Stochastic Integration and Ito s Formula

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1 CHAPTER 3 Stochastic Itegratio ad Ito s Formula I this chapter we discuss Itô s theory of stochastic itegratio. This is a vast subject. However, our goal is rather modest: we will develop this theory oly geerally eough for later applicatios. We will discuss stochastic itegrals with respect to a Browia motio ad more geerally with respect to a cotiuous local martigale. Istead of attemptig to describe the largest possible class of itegrad processes, we will oly sigle out a class of itegrad processes sufficietly large for our later applicatios. For example, i the case of a cotiuous local martigale as a itegrator, we oly restrict ourselves to cotiuous adapted itegrad processes, a class of processes sufficietly large for most applicatios. This of course does ot mea that itegrads which are ot cotiuous caot be itegrated with respect to a cotiuous martigale. It is usually the case that whe dealig with a discotiuous itegrad, we ca quickly decide the ad there whether the itegral has a meaig. Sice may excellet textbooks o stochastic itegratio are available (McKea [8, Ikeda ad Wataabe [6, Chug ad Williams [3, Oksedal [1, Karatzas ad Shreve [7, to cite just a few), there is little motivatio o the part of the author to go beyod what will be preseted i this chapter. 1. Itroductio If A is a process of bouded variatio ad f : R R is fuctio such that s f (s, ω) is Borel measurable fuctio, the f (s) da s ca be iterpreted as a pathwise itegral if proper itegrability coditios are assumed ad its value at ω Ω is the usual Lebesques Stieljes itegral f (s, ω) da s (ω). I theoretical ad applied problems, there is a obvious eed to make sese out of a itegral of the form f (s) db s, where B is a Browia motio. As we kow, Browia motio sample paths are ot fuctios of bouded variatio (see REMARK 7.2). For this 43

2 44 3. STOCHASTIC INTEGRATION AND ITO S FORMULA reaso i geeral there is o easy ad direct pathwise iterpretatio of the above itegral. However, i some special situatio, a simple iterpretatio is possible. For example, if s f (s, ω) itself has bouded variatio for each ω, we ca defie the above itegral by itegratio by parts: f (s) db s = f (t)b t B s d f (s). Such stochastic itegrals are rather limited i its scope of applicatio. Itô s theory of stochastic itegratio greatly expads the class of itegrad processes, thus makig the theory ito a powerful tool i pure ad applied mathematics. We first defie the itegratio of a step (ad determiistic) process with respect to a Browia motio. Let : = t < t 1 < t = t be a partitio of [, t. If f is a step process: f (s) = f j 1, t j 1 t < t j, the the atural defiitio of its stochastic itegral with respect to Browia motio B is 1 [ f (s) db s = f j 1 B tj B tj 1. Usig the property of idepedet icremets for Browia motio, its secod momet is give by [ 2 [ (Bti ) ( ) E f (s) db s = f i1 f j 1 E B ti 1 B tj B tj 1 This gives (1.1) E 1 = = f (s) db s 2 = i, If f ad g are two step fuctios, the E f (s) db s f j 1 2 E [ ( B tj B tj 1 f j 1 2 ( t j t j 1 ). ) 2 f j 1 2 (t j t j 1 ) = f (s) 2 ds. g(s) db s 2 = f (s) g(s) 2 ds = f g 2 2. For a arbitrary fuctio f : [, t R, it is ow clear that as log as there is a sequece of step fuctios f such that f f 2, we ca defie

3 the stochastic itegral as the limit f (s) db s = lim 1. INTRODUCTION 45 f (s) db s. It is well kow from real aalysis that this approximatio property is shared by all Borel measurable fuctios f such that f 2 <. We have thus greatly elarged the space of determiistic fuctios which ca be itegrated with respect to a Browia motio. The key observatio i Itô s theory of stochastic itegratio is that we ca pass to a radom step process as log as f j F tj. Note that ow the itegrad has the form f (s) = f j 1, t j 1 s < t j, f j 1 F tj 1, Namely, i the time iterval [t j 1, t j, the itegrad is measurable with respect to the σ-algebra at the left edpoit of the time iterval. For this reaso above argumet still applies as log as we replace the equality (1.1) by the more geeral equality E 1 f (s) db s 2 = E f j 1 2 (t j t j 1 ) = E f (s) 2 ds. The key step i establishig this equality is the vaishig of the off diagoal term [ E f i 1 f j 1 (B ti B ti1 )(B tj B tj 1 ) =. This holds because if i < j, the by coditioig o F tj 1, we have [ E f i 1 f j 1 (B ti B ti1 )(B tj B tj 1 ) F tj 1 = f i 1 f j 1 (B ti B ti1 )E [B tj B tj 1 F tj 1 =. For the diagoal term, by coditioig o F tj 1 agai, we have E [ f i 1 2 (B ti B ti 1 ) 2 = E f i 1 2 (t i t i 1 ). A importat property of stochastic itegrals is that M t = f (s) db s is a cotiuous martigale. Let us see why this is so if the itegrad process is a step process We ca write f (s) = f j 1, t j 1 < s t j, f j 1 L 2 (Ω, F tj 1, P). M t = f (s) db s = f j 1 (Bt j t B tj 1 t ).

