Ito s Stochastic Calculus

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1 Chapter 6 Ito s Stochastc Calculus 6.1 Itroducto I Bacheler s frst applcatos of the Weer process o the modelg the fluctuato of asset prces, the prce of a asset at tme t, X t, has a ftesmal prce cremet dx t that s proportoal to the cremet dw t of the Weer process, dx t σ dw t, where σ s a postve costat. As a result, a asset wth tal prce X( x worths X t x + σw t at tme t. Ths model suffers from oe serous flaw: for ay t > the prce X t ca be egatve wth o-zero probablty (but actual stock prce are ever egatve. To tackle ths problem, later researchers assumed that the relatve prce dx t /X t of a asset s proportoal to dw t,.e., dx t σx t dw t. Formally, ths equato looks lke a dfferetal equato, but ths leads to a dffculty because the paths of W t are ot dfferetable (Theorem 5.. A way aroud the obstacle was foud by Ito the 194s. I hs theory of stochastc tegral ad stochastc dfferetal equatos (SDEs. Ito gave a rgorous meag to equatos such as the above, by wrtg them as equatos volvg a ew kd of tegrals. I partcular, the above equato ca be wrtte as t X t x + σ X s dw s, (6.1 where the tegral wth respect to W t o the rght-had sde s called the Ito stochastc tegral to be defed the ext secto. 99

2 1 6 Ito s Stochastc Calculus 6. Ito Stochastc Itegral 6..1 Motvato Motvated by (6.1, we costruct the Ito stochastc tegral the form t f s dw s for some stochastc process/radom fucto f (s ( f (s,ω, to be precse. We follow a approach smlar to costructg Rema tegral,.e., defe the tegral by the lmt of the dscretzed verso f (s (W t+1 W t, (6. where s [t,t +1 ]. The major dffereces betwee Rema ad Ito tegrato are 1. Rema tegrato results a real umber, but Ito tegrato results a radom varable (sce W t s radom. Thus, whle defg Rema tegral t volves covergece of real umbers, defg Ito tegral t requres the covergece of radom varables (6., whch s cosderably more dffcult.. If a Rema tegral exsts, the s ca be a arbtrary pot [t,t +1 sce the upper ad lower Rema sum coverges. However, Ito tegral, the lmt wll be dfferet depedg o the choce of s. Ths s due to the o-zero quadratc varato of the Browa moto W t. See Exercse 6.1. To avod the ambguty, the defto of stochastc tegrals wll fx the choce s t for each the approxmatg sum (6.. The choce s t s atural f we regard f (t as the tradg strategy ad W t s the stock prce: For the + 1-th perod, the tradg strategy should oly deped o the formato up to tme t. Hece, f (t s vested to the stock, yeldg a proft of f (t (W t+1 W t. Therefore, f (t (W t+1 W t represets the total proft. Remark 6.1. (Prevsble To be precse, defg stochastc tegral we requre the Itegrad f (t to be prevsble or predctable,.e., f (t s F t for all t where F t s<t F s. However, t ca be show that f f (t s cotuous ad adapted to F t, the f (t s automatcally prevsble. Sce we maly deal wth cotuous tegrad, we do ot dstgush betwee prevsble ad adapted process. 6.. Ito Itegral The Ito tegral s a radom varable sce W t ad the tegrad f (t are radom. To esure regularty of the Ito tegral (such as the exstece of the frst ad the secod momets, we restrct f (t to the followg class of stochastc processes. Defto 6.1. (M Stochastc Process Deote MT to be the class of stochastc processes f (t, t, such that

