral hen fllws sraigh frm he definiin. Nw fr () we have E[I (X)] = E[X j1 X j (B B j1 +1 j1 )(B B j +1 j )]. When j 1 < j we have j 1,j n 1 E[X j1 X j (B j1 +1 B j1 )(B j +1 B j )] = which we bain by cndiining n F j, using he wer prpery and bserving ha all f he randm variables invlved excep fr B j +1 are measurable wih respec F j (recall ha F j1 F j ).

3 Nw when j 1 = j = j we have Cmbining, we bain E[X j (B j+1 B j ) ] = E[X j E[(B j+1 B j ) F j ]] = E[X j ( j+1 j )]. E[I (X)] = E[X ( j j+1 j )] = E[ X ( j j+1 j )] = E[ j j X s ds]. Le us shw (3). We already knw ha he prcess I (X) is cninuus. Frm I ismery i fllws ha E[I (X)] <. I remains shw ha i is a maringale. Thus fix s <. Define n = and define j = max{j : j s}. E[I (X) F s ] = E[ X j (B j+1 B j ) F s ] j n 1 = E[ X j (B j+1 B j ) F s ] + E[X j (B s B j ) F s ] j j 1 + E[X j (B B j +1 s) F s ] + E[ X j (B j+1 B j ) F s ] j>j = E[ X j (B j+1 B j ) F s ] + E[X j (B s B j ) F s ] j j 1 = I s (X). (hink abu jusifying las w equaliies). Cnsrucing I inegral fr general square inegrable prcesses The idea fr defining I inegral XdB fr general prcesses in L is ap prximae X by simple prcesses X (n) and define XdB as a limi f X (n) db, which we have already defined. Fr his purpse we need shw ha we can indeed apprximae X wih simple prcesses apprpriaely. We d his in 3 seps. Sep 1. Prpsiin 1. Suppse X L is an a.s. bunded cninuus prcess in he sense M s.. P(ω : sup X (ω) M) = 1. Then fr every T > here 3

4 exiss a sequence f simple prcesses X n L such ha lim E[ (X n X ) d] =. (5) n Prf. Fix a sequence f pariins Π n = { n } f [, T ] such ha Δ n = max( n n j ) as n. Given prcess X, cnsider he mdified prcess Xn = X n j fr all [ n j, n j+1 ). This prcess is simple and is adaped F. Since X is a.s. cninuus, hen a.s. X (ω) = lim n X n (ω) (nice ha we are using lef- cninuiy par f cninuiy). We cnclude ha a sequence f measurable funcins X n : Ω [, T ] R a.s. cnverges X : Ω [, T ] R. On he her hand P(ω : sup T X n (ω) M) = 1. Using Bunded Cnvergence Therem, he a.s. cnvergence exends inegrals: E[ T (X n X ) d]. j j+1 Sep. Prpsiin. Suppse X L is a bunded, bu n necessarily cninuus prcess: X M a.s. Fr every T >, here exiss a sequence f a.s. bunded cninuus prcesses X n such ha lim E[ (X n X ) d] =. (6) n Prf. We use a cerain regularizain rick urn a bunded prcess in a bunded cninuus apprximain. Le X n = n 1/n X s ds. We have X n n(1/n)m = M and X n ; X n n ' M (verify his), implying ha X n is a.s. bunded cninuus. Since X is a.s. Riemann inegrable, hen fr alms all ω, he se f discninuiy pins f f X (ω) has measure zer and fr all cninuiy pins by Fundamenal Therem f Calculus, we have lim n X n (ω) = X (ω). We cnclude ha X n : Ω [, T ] R cnverges a.s. X n he same dmain. Applying he Bunded Cnvergence Therem we bain he resul. Sep 3. Prpsiin 3. Suppse X L. Fr every T > here exiss a sequence f a.s. bunded prcesses X n L such ha lim E[ (X n X ) d] =. (7) n 4

5 Prf. Define X n by X n = X when n n, X n X = n, when X < n and X n = n, when X > n. We have X n X a.s. w.r. bh ω and [, T ]. Als X n X implying (X n X ) d (X n ) d + X d 4 X d. T Since E[ X d] <, hen applying Dminaed Cnvergence Therem, we bain he resul. Exercise 1. Esablish (7) by applying insead Mnne Cnvergence Therem. 3 Addiinal reading maerials Karazas and Shreve [1]. Øksendal [], Chaper III. References [1] I. Karazas and S. E. Shreve, Brwnian min and schasic calculus, Springer, [] B. Øksendal, Schasic differenial equains, Springer,

6 MIT OpenCurseWare hp://cw.mi.edu 15.7J / 6.65J Advanced Schasic Prcesses Fall 13 Fr infrmain abu ciing hese maerials r ur Terms f Use, visi: hp://cw.mi.edu/erms.