4 46 3. STOCHASTIC INTEGRATION AND ITO S FORMULA Note that sum has oly fiitely may terms for each fixed t. The process M has cotiuous sample paths because Browia motio does. Suppose that s < t. If s ad t are ot amog the poits t j we may simply isert them ito the sequece. Now s ad t are amog the sequece we have M t M s = s t j 1 t ( ) f j 1 B tj B tj 1. The geeral term i the sum vaishes after we coditio it o F tj 1, hece it also vaishes after we coditio o F s because F s F tj 1. It follows that E [M t M s F s = ad M t is a cotiuous martigale. We ow calculate the quadratic variatio process for the cotiuous martigale We claim that M t = M, M t = f (s) db s. f (s) 2 ds. We agai verify this for a step process f. As before, we assume that s < t are amog the poits t j. We have M 2 t M 2 s = f i 1 f j 1 (B ti B ti 1 )(B tj B tj 1 ), where the sum is over those i ad j such that t i t, t j t ad at least oe of t i 1 ad t j 1 is greater or equal o s. For a off-diagoal term, say, i < j, the term vaishes after coditioig o F tj 1, hece also vaishes after coditioig o F s. For a diagoal term i = j, sice t j 1 s, by first coditioig o F tj 1 ad the coditioig o F s we have E [ f j 1 2 (B tj B tj 1 ) 2 F s = E [ f j 1 2 (t j t j 1 ) F s. It follows that This shows that is a martigale. E [ [ Mt 2 Ms 2 F s = E f (s) 2 ds F s. Mt 2 f (s) 2 ds s

5 2. STOCHASTIC INTEGRALS WITH RESPECT TO BROWNIAN MOTION Stochastic itegrals with respect to Browia motio The geeral settig is as follow. We have a Browia motio B with respect to a filtratio F defied o a filtered probability space (Ω, F, P). We assume that the filtratio F satisfies the usual coditios. DEFINITION 2.1. A real-valued fuctio f : R + Ω R is called a step process if there exists a odecreasig sequece = t < t 1 < t 2 < icreasig to ifiity ad a sequece of square-itegrable radom variables f j 1 F tj 1 such that f t = f j 1, t j 1 t < t j. The space of step processes is deoted by S. The space of step processes o [, t is deoted by S t. Note that o a iterval [t j 1, t j ) where f is costat, it is measurable with respect to the σ-algebra o the left edpoit of the iterval, i.e., f = f j 1 F tj 1. Let f be a step process as above. The stochastic itegral of f with respect to Browia motio B is the process I( f ) t = f (s) db s = ) f j 1 (B tj t B tj 1 t. Of course this is a fiite sum for each fixed t. If t i t < t i+1, the I( f ) t = i f j 1 (B tj B tj 1 ) + f i (B t B ti ). It is easy to see that the defiitio of I( f ) t is idepedet of the partitio { tj }. As we have show i the last sectio I( f ) is a cotiuous martigale with the quadratic variatio I particular we have I( f ), I( f ) t = f (s) 2 ds. [ E I( f ) t 2 = E f (s) 2 ds. This relatio allows us to exted the defiitio of stochastic itegrals to more geeral itegrads by a limit procedure. Sice the quadratic variatio process of a Browia motio correspods to the Lebesgue measure o R +, a very wide class of processes ca be used as itegrad processes. DEFINITION 2.2. A fuctio f : R + Ω R is called progressively measurable with respect to the filtratio F if for each fixed t the restrictio f : [, t Ω R is measurable with respect to the product σ-algebra B[, t F t.