3 6. Ito Stochastc Itegral 11 ( E f (t dt <. Let M be the class of stochastc processes f (t such that f (t M for ay T >. Recall that we say a radom varable X s L, or X L, f E X <. Both M ad L are related to the exstece of secod momet, but M s for a stochastc process ad L s for a radom varable. Sce the Ito tegral s a radom varable ad the tegrad s a radom fucto (stochastc process, we eed to defe the measure of legth ad the mode of covergece defg the tegral by a lmt of a dscretzato (6.. Defto 6.. (L ad MT Norm For a radom varable X ad a stochastc process f f (t, the L ad MT Norm are gve respectvely by ( X L E(X ad f M E f (t dt. (6.3 T Defto 6.3. (L ad MT {X } coverges L to X f Covergece A sequece of radom varables X X L E( X X. A sequece of radom fuctos / stochastc processes { f (t} coverges MT to f f for ay T, ( f f M E f (t f (t dt. T Smlarly, { f (t} coverges M to f f { f (t} coverges to f MT for all T. Wth defto 6.1 ad 6.3, we defe the Ito Itegral o the class of stochastc process M as follows: Defto 6.4. (Ito Itegral For ay T > ad ay stochastc process f M, the stochastc tegral of f o [,T ] s defed by I T ( f lm j f (t j ( W(t j+1 W(t j, where ( t < t 1 < < t < t T s ay partto of (,T wth max j t j t j 1 as. We wrte I( f as I T ( f f (tdw t, ( or f dw.

4 1 6 Ito s Stochastc Calculus The followg theorem justfes the above defto of Ito tegral. Theorem 6.1. (Exstece ad Uqueess of Ito Itegral Suppose that a fucto f M satsfes the followg assumptos A1 f (t s almost surely cotuous,.e., P(lm ε f (t + ε f (t 1, A f (t s adapted to the fltrato {F t }, where F t σ({w s,s < t}. The, for ay T >, the Ito tegral exsts ad s uque almost everywhere. Proof. The proof proceeds three steps: I T ( f f (tdw t 1 Costruct a sequece of adapted stochastc processes f such that f f M. Show that I T ( f I T ( f L. 3 Show the a.s. uqueess of the lmt I T ( f. To beg, 1 Frst we fd a sequece of M T fuctos f 1, f,... such that f f M T E( f (t f (t dt. Defe f (t { k k 1 f s ds f t [ k, k+1 for k 1,,...,[T ] 1, otherwse. The by costructo f (t s adapted to F t ad s a step fucto for each ω. (Fucto of ths kd s called radom step fucto. Sce t s radom, to be precse, t has to be deoted by f (t f (t,ω. Note that k+1 k f (t dt k k 1 f (tdt k f (t dt,a.s. (6.4 k 1 by Cauchy Schwartz (CS equalty. The above equalty holds almost surely as CS s appled for each ω f (t,ω. Next we show f f M T E( f (t f (t dt. By the a.s. cotuty of f (t, we have lm f (t f (t dt a.s. (6.5 Let Y f (t f (t a.s. dt. Note that (6.5 meas that lm Y. Note also that ( Y f (t + f (t T dt 4 f (t dt Ȳ,

5 6. Ito Stochastc Itegral 13 where the secod equalty follows from summg over all k (6.4. Sce a.s. Y ad Y Ȳ a.s., we ca apply Domat Covergece Theorem (DCT to obta E(Y,.e., f f M E( T f (t f (t dt. (6.6 To show the exstece of Ito Itegral, we eed to show that I T ( f f (tdw t coverges to a elemet L. Sce f (t s a step fucto takg costat values [ k, k+1, we ca wrte Note that ( k ( I T ( f f Wk+1 Wk. k 1 E E I T ( f L ( k 1 ( k 1 j 1 f ( k ( Wk+1 f ( k ( k f E( k 1E ( ( k E f E k 1 f (t dt Wk f ( j ( Wk+1 Wk+1 1 Wk Wk (W j+1 W j (Idepedet Icremet ad f ( k (sce E(W t t F k f M T. (6.7 Smlarly, t ca be show that I T ( f I T ( f m L f f m M T. (6.8 From (6.6, for ay ε >, there s a N such that f f M < ε T for all > N. Thus for,m > N, I T ( f I T ( f m L f f m M T f f M T + f m f M T (Tragular Iequalty ε + ε ε.

