Stochastic Analysis I S.Kotani April 2006

Transcription

1 Stochastic Analysis I S.Kotani April 6 To describe time evolution of randomly developing phenomena such as motion of particles in random media, variation of stock prices and so on, we have to treat stochastic processes. The mathematical tools of analyzing stochastic processes have been well developed from various points of view. The purpose of this lecture is to give a basic knowledge on Ito s calculus based on the martingale theory. 1 Conditional expectations 1.1 σ fields and information In the modern probability theory the notion of σ fields is important because it is supposed to express a certain aspect of information. Let Ω, F, P be a probability space, that is, Ω is a set finite or infinite, F is a σ field and P is a probability measure. Recall that a σ field F is a family of subsets of Ω satisfying the three properties: 1 φ, Ω F A F = A c F 3 A n F n = 1,, = A n F. n=1 Suppose we have two sub σ fields G 1, G of F such that G 1 G. Then this can be interpreted as G contains more information than G 1. This is plausible in the following reason. If G 1 and G consist of finite families of subsets of Ω, then there exist partitions {A 1, A,, A m },{B 1, B,, B n } of Ω such that { G 1 = { i I G = } A i ; I is arbitrary subset of {1,,, m} B j ; J is arbitrary subset of {1,,, n} j J }. Then G 1 G if and only if the partition {B 1, B,, B n } is finer than the partition {A 1, A,, A m }, that is, any B j is included in some A i. Example 1 Let Ω be the set of all Japanese and consider two partitions classifications of Ω. One classification is by sex and another one is by sex, weight. Define appropriately G 1, G associated with these two classifications. The notion of random variables also should be understood in this context. A random variable X on Ω, F, P is an F measurable function on Ω, that is, X 1, a] {ω Ω; Xω a} F for any a R. 1

2 Suppose a sub σ-field G of F is generated by a finite partition {B 1, B,, B n } of Ω. Then a random variable X is G measurable if and only if X is a constant on each B j. In another word, a finite partition is equivalent to a random variabale taking only finitely many values. Generally a σ-field σx generated by a random variable X is defined by σx = { X 1 F ; F B R }, where B R is the Borel σ-field of R. Analogously a σ-field σx 1, X,, X m generated by random variables {X 1, X,, X m } is defined by σx 1, X,, X m = { X 1 F ; F B R m } with X = X 1, X,, X m. This σ-field is supposed to contain the information of X. 1. Conditional expectations as random variables In elementary probability theory, for A, B F, the probability of A conditioned on B is defined by P A B P A B =. P B However in modern probability theory it is supposed that information is nothing but a σ-field, therefore it is reasonable to define a conditional probability based on a sub σ-field G of F. For simplicity assume G consists of a finite partition {B 1, B,, B n } of Ω. Then we define P A G as a random variable by P A G ω = P A B j if ω B j P B j n P A B j = I Bj ω, P B j j=1 where I B is the characteristic function of B, that is, { 1 if ω B, I B ω = if ω B c. It should be remarked that P G ω is a probability measure on Ω, F for each fixed ω Ω. Analogously the conditional expectation E X G for a random variable X with finite expectation is defined by E X G ω = E I Bj X if ω B j P B j n E I Bj X = I Bj ω P B j = j=1 n j=1 B j Xω P dω I Bj ω = P B j Ω Xω P dω G ω.

3 Therefore the conditional expectation E X G also is a random variable measurable with respect to G. If X = I A, then E I A G = P A G. In this way, when a sub σ-field G is generated by a finite partition, its conditional expectation is easily defined. However, if it is not so, we have to take another approach. To see this, we pay attention to the following properties of the conditional expectation introduced above. Set Y = E X G. Then 1 Y is measurable with respect to G. Y has a finite expectation and E I B Y = E I B X holds for any B G. Theorem For any random variable X with finite expectation there exists a unique random variable Y satisfying 1,. Proof. To prove the uniqueness let Z be another random variable satisfying 1,. Then E I B Y Z = E I B X E I B X = for any B G. Choose B + = {ω Ω; Y ω Zω }. Then B + G and I B+ Y Z, hence we have E I B+ Y Z = = I B+ Y Z = a.s.. Similarly introducing B = {ω Ω; Y ω Zω < }, we see I B Y Z = a.s. Combining these two equalities, we can conclude Y = Z a.s.. To prove the existence we define a singed measure Then it clearly satisfies µb = E I B X for B G. µb = if P B =, that is, µ is absolutely continuous with respect to P. Applying the Radon- Nikodym theorem, we see that there exists a G measurable random variable Y with finite expectation such that µb = Y ωp dω = E I B Y, B which comletes the proof. This unique random variable Y is denoted by E X G. In addition to the properties 1,, this satisfies 3 For any bounded G measurable random variable G it holds E GX G = GE X G. 4 X 1 X = E X 1 G E X G, in particular E X G E X G. 5 G 1 G = E E X G G 1 = E X G 1. 6 If X is independent to G, then E X G = EX 3