6 48 3. STOCHASTIC INTEGRATION AND ITO S FORMULA EXAMPLE 2.3. Here are some examples of progressively measurable processes: (1) A step process is progressively measurable. (2) A cotiuous ad adapted process is progressively measurable. (3) Let τ be a fiite stoppig time. The the process f (t, ω) = I [,τ(ω) (s) is progressively measurable. For a discrete τ, this ca be verified directly. For a geeral τ, let τ = ([2 τ + 1)/2. The τ τ ad I [,τ (ω)(s) I [,τ(ω) (s) for all (s, ω) R + Ω. For a progressively measurable process f, we defie for each fixed T, [ T f 2 2,T = E f (s) 2 ds. The progressive measurability assures that the itegral o the right side has a meaig. We use LT 2 to deote the space of progressively measurable processes f o [, T Ω with f 2,T <. The orm 2,T makes LT 2 ito a complete Hilbert space. We use L 2 to deote the space of progressively measurable processes f such that f 2,T is fiite for all T. It ca be made ito a metric space by itroducig the distace fuctio d( f, g) = =1 f g 2, 1 + f g 2,. Note that S T L 2 ad S L 2, ad there is a obvious restrictio map L 2 Lt 2. We ow show how to defie the stochastic itegral I( f ) t = f (s) db s for every process i L 2. We eed the followig approximatio result. LEMMA 2.4. S T is dese i LT 2. I other words, for ay square itegrable progressively measurable process f LT 2, there exists a sequece { f } of step processes such that f f 2,T. PROOF. First ote that the set of uiformly bouded elemets i LT 2 is dese. Suppose that f is a progressively measurable process uiformly bouded o [, T Ω. Defie f h (t, ω) = 1 h t h f (s, ω) ds. Each f h is cotiuous ad adapted. From real aalysis we also kow that T lim h f h (t, ω) f (t, ω) 2 ds =.

7 2. STOCHASTIC INTEGRALS WITH RESPECT TO BROWNIAN MOTION 49 It follows that f h f 2,T as h. This shows that the set of uiformly bouded cotiuous ad adapted processes is dese i LT 2. Fially for a cotiuous, adapted, ad uiformly bouded process f o [, T we defie ( j 1 f (t) = f ), j 1 t < j. Each f S T ad f (t, ω) f (t, ω) for all (t, ω) [, T Ω by cotiuity. Hece f f 2,T by the domiated covergece theorem. The lemma is proved. We are ow ready to defie the stochastic itegral of a progressively measurable ad square itegrable process with respect to a Browia motio. Suppose that f L 2. By restrictig to [, t we ca regard it as a elemet i Lt 2 for ay t. Let f be a sequece of step processes o [, T such that f f 2,T. The stochastic itegral I( f ) t is well defied for t T as a Riemaa sum ad is a square itegrable martigale. We have I( f m ) t I( f ) t = [ f m (s) f (s) db s. By Doob s martigale iequality, [ E max I( f m) t I( f ) t 2 4E I( f m f ) T 2 t T [ T = 4E f m (s) f (s) 2 ds = 4 f m f 2 2,T. This shows that I( f ) t is a Cauchy sequece i L 2 (Ω, P). By the completeess of L 2 (Ω, P), the limit I( f ) t = lim I( f ) t exists ad we defie f (s) db s = lim f (s) db s. Sice f f 2,T1 implies f f 2,T2 for T 2 T 1, it is easy to see that I( f ) t thus defied is idepedet of T ad the choice of the approximatig sequece { f }. We have oly defied {I( f ) t, t } as a collectio of radom variables. We ca do more. THEOREM 2.5. Suppose that f L 2 is a square itegrable, progressively measurable process. The the stochastic itegral I( f ) t = f (s) db s is a cotiuous martigale. Its quadratic variatio process is I( f ), I( f ) t = f (s) 2 ds.

8 5 3. STOCHASTIC INTEGRATION AND ITO S FORMULA PROOF. The martigale property of I( f ) is iherited from the same property of I( f ) because this property is preserved whe passig to the limit i L 2 (Ω, F, P) as. We show that it has cotiuous sample paths. From [ E max I( f m) t I( f ) t 2 4 f m f 2 2,T t T we have by Chebyshev s iequality, [ P max I( f m) t I( f ) t ɛ 4 t T ɛ 2 f m f 2 2,t. Sice f m f 2,T as m,, by choosig a subsequece if ecessary, we may assume that f f +1 2,T 1/ 3, hece { P max I( f ) t I( f +1 ) t 1 } t T By the Borel Catelli lemma, there is a set Ω with P {Ω } = 1 with the followig property: for ay ω Ω, there is a (ω) such that max t T I( f ) t (ω) I( f +1 ) t (ω) 1 2 for all (ω). It follows that I( f ) t (ω) coverges uiformly o [, T. Sice each I( f ) t (ω) is cotiuous i t, the limit, which ecessarily coicide with I( f ) t (ω) with probability 1 for all t, must also be cotiuous. We therefore have show that the stochastic itegral I( f ) t has a versio with cotiuous sample paths. We have show before that for the step process f the process I( f ) 2 t f (s) 2 ds is a martigale. For each fixed t, as, the above radom variable coverges i L 1 (Ω, P) to I( f ) 2 t f (s) 2 ds. Therefore the martigale property ca be passed to the limitig process, which idetifies the quadratic variatio process of the stochastic itegral as stated i the propositio. COROLLARY 2.6. Suppose that f, g L 2. The I( f ), I(g) t = PROOF. This follows from the equality f (s)g(s) ds. M, N t = M + N, M + N t M N, M N t 4 for two cotiuous martigales M ad N.