6 14 6 Ito s Stochastc Calculus The sequece {I T ( f } 1 wth the property I T ( f I T ( f m L for,m s called a (Cauchy sequece. It s well kow Mathematcal Aalyss that ay Cauchy Sequece L has a lmt. Call the lmt I T ( f. 3 Fally, we show the uqueess of the lmt. Suppose that there are sequeces of processes { f (1 } 1 ad { f ( } 1 satsfyg E( ( j f (t f dt for j 1,. We eed to show that I T ( f ( j coverge to the same lmt for j 1,. Itroduce a ew sequece {g } { f (1 1, f ( 1, f (1, f (, f (1 3,...}. The {g } satsfes E( f (t g dt. Therefore, the above proof shows that I(g coverges to some lmt. Note that f a sequece coverges, the every subsequece coverges to the same lmt (Exercse 6.. Thus I T ( f (1 ad I T ( f ( do coverge to the same lmt, completg the proof of uqueess. Note that the above uqueess s proved L sese,.e., I T ( f (1 I T ( f ( L, sce I T ( f (1 I T ( f ( L I T ( f (1 I T (g L + I T (g I T ( f ( L for all ad the quatty o the rght s arbtrarly small. However, I T ( f (1 I T ( f ( L mples that I T ( f (1 I T ( f ( almost surely (See Exercse Thus the defto of Ito s tegral s uque a.s.. Example 6.1. To show the exstece of W t dw t, we eed to show that the Weer process W t belogs to M. Sce for all T ( E W t dt E( W t dt t dt <. Thus W t belogs to M. Also, we have see from Chapter 5 that W t satsfes Assumptos A1 ad A of Theorem 6.1. Hece the exstece of the Ito tegral W t dw t s justfed. Example 6.. We derve the formula W t dw t 1 W t 1 T drectly from the defto,.e. by approxmatg the tegrad by radom step fuctos. Fx T > ad t T, set f W t 1 [t,t +1. The the sequece f 1, f, MT approxmates f, sce

7 6.3 Propertes of the Stochastc Itegral 15 ( E f t f (t dt t +1 t t +1 t E ( W t W(t dt (t t dt 1 ( t +1 t 1 T as. From Theorem 6.1, lm I T ( f exsts L sese. To fd a explct formula of the lmt, ote from the equato a(b a 1 (b a 1 (b a that I T ( f 1 W t ( W t +1 W t [ 1 W T 1 ( Wt +1 W t (W t +1 W t ] (W t +1 W t 1 W T 1 T L as. The last covergece follows from the fact that the quadratc varato of Weer process s T. Therefore, we coclude that W t dw t I T ( f 1 W T 1 T. 6.3 Propertes of the Stochastc Itegral The basc propertes of the Ito tegral are summarzed the theorem below: Theorem 6.. The followgs propertes hold for ay f,g M, ay α,β R ad ay s < t: 1. Learty: t. Isometry: t t (α f (r + βg(r dw(r α f (rdw(r + β g(rdw(r; (6.9 t E( ( t f (rdw(r E f (r dr ; (6.1

8 16 6 Ito s Stochastc Calculus 3. Martgale Property: ( t E f (rdw(r F s I partcular, E ( t f (rdw(r. s f (rdw(r. Proof. 1. If f ad g belog to M, the from the proof of Theorem 6.1, they ca be approxmated by some sequeces f 1, f, ad g 1,g,. From Defto 6.4, t s clear that, for each, I(α f + βg j ( f (t j + g(t j (W t j+1 W t j αi( f + βi(g, where {t j } j1,... s the grd where both f ad g are costat each terval (t j 1,t j. Takg lmt o both sdes of ths equalty as, we obta I(α f + βg αi( f + βi(g. whch s (6.9.. The word Isometry meas a equalty uder dfferet metrcs. Note that (6.1 ca be wrtte as I t ( f L f M T, whch meas the Ito tegral coects the two metrcs L ad M. The proof of (6.1 follows by observg that I t ( f I t ( f L, f f M ad (6.7 the proof of Theorem Followg the proof of Theorem 6.1, we ca fd a radom step fucto f approxmatg f such that I t ( f I t ( f L. W.L.O.G., we ca assume s t k for some k. (addg a extra pot stll gves a step fucto. Thus E E(I t ( f F s ( j f (t j 1 (W t j W t j 1 F s k f (t j 1 (W t j W t j 1 + E( f (t j 1 (W t j W t j 1 F s j jk+1 k f (t j 1 (W t j W t j 1 (by depedet cremet ad t k s j I s ( f. (6.11 The same argumet the proof of Theorem 6.1 yelds I s ( f I s ( f L. O the other had, we have E(I t ( f F s L E(I t ( f F s, sce