9 2. STOCHASTIC INTEGRALS WITH RESPECT TO BROWNIAN MOTION 51 If Z F s, we have Z f u db u = Z f u db u. s s This ca be cosidered to be self-evidet, eve though it requires a approximatio argumet to verify. The same remark ca be said of the idetity s f u db u = s f u db u f u db u. Either we defie the left side by the right side or we cosider the left side as the stochastic itegral with respect to {B s+u, u }, which is a Browia motio with respect to the filtratio F s+ = {F s+u, u }, the ed result is the same. However, if we replace s or t by a radom time, the meaig of the two ad other similar equalities may become ambiguous, especially whe the radom time is ot a stoppig time. I this respect, if we restrict ourselves to stoppig times, the followig results may help us clarify the situatio. PROPOSITION 2.7. Suppose that f LT 2 (1) We have ad τ T a stoppig time. τ f s db s = T I [,τ (s) f s db s. (2) If Z F τ is a bouded bouded radom variable, the T T Z f db s = ZI [τ, ) (s) f s db s. τ PROOF. We will verify the equalities for discrete stoppig times ad step processes. I geeral we ca approximate the stoppig times σ ad τ by discrete stoppig times from above ad the itegrad process f by step processes i the 2,T orm. The equalities are clearly preserved whe passig to the limit uder such approximatios. Let = t < t 1 < < t = T be a partitio such that f = f j 1 F tj 1 for t j 1 t < t j ad τ takes values oly i the sequece { t j }.

10 52 3. STOCHASTIC INTEGRATION AND ITO S FORMULA (1) Note that I {s<τ} is a step process. We have I( f ) τ = = = = = I {τ=ti } T I {τ=tj} i f s db s i f j 1 (B tj B tj 1 ) f j 1 (B tj B tj 1 ) I {τ=ti } i=j I {tj 1 <τ} f j 1(B tj B tj 1 ) I [,τ) (s) f s db s. (2) Note that ZI {τ s} is a step process ad belogs to F s for fixed s. We have T i Z f s db s Z ZI {τ=ti } f s db s σ = = = = T I {τ=ti } Z f j 1 (B tj B tj 1 ) j=i+1 j 1 Z f j 1 (B tj B tj 1 ) I {τ=ti } ZI {τ tj 1} f j 1(B tj B tj 1 ) ZI [τ, ) (s) f s db s. EXAMPLE 2.8. Cosider the stochastic itegral B s db s = lim B tj 1 (B tj B tj 1 ). Usig the idetity 2a(b a) = b 2 a 2 + (b a) 2 we have 2 B tj 1 (B tj B tj 1 ) = B 2 t + (B tj B tj 1 ) 2. The last sum coverges to the quadratic variatio B, B t = t. Hece we have 2 B s db s = Bt 2 t, which is ideed a martigale.

11 3. EXTENSION TO MORE GENERAL INTEGRANDS 53 EXAMPLE 2.9. Oe may woder what will happe if i the approximatig sum of a stochastic itegral we do ot take the value of the itegrad at the left edpoit of a partitio iterval. For example, what is the sum B tj (B tj B tj 1 )? This is ot hard to figure out. The differece betwee this sum ad the usual sum takig the left edpoit value of the itegrad is just the quadratic variatio alog the partitio hece we ave lim (B tj B tj 1 ) 2, B tj (B tj B tj 1 ) = B s db s + t. This example shows where to take the value of the itegrad o each partitio iterval really matters. 3. Extesio to more geeral itegrads We have defied stochastic itegral with respect to a Browia motio for itegrad process f L 2. These are progressively measurable processes f such that [ E fs 2 ds < for all t. This itegrability is too restrictive. Without too much effort we ca defie a much wider class of itegrad processes. DEFINITION 3.1. We use L 2 to deote the space of progressively measurable processes f such that [ P fs 2 ds < = 1 for all t. This is a class of process wider tha L 2 ad is sufficiet for our applicatios. For example, all adapted cotiuous processes belog to Lloc 2. We will exted the defiitio of stochastic itegrals to processes i Lloc 2. The resultig stochastic itegral process is o loger a square itegrable martigale but a local martigale. Recall the defiitio of a local martigale. DEFINITION 3.2. We say that M = {M t, t } is a local martigale if there exists a sequece of stoppig times τ such that the stopped processes M τ = {M t τ, t } are martigales.

12 54 3. STOCHASTIC INTEGRATION AND ITO S FORMULA For cotiuous local martigale M with M =, we ca always take τ = if {t : M t }. Suppose that f Lloc 2. Defie the stoppig time { } τ = if t > : f (s) 2 ds. the assumptio that f L 2 loc implies that τ (with probability oe, if oe wats to be very precise). Now cosider the process f (s) = f (s)i {s τ }. It is clear that f L 2, i fact f 2,T for all T. The stochastic itegral I( f ) is well-defied ad is a square itegrable cotiuous martigale. We ow defie the stochastic itegral I( f ) t as follows: I( f ) t = I( f ) t, t τ. I order that this defiitio make sese, we have to verify the cosistecy: if m, the I( f ) t = I( f m ) t, t τ m. This is immediate. Ideed, I( f ) t τm = From defiitio we have f I {s τm } db s = I( f ) t τ = I( f ) t, f m db s = I( f m ) t. which shows that I( f ) is a cotiuous local martigale. [ EXAMPLE 3.3. The process Be B2 L 2 because E B t 2 e B2 t is ifiite for t 1/2. However, sice it is cotiuous it belogs to Lloc 2 ad its stochastic itegral with respect to Browia motio is well defied. I fact, Itô s formula will show that [ 2 B s e B2 s db s = e B2 t 1 + 2B 2 s e Bs 2 ds. 4. Stochastic itegratio with respect to cotiuous local martigales We have discussed stochastic itegratio with respect to Browia motio so that oly the martigale properties of Browia motio is used. For this reaso ot much eeds to be chaged if we replace Browia motio by a geeral cotiuous martigale. All we eed to do is to replace the quadratic variatio process of Browia motio B, B t = t by M.M t, which is a cotiuous icreasig process. More specifically, the theory of stochastic itegratio with respect to Browia motio is based o several properties of Browia motio: (1) B is a cotiuous square itegrable martigale; (2) The quadratic variatio process is B, B t = t.

13 4. STOCHASTIC INTEGRATION WITH RESPECT TO CONTINUOUS LOCAL MARTINGALES55 (3) If f is a step process, the [ ( ) 2 [ E f db s = E f s 2 ds. (4) The space of step processes S T is dese i the space of progressively measurable ad square itegrable processes LT 2. For a theory of stochastic itegratio with respect to a geeral cotiuous martigale M, we make the followig observatios: (1) By the usual stoppig time argumet, we ca restrict ourselves iitially to uiformly bouded cotiuous martigales. (2) The quadratic variatio process M, M t is cotiuous, adapted, ad icreasig. Therefore itegratio of a progressively measurable process with respect to M, M is well defied. (3) For a bouded step process f we have 2 [ E f s dm s = E f 2 s d M, M s. (4) We ca itroduce the orm [ f 2,T;M = E fs 2 d M, M s. There does ot seem to be a good descriptio of the closure of the space of S T uder this orm that is good for all M, but at least all progressively measurable ad cotiuous processes with fiite orm ca be approximated by step processes. It is also possible develop a theory of stochastic itegratio with respect to a geeral, ot ecessarily cotiuous martigale. Sice we will mostly oly deal with cotiuous martigales, we will restrict ourselves to cotiuous local martigales. Suppose that M is a cotiuous local martigale ad f S is a step process. The stochastic itegral of f with respect to the martigale M is I( f ) t = f dm s = j f j 1 (M tj 1 t M tj t Just as i the case of Browia motio we have 2 [ E f s dm s = E f s 2 d M, M. Let L 2 (M) be the space of progressively measurable processes f such that [ T f 2 2,T;M = E f s 2 d M, M s < for all t. It ca be show that S T is dese i L 2 (M) T (see Karatzas ad Shreve [7). Thus by the usual approximatio argumet, we see that ).

14 56 3. STOCHASTIC INTEGRATION AND ITO S FORMULA the stochastic itegral I( f ) t = f s dm s is well defied for all f L 2 (M) ad the quadratic variatio process is I( f ), I( f ) t = f s 2 d M, M s. Although it is ot easy to fid a good descriptio of L 2 (M) that fits all cotiuous martigales, it does cotai all cotiuous adapted processes such that f 2,T;M is fiite for all T. By the stadard stoppig time argumet we have used i the case of Browia motio, the stochastic itegral I( f ) t = f s dm s ca be defied for a cotiuous local martigale M ad a progressively measurable process f such that [ P f s 2 d M, M s < = 1 for all t. Uder these coditios, the stochastic itegral I( f ) is a cotiuous local martigale. EXAMPLE 4.1. Let M be a uiformly bouded cotiuous martigale. As i the case of Browia motio we have M 2 t M 2 = 2 M tj 1 (M tj M tj 1 ) + (M tj M tj 1 ) 2. The sum after the equal sig coverges to the stochastic itegral M s dm s. Therefore the secod sum must also coverge. It is clear that the limit of the secod sum is a cotiuous, icreasig, ad adapted process. Hece by the Doob-Meyer decompositio theorem, the limit must be the quadratic variatio process, i.e., ad lim (M tj M tj 1 ) 2 = M, M t Mt 2 = M M s dm s + M, M t. The covergece of the quadratic variatio process takes place i L 2 (Ω, P). For a geeral local martigale, it ca be show by routie argumet that the covergece takes place at least i probability.