9 6.4 Ito s Lemma 17 E [ (E(I t ( f F s E(I t ( f F s ] E [ (E(I t ( f I t ( f F s ] E ( E((I t ( f I t ( f F s (Jese s Iequalty E((I t ( f I t ( f (Tower Property. By takg lmt o both sdes of Equato (6.11, we have completg the proof. E(I t ( f F s I s ( f, ( Ito s Lemma I ths secto we prove the Ito s Lemma whch s the foudato of mathematcal face ad stochastc calculus The case F(t,W t Theorem 6.3. (Ito s Lemma Suppose that F(t, x s a real valued fucto wth cotuous partal dervatves F t (t,x, F x (t,x ad F xx (t,x for all t ad x R. Assume also that the process F x (t,w t belogs to M. The F(t,W t satsfes F(T,W t F(,W( [ F t (t,w t + 1 ] F xx(t,w t dt + F x (t,w t dw t, a.s. (6.13 I dfferetal otato, (6.13 ca be wrtte as [ df(t,w t F t (t,w t + 1 ] F xx(t,w t dt + F x (t,w t dw t. (6.14 Remark Compare (6.14 wth the usual cha rule df(t,x t F t (t,x t dt + F x (t,x t dx t for a dfferetable fucto x t. The addtoal term 1 F xx(t,w t dt (6.14 s called the Ito correcto.. Equato (6.14 s ofte wrtte the abbrevated form

10 18 6 Ito s Stochastc Calculus ( df F t + 1 F xx dt + F x dw t. (6.15 Strctly speakg, the dfferetal otato (6.14 does ot make sese due to the o-dfferetablty of Browa Moto (dw t s udefed. Proof. We frst prove oly the case where F, F x, F xx are all bouded by some C >. Cosder a partto of [,T ], t < t 1 < < t T, where t T. Deote W t by W ; the cremets W +1 W by W; ad t +1 t by t. Usg Taylor s expaso, there s a pot W each terval [W,W +1 ] ad a pot t each terval [t,t +1 ] such that F(T,W t F(,W( [ F(t +1,W+1 F(t,W ] [ F(t +1,W+1 F(t,W+1 ] + F( t,w +1 t + F( t,w+1 t + 1 F x (t,w W + 1 F xx (t,w [ F(t,W +1 F(t,W ] t + F xx (t, W ( W F x (t,w W + 1 [ ] F xx (t,w ( W t + 1 [ ] F xx (t, W F xx (t,w ( W. A 1, + A, + A 3, + A 4, + A 5,, (6.16 say. Note that as F t,f x, F xx ad W t are cotuous ad bouded fuctos, we have lm sup 1,..., lm sup 1,..., lm sup F t ( t t [t,t +1 ],W F t (t,w t a.s., (6.17 sup F xx (t t [t,t +1 ],W F xx (t,w t, a.s., (6.18 sup 1,..., Now we deal separately wth each sum (6.16: F xx (t, W F xx (t,w a.s.. ( From the cotuty of the F t ad F xx (6.17 ad (6.18, we have the covergece of the Rema tegral lm A 1, lm lm A, lm F t ( t,w+1 t F t (t,w t dt F xx (t,w t F xx (t,w t dt a.s., ad a.s.