15 5. ITÔ S FORMULA Itô s formula Itô s formula is the fudametal theorem for stochastic calculus. Let us recall that the fudametal theorem of calculus states that if F is a cotiuously differetiable fuctio the F(t) F() = F (s) ds. Let = t < t 1 < < t = t be a partitio of the iterval [a, b. We have F(t) F() = [ F(tj ) F(t j 1 ). Sice F is cotiuous, by the mea value theorem, there is a poit ξ i [t j 1, t j such that F(t j ) F(t j 1 ) = F(ξ j )(t j t j 1 ). Hece, by the defiitio of Riema itegrals we have as, F(b) F(a) = F (ξ j )(t j t j 1 ) F (s) ds. What happes to this proof if we replace t by B t. I this case ξ j will be a poit betwee B tj 1 ad B tj. We have to deal with the sum F (ξ j )(B tj B tj 1 ). Here the argumet has to depart from what we have doe above. As EX- AMPLE 2.9 shows, the place where we take the value of the itegrad o each partitio iterval ca chage the limit of the Riema sum. Therefore we should ot expect that the above sum will coverges to the stochastic itegral F (B s ) db s, for which the left edpoit value of the itegrad is used i each partitio iterval. The method to remedy this situatio is to take oe more term of the Taylor expasio F(B tj ) F(B tj 1 ) = F (B tj 1 )(B tj B tj 1 ) F (ξ j )(B tj B tj 1 ) 2. The sum of the first term o the right side will coverge to the stochastic itegral as usual. The crucial observatio is that because Browia motio has fiite quadratic variatio, the sum of the secod term o the right side F (ξ j )(B tj B tj 1 ) 2 F (B s ) ds as. The fial result is the well kow Itô s formula F(B t ) = F(B ) + F (B s ) db s F (B s ) ds.

16 58 3. STOCHASTIC INTEGRATION AND ITO S FORMULA O the right side, the first term is a cotiuous martigale, ad the secod term is a process of bouded variatio. Thus if F is cotiuous ad uiformly bouded together with its first ad secod derivatives, the the compositio F(B t ) is a semimartigale ad Itô s formula gives a explicit expressio for its Doob-Meyer decompositio. The proof of Itô s formula for for a geeral semimartigale Z will ot be a lot more difficult tha the case of Browia motio. For this reaso, i this sectio we will prove it i this geerality. THEOREM 5.1. Suppose that F C 2 (R) ad Z is a semimartigale. The (5.1) F(Z t ) = F(Z ) + F (Z s ) dz s + 1 F (Z s ) d Z, Z s. 2 Recall that a semimartigale has the form Z t = M t + A t, where M is a local martigale ad A a process of bouded variatio. We first use a stoppig time argumet to reduce the proof to the case where Z, M, M, M, ad A are all uiformly bouded ad F together with its first ad secod derivatives are uiformly bouded ad uiformly cotiuous. First of all, let Ω N = { Z N} ad defie P N (C) = P(C Ω N). P(Ω ) It is clear that uder the P N, the process Z is still a semimartigale with the same decompositio Z = M + A ad P N { Z N} = 1, i.e., Z is bouded. Next, defie the stoppig time τ N = if {t : M t M + M, M t + A t N}. The stopped processes Z τ N = M τ N + A τ N ad M τ N, M τ N t = M, M τ N t have the bouded properties we desired. Defie a fuctio F N C 2 (R) such that it coicides with F o [ 2N, 2N ad vaishes outside [ 3N, 3N. O the probability space (Ω N, F Ω N, P N ) suppose that we have the formula F N (Z τ N ) = F N (Z τ N ) + F N(Z τ N s ) dz s Replacig t there by t τ N we see that F(Z t τn ) = F(Z ) + τn F (Z s ) dz s F N(Z τ N s ) d Z τ N, Z τ N s. τn F (Z s ) d Z, Z s. Here we have used the defiitio of stochastic itegral with respect to a local martigale. This shows that the formula (5.1) holds o Ω N ad t < τ N. Fially from P {Ω N } 1 ad P {τ N } = 1, we see that Itô s formula holds without ay extra coditios.

17 5. ITÔ S FORMULA 59 After these reductios, we start the proof of Itô s formula itself. By Taylor s formula with remaider we have F(Z tj ) F(Z tj 1 ) = F (Z tj 1 )(Z tj Z tj 1 ) F (ξ i )(Z tj Z tj 1 ) 2, where ξ is a poit betwee Z tj 1 ad Z tj. For simplicity let us deote Z j = Z tj Z tj 1, M j = M tj M tj 1, M j = M tj M tj 1. With these otatios we ca write F(Z t ) F(Z ) = F (Z tj 1 ) Z j F (ξ i )( Z j ) 2. The first sum coverges to the stochastic itegral i L 2 (Ω, P): F (Z tj 1 ) Z j The real work starts with the proof of (5.2) F (ξ i )( Z j ) 2 F (Z s ) dz s. F (Z s ) d Z, Z s. We will prove this through a series of three replacemets F (ξ i )( Z j ) 2 F (ξ i )( M j ) 2 F (Z tj 1 )( M j ) 2 F (Z tj 1 ) M j. We will show that each replacemet produces a error which will vaish i probability as the mesh of the partitio. After these replacemets we will have F (Z tj 1 ) M j F (Z s ) d M, M s, which will complete the proof. (1) From ( Z j ) 2 ( M j ) 2 = A j ( Z j + M j ) we have F (ξ i ) { ( Z j ) 2 ( M j ) 2} F A t max 1 i Z j + M j. By sample path cotiuity we have lim max Z j + M j = 1 i almost surely. Hece the error produced by the first replacemet vaishes i probability as. (2) We have [ F (ξ i ) F (Z tj 1 ) ( M j ) 2 max F (ξ i ) F (Z tj 1 ) ( M j ) 2. 1 i

18 6 3. STOCHASTIC INTEGRATION AND ITO S FORMULA Agai by sample path cotiuity we have max F (ξ i ) F (Z tj 1 ) 1 i almost surely. O the other had, by EXAMPLE 4.1, lim ( M j ) 2 = M, M t i L 2 (Ω, P). Therefore the error produced by this replacemet vaishes i probability as. (3) This is the most delicate replacemet of the three. The error is E = F (Z tj 1 ) { ( M j ) 2 } M j. The square E 2 is a sum of 2 terms. From EXAMPLE 4.1 ( M j ) 2 M j = 2 j t j 1 (M s M tj 1 ) dm s. This is a stochastic itegral with respect to a martigale, hece the coditioal expectatio of this expressio with respect to F tj 1 is clearly zero. For a off-diagoal term, say i < j, i E 2, its coditioal expectatio with respect to F tj 1 is therefore zero. Hece all off-diagoal term has expectatio value zero. For the diagoal term we have [ ( tj ) 2 [ j E (M s M tj 1 ) dm s F tj 1 = E (M s M tj 1 ) 2 d M, M s. t j 1 t j 1 Therefore the expected value of this off-diagoal term is bouded by [ F 2 E M j max M s M tj 1. t j 1 s t j Addig the diagoal terms together we obtai [ E E 2 F 2 E M t max max M s M tj 1. t j 1 s t j 1 j Note that uder our assumptio, the expressio uder the expectatio is uiformly bouded. This expectatio coverges to zero by sample path cotiuity ad the domiated covergece theorem. With this fial replacemet, we have completed the proof of (5.2) ad also the proof of Itô s formula. We ed this sectio with the statemet of Itô s formula for multidimesioal semimartigales. The basic igrediets of its proof have already explaied i great detail i the proof of the oe dimesioal case. Sice othig will be leared from its tedious proof, we will justifiably omit it.

19 6. DIFFERENTIAL NOTATION AND STRATONOVICH INTEGRALS 61 THEOREM 5.2. Suppose that Z = (Z 1,..., Z ) be a R valued semimartigale. The for ay F C 2 (R ). F(Z t ) = F(Z ) + F xi (Z s ) dz i s F xi x j (Z s ) d Z i, Z j s. i, 6. Differetial otatio ad Stratoovich itegrals This sectio does ot cotai ew results. We itroduce differetial otatio for stochastic calculus ad Stratoovich itegrals, a form of stochastic itegral more restrictive tha Itô itegrals. We ofte work with the followig classes of processes: M = the space of cotiuous local martigales; A = the space of cotiuous processes of bouded variatio; I = the space of icreasig processes; Q= the space of semimartigales H = the space of itegrad processes. Every process of bouded variatio is the differece of two icreasig processes: A = I I. A semimartigale is the sum of a local cotiuous martigale ad a process of bouded variatio: Q = M + A It is ofte coveiet to write equalities amog these processes i a d- ifferetial form. We have defied the meaig of a expressio such as H s dx s for H H ad X Q. What exactly is the meaig a differetial expressio such as dx? We should cosider dx as a symbol for the equivalece class of semimartigales Y such that X t X s = Y t Y s for all s t. The equivalet classes of semimartigale differetials is deoted by dq. We ca carry out some purely symbolic calculatios. To start with we ca set dx u = X t X s. We ca defie a multiplicatio i Q by settig s dx dy = d X, Y. We also defie the multiplicatio of a elemet from Q by a elemet from H : HdX is the equivalece class cotaiig the semimartigale Here are some properties of these operatios: (1) dq dq da ; (2) dq da = ; (3) H(dX dy) = (H dx) dy; (4) H 1 (H 2 dx) = (H 1 H 2 )dx. H s dx t s.