11 6.4 Ito s Lemma 19. From the assumpto F x M, we have lm A 3, lm F x (t,w W F x (t,w t dw t L from Theorem We show the L covergece E(A 4,. To be specfc, E(A 4, E ( [ ] F xx (t,w ( W t EF xx (t,w [ ] ( W t (cross term has expectato E F xx (t,w E( W t (depedet cremet C E( W t (boudedess of F xx C ( t C T T C as. 4. Note that ( W t L ad thus probablty (See Exercse 6.13 sce the left quatty s the quadratc varato of Browa Moto. Together wth the cotuty result (6.18, we have the followg covergece ( probablty: A 5, sup [ ] F xx (t, W F xx (t,w ( W F xx (t, W +1 F xx(t,w+1 ( W P. Note that the covergece of A,, 1,...,5 volve dfferet modes: A 1, ad A, coverge almost surely, A 3,, A 4, coverge L, ad A 5, coverges probablty. To combe the results, ote from Exercse 6.13 that coverget L mples coverget probablty. Thus all A 3,, A 4, ad A 5, coverge probablty. Note also from Exercse 6.14 that there s a subsequece { k } k1,,... such that {A 3,k } k1,,... coverge a.s. Alog ths subsequece, we ca fd a further subsequece kl such that A 4,kl coverges a.s., ad so forth. Fally, all A j,, j 1,...,5 coverge a.s. wth respect to some subsequece m 1 < m <, say. The

12 11 6 Ito s Stochastc Calculus F(T,W t F(,W( { mk 1 lm k F( t m k,w m k +1 m k t + 1 m k m k 1 m k 1 F xx (t m k,w m k + 1 F xx (t m k [ F t (t,w t + 1 F xx(t,w t [, W m k ] [ ( m k F xx (t m k,w m k m m k 1 k t + W ] m k t ] F xx (t m k,w m k ( m k W }. dt + F x (t,w t dw t, completg the proof. The geeral case where F deferred to Exercse 6.6. t,f a.s., F x (t m k,w m k m k W x ad F xx are ot bouded s Example 6.3. For F(t,x x, we have F t (t,x, F x (t,x x ad F xx (t,x. Ito formula gves dwt dt + W t dw t, provded that W t M (whch has bee verfed Example Geeral Case Note that, Ito s Lemma ams at provdg a frst order approxmato for df(t,w t usg the term dt ad dw t. I the Taylor s seres expaso of F(t,W t up to the terms wth partal dervatves of order two, dt,dw t,dtdw t,(dt,(dw t,... are volved. Sce the quadratc varato of W t s [W] t t, t follows that the term (dw t cotrbuto to a addtoal dt term. Other terms such as dtdw t,(dt,(dw t 3 wth order smaller tha dt ca thus be omtted. It s coveet to remember the results usg the so-called Ito multplcato table (Table 6.1: dt dw t dt dw t dt Table 6.1 Ito multplcato table. Lookg closely to the proof of Theorem 6.3, t ca be see that Ito s Lemma ca be exteded from F(t,W t to F(t,X t where X t s a arbtrary process wth quadratc varato [X] t satsfyg d[x] t g(tdt, for some g(t M. For example, f X t s a Ito Process gve by dx t a t dt + b t dw t, (6.

13 6.4 Ito s Lemma 111 where a t L 1 ad b t M, the t ca be checked (Exercse 6.15 that d[x] t b t dt. (Iformally, we ca use Table 6.1 to see that d[x] t (dx (a t dt +b t dw a t (dt +a t b t (dt(dw+b t (dw b t dt. The followg Theorem gves the geeral case of Ito s Lemma for F(t,X t, where X t s a arbtrarly process stead of Browa moto. Theorem 6.5. (Ito formula, geeral case Let X t be a stochastc process wth quadratc varato [X] t satsfyg d[x] t g(tdt where g(t M. Suppose that F(t,x, F t (t,x, F x (t,x ad F xx (t,x are cotuous for all t ad x R. Also the process g t F x (t,x t M. The F(t,X t s ca be expressed as df(t,x t F t (t,x t dt + F x (t,x t dx t + 1 F xx(t,x t d[x] t. (6.1 Example 6.4. If X t s a Ito process satsfyg (6., the (6.1 reduces to (wrtg F(t,X t as F df F t dt + F x dx + 1 F xx d[x] t F t dt + F x (a t dt + b t dw + 1 F xxbt dt ( F t + F x a t + 1 F xxbt dt + F x b t dw. We ed ths secto wth the aalog of the product rule d f (xg(x f (xdg(x+ g(xd f (x stochastc calculus. Aga, there s a addtoal term of dt cotrbuted by the quadratc varato process. Corollary 6.1. (Product Rule If X t ad Y t are process satsfyg dx t a x (tdt + b x (tdw t ad dy t a y (tdt + b y (tdw t, the Proof. For a small t, dx t Y t X t dy t +Y t dx t + b x (tb y (tdt. X t+ t Y t+ t X t Y t X t (Y t+ t Y t +Y t (X t+ t X t + (Y t+ t Y t (X t+ t X t As, by defto, X t+ t Y t+ t X t Y t dx t Y t, X t+ t X t dx t ad Y t+ t Y t dy t. Thus Corollary 6.1 follows by otg from Table 6.1 that (Y t+ t Y t (X t+ t X t (a y (tdt + b y (tdw t (a x (tdt + b x (tdw t b x (tb y (tdt. Note that the formal proof ca be obtaed by smlar argumet as Theorem 6.1.