20 62 3. STOCHASTIC INTEGRATION AND ITO S FORMULA With these formal symbolic otatios, the multi-dimesioal Itô s formula ca be writte as d { f (Z t )} = f xi (Z t ) dzt i f xi x j (Z t ) dx i dx j. i, If we are willig to adopt Eistei s summatio covetio, we may eve drop the two summatio sigs. Suppose that X ad Y are two semimartigales. We defie a ew semimartigale differetial by This meas that X dy = X dy + 1 dx dy. 2 X s dy s = X s dy s X, Y t. This is called the Stratoovich itegral. Ulike Itô itegral, the Stratoovich itegral requires that the itegrad process is also a cotiuous semimartigale, for i its defiitio we eed the quadratic covariatio of X ad Y. I this sese Stratoovich itegral is a weaker form of stochastic itegral ad its use is much more limited tha Itô itegrals. The mai advatage of S- tratoovich itegrals is that Itô s formula takes a form similar to that of the fudametal theorem of calculus. PROPOSITION 6.1. Let F C 3 (R ) ad Z = (Z 1,..., Z ) a -dimesioal semimartigale. The d { f (Z t )} = PROOF. The right side is equal to F xi (Z t ) dzt. i (6.1) F xi (Z t ) dz j t F x i x j (Z t ) dz j t. Usig It&o s formula we have df xi (Z t ) = F xi x j (Z t ) dz i t F x i x j x k (Z t ) dz j dz k. Thus the last term i (6.1) becomes 1 2 F x i x j (Z t ) dz i t dz j t F x i x j x k (Z t ) dz i t dz j t dzk t. The last triple sum is equal to zero because dzt i dzj t dzk t =. Therefore the equality we wated to prove is just Itô s formula. Fially we metio that Stratoovich itegral ca also be approximated by a Riema sums. It is istructive to compare this Riema sum with the oe that approximates the correspodig Itô itegral. The differece is subtle but crucial i makig Itô itegratio such a successful theory.

21 7. THIRD ASSIGNMENT 63 THEOREM 6.2. Let X ad Y be cotiuous semimartigales. The X tj 1 + X tj X s dy s = lim [Y tj Y tj 1, where the covergece is i probability. PROOF. Rewrite the summatio as [ X tj 1 Y tj Y tj [X tj X tj 1 [Y tj Y tj 1. 2 The limit of the first term is the Itô itegral 2 term is exactly the quadratic covariatio X, Y t /2. 7. Third Assigmet X s dy s, that of the secod EXERCISE 3.1. Let f be a (oradom) fuctio of boud variatio. Show that f s db s = f t B t B s d f s, where the last itegral is uderstood to be a Lebesgue-Stieljes itegral. EXERCISE 3.2. Let F be a etire fuctio i the complex plae C ad Z = X + iy be the complex Browia motio (meaig that X ad Y are idepedet stadard Browia motio). We have F (Z s ) dz s = F (Z s ) dz s = F(Z t ) F(Z ). EXERCISE 3.3. The price S t of a stock with average rate of retur µ ad volatility σ is usually described by the stochastic differetial equatio ds t = S t (µ dt + σ db t ). Usig Itô s formula to show that the solutio of the above stochastic differetial equatio is [ ) S t = S exp σb t + (µ σ2 t. 2 EXERCISE 3.4. Let B = (B 1, B 2, B 3 ) be a 3-dimesioal Browia motio which does ot start from zero. Usig Itô s formula to show that 1/ B t is a local martigale. EXERCISE 3.5. Let B be a -dimesioal Browia motio startig from zero ad X t = B t = Bt i 2

22 64 3. STOCHASTIC INTEGRATION AND ITO S FORMULA be the radial process. Show that X satisfies the followig Itô type stochastic differetial equatio: X t = W t + 1 ds, 2 X s where W is a oe-dimesioal Browia motio. EXERCISE 3.6. Let M ad N be two cotiuous local martigales. Show that M, N t M t N t. EXERCISE 3.7. Suppose that X, Y ad Z be three semimartigales. Are the followig relatios true? (1) X(Y dz) = (XY) dz; (2) X (Y Z) = (XY) dz. EXERCISE 3.8. A Browia bridge is defied by dx t = db t X t dt 1 t, X =. Show that X is a Gaussia process with mea zero ad E [X s X t = mi {s, t} st. Show that {X t, t 1} ad the reversed process {X 1 t, t 1} have the same law. EXERCISE 3.9. Suppose that M ad N are two bouded cotiuous martigales which are idepedet. Show that MN is a cotiuous martigale with respect to the filtratio F M,N = σ {M s, N s ; s t} geerated by M ad N. EXERCISE 3.1. Suppose that M is a strictly positive local martigale. Show that there is a local martigale N such that [ M t = M exp N t 1 2 N, N t.