14 11 6 Ito s Stochastc Calculus 6.5 Stochastc Dfferetal Equatos I ths secto, we cosder stochastc dfferetal equato of the form dx t f (X t dt + g(x t dw t wth the tal codto X x. The goal s to obta a explct formula for X t. Example 6.5. Cosder the tal value problem { dxt αx t dt + σ dw t, X x R, Take F(t,x e αt x, wth F(,X, by Ito s Lemma we have df(t,x t [ αe αt X t αe αt X t ] dt + σe αt dw t σe αt dw t. It s easy to see that e αt M. It follows that F(t,X t x + σ t eαs dw s ad t X t e αt F(t,X t e αt x + σe αt e αs dw s. Example 6.6. Suppose X t x e at+bw t for t. By Ito formula, ( dx t d x e at+bw t (ax e at+bw t + b x e at+bw t dt + bx e at+bw t dw t (a + b X t dt + bx t dw t. The problem of checkg that bx t M for all T > s left to the reader Exercse Ths mples the soluto of the tal value problem { ( dx t a + b X t dt + bx t dw t, X x s X t x e at+bw t. From the above examples, to solve a SDE, we eed to guess a soluto ad use Ito s Lemma to verfy that the soluto satsfes the SDE. The below table summarzes some examples ad solutos of SDE. I the table, c stads for a costat ad sh (e x e x /.

15 6.5 Stochastc Dfferetal Equatos 113 Name SDE Soluto Orste-Uhlebeck(OU process dx t αx t dt + σw t c + σe αt t eαs dw s Mea revertg OU dx t (m αx t dt + σw t m (c me t + σ t es t db s Geometrc Browa Moto dx t ax t dt + bx t dw t ce (a b /t+bw t Browa Brdge dx t b X t 1 t dt +W t a(1 t + bt + (1 t t dw s ( 1 s dx t 1 + X t + 1 X t dt Xt dw t sh(c +t +W t dx t Xt 3 dt + Xt 1 dw t 1 W t dx t 1 X tdt + 1 Xt dw t s(c +W t dx t 1+t 1 X tdt + 1+t 1 dw t W t /(1 +t dx t rdt + αx t dw t ce αw t 1 αt + r t eα(b t B s 1 α (t s ds

16 14 6 Ito s Stochastc Calculus 6.9 Exercses Exercse 6.1. Let t < t 1 < < t T, where t j jt, be a partto of the terval [,T ] to equal parts. Usg the detty a(b a 1 ( b a 1 (a b ad Theorem 5.14, show that lm j Usg the detty b(b a 1 Fd the lmt for p (,1. lm j lm j W(t j ( W(t j+1 W(t j 1 W t T. (6.44 ( b a + 1 (a b ad Theorem 5.14, show that W(t j+1 ( W(t j+1 W(t j 1 W t + T. (6.45 W(pt j+1 + (1 pt j ( W(t j+1 W(t j (6.46 Exercse 6.. Show that f a sequece coverges to some lmt, the every subsequece coverges to the same lmt. Exercse 6.3. Show that W t Exercse 6.4. Verfy the equaltes ad belogs to M, where W t s a Weer process. t dw t TW t W t dt, (6.47 Wt dw t 1 3 W t 3 W t dt. (6.48 Note that the tegral o the rght-had sde s a Rema tegral defed path-wse,.e. separately for each ω Ω. Exercse 6.5. Suppose that g(x s a cotuous, ad wth t T /, t < t 1 < < t T s a partto of [,T ]. Show that lm sup 1,..., sup t [t,t +1 ] g(t g(t. If T, does the covergece hold? Prove t or gve a couter example. Therefore the Ito s Lemma ca be appled to F. Exercse 6.6. (Geeral Case of Ito s Lemma Suppose that ψ (x s a smooth fucto satsfyg ψ (x 1 for ay x [,] ad ψ (x for x / [ 1, + 1]. Let F (t,x ψ (xf(t,x, show that F (t,x satsfes the codtos of Theorem 6.3 ad has bouded partal dervatves (F x ad (F xx for each.

17 6.9 Exercses 15 Sce F(t,x F (t,x for every t [,T ] ad x [,], the Ito s Lemma holds for F o the set A {sup t [,T ] W t < }. By show that lm P(A 1, deduce that Ito s Lemma holds for F almost surely. Exercse 6.7. For F(t,x x 3 we have F t (t,x, F x (t,x 3x ad F xx (t,x 6x. Verfy that 3Wt MT ad use Ito s Lemma to show the equalty d ( Wt 3 3Wt dt + 3Wt dw t. Exercse 6.8. Show that the expoetal martgale X t exp { W t t } s satsfes dx t X t dw t. Exercse 6.9. Show that the tal value problem { dxt ax t dt + bx t dw t, X( x {( has a soluto gve by X t x exp Exercse 6.1. Show Equato 6.3. a b t + bw t }. Exercse Show that for ay two radom varables X ad Y, X Y L mples X Y a.s.. Cosder a sequece of radom varables X. If X X L, does t mply X X a.s.? Prove or gve couter examples. Exercse 6.1. Use Ito s Lemma to compute E(W 6 t. Exercse Let X ad {X (ω} 1 be radom varables o a probablty space (Ω,F,P. Recall the three mode of covergece of a sequece of radom varables {X } to X. (Almost Surely, a.s.: P(ω : lm X (ω X(ω 1. (Probablty: For each ε >, lm P( X (ω X(ω > ε. (L : lm E( X X. Show that 1. By rewrtg the lmt otato as uo/tersect operato, show that coverget a.s. mples coverget probablty.. By cosderg the equalty 1 { Y >a} Y 1 a { Y >a} Y a.s., show that coverget L mples coverget probablty. a 3. Fd a example where X coverges to X a.s. but ot L. Hts: X (ω takes o-zero values a decreasg porto of Ω, but the value s huge whe o-zero. 4. Fd a example where X coverges to X L but ot a.s.. Hts: X (ω takes o-zero values a decreasg porto of Ω. The value s bouded whe X (ω s o-zero. But the subset of Ω where X (ω s o-zero s movg aroud so that for each ω there are fte may such that X (ω s o-zero.

18 16 6 Ito s Stochastc Calculus Exercse Use the otato Exercse Recall the Frst Borel-Catell Lemma that P( 1 j A j f 1 P(A <. 1. Let A j { X j X > j }. Show that P( 1 j A j mples X j a.s. X.. Suppose that X p X. Show that there s a subsequece X j of X such that P( X j X > j < j alog the subsequece. 3. Show that, f X coverges to X probablty, the there s some subsequece such that X j coverges to X a.s.. Exercse Fd the quadratc varato of X t where X t s a Ito s process satsfyg (6.. Show that d[x] t b(tdt. Exercse Gve X t x e at+bw t, where a,b R, verfy that X t M. Exercse Let S t follows the Geometrc Browa Moto model (6.4, prove that 1. the desty fucto f (x of S t S s gve by 1 f (x exp πtσx [ ( lx µ σ σ t t ] 1 R +(x.. ES t S exp{µt}. 3. Var(S t S exp{µt}( exp { σ t } 1.