4 Martingalesdiscrete time parameter.1 Definitions The notion of martigales was introduced by Doob in 195 s, which plays an indispensable role in the theory of stochastic analysis nowadays. This is a generalization of sums of indipendent random variables and is effective because it is flexible under many non-linear transformations. A family of sub σ fields {F n ; n =, 1,, } of F is called a filtration if F F 1 F n F. A stochastic process one parameter family of random variables {X n } n = {X n ; n =, 1,, } is called adapted to {F n } n, if X n is measurable with respect to F n for each fixed n. An adapted random variables {X n } n is called a martingale with respect to {F n } n, if it satisfies 1 E X n < for n =, 1,,. E X n+1 F n = X n for n =, 1,,. From this definition martingales can be understood as a mathematical expression of a fair game, which will be ensured in the optional stopping theorem below. Example 3 Let {Y n } n 1 be a family of independent random variables with finite expectations. { Xn = Y i 1 EY 1 + Y EY + + Y n EY n for n 1 X = is a martingale with respect to F n = σx 1, X,, X n and F = {φ, Ω}. ii Moreover assume E Y n < and EY n = for n 1. Then { Xn = Y 1 + Y + + Y n EY 1 + EY + + EY n for n 1 X = is a martingale. iii Suppose E Y n = 1 for all n. Then { Xn = Y 1 Y Y Y n for n 1 is a martingale. X = 1 Example 4 Let X be a random variable with finite expectation and {F n } n be a filtration. Then X n = E X F n becomes a martingale. 4

5 Example 5 martingale transformation=discrete stochastic integral Let {M n } n be a martingale and {f n } n be random variables such that f n is bounded and measurable with respect to F n for n =, 1,,. Then f X n = k M k+1 M k if n 1 k= if n =. becomes a martingale. It is convenient to introduce wider notions. {X n } n is called a submartingalesupermartingale with respect to {F n } n, if in addition to the property 1 it satisfies 3 E X n+1 F n X n for n. E X n+1 F n X n for n. Example 6 Let {X n } n be a martingale and f be a convex function, that is, fαx + βy αfx + βfy if α + β = 1, α, β and x, y R. Suppose E f X n < submartingale. for n =, 1,,. Then {f X n } n becomes a Example 7 Let {Y n } n be a Markov process on S with transition kernel {px, dy} and f be a subharmonic function on S for this Markov process, that is, fx fypx, dy. If E fy n <, then X n = fy n is a submartingale.. Martingale inequality S In order to establish the theorem of law of large numbers in its full generality, Kolmogorov used a maximal inequality for sums of independent random variables with means as an analogy of an inequality in Fourier analysis. Doob pointed out the inequality is valid also for submartingales, and it became a very useful tool in stochastic analysis. Theorem 8 martingale inequaly Let {X n } n be a submartingale. Then for any λ > it holds that P max X k λ λ 1 E X n ; max X k λ λ 1 E X + n. k n k n Proof. Since the last two inequalities are trivial, we show only the first one. Set { } A = max X k λ k n. A m = {X < λ, X 1 < λ,, X m 1 < λ, X m λ} 5

6 Then {A m } m n are disjoint and n A = A m, A m F m. m= The definition of submartingales3 of Section.1 implies for n m EX n ; A m EX m ; A m. Hence n n EX n ; A = EX n ; A m EX m ; A m λp A. m= m= Corollary 9 Suppose {X n } n is a martingale. Then i P max X k λ λ p E X n p for p 1 Kolmogorov-Doob, k n p ii E max X k p p E X n p for p > 1. k n p 1 Proof. Since fx = x p is convex if p 1, { X n p } n becomes a submartingale due to Example8. Applying Theorem11, we see P max X k λ = P max X k p λ p λ p E X n p. k n k n To prove ii, set Y = max k n X k. Observe for a non-negative random variable Y EY p = E pλ p 1 I Y λ dλ = holds. Thus from i it follows that EY p pλ p 1 P Y λ dλ pλ p 1 λ 1 E X n ; Y λ dλ = p p 1 E X n Y p 1 p p 1 {E X n p } p 1 {EY p } 1 p 1, hence EY p p p E X n p. p 1 6

7 .3 Optional stopping theorem For further investigation of stochastic processes, the notion of stopping times is crucial. A stopping time is not simply a non-negative random variable, but it has to be non-anticipating, which will be specified soon. Let {F n } n be a filtration. A non-negative integer valued random variable τ taking possibly is called a stopping time with respect to {F n } n if it satisfies {τ n} F n for n {τ = n} F n for n. Typical examples of stopping times are given by hitting times for adapted stochastic processes. Example 1 Let {X n } n be a stochastic process adapted to {F n } n and F BR. Define { inf {n ; Xn F } if X τ = n F for some n < otherwise. Then τ is called the first hitting time to F and becomes a stopping time with respect to {F n } n. On the other hand, the last hitting time { sup {n ; Xn F } if X τ = n F for some n < otherwise, can not be a stopping time. We remark the following Lemma 11 Let σ, τ be stopping times. Then σ τ, σ τ and σ + τ are also stopping times. Proof. We prove the statement only for σ + τ. Observe {σ + τ = n} = which concludes the proof. Now the theorem is n {σ = k, τ = n k} F n, k= Theorem 1 Optional stopping theorem Let {X n } n be a submartingale. Then for any stopping times τ, σ such that τ σ, {X n τ X n σ } n becomes a submartingale. In particular, if τ is bounded, then E X τ E X σ. Proof. Set f k = I τ>k σ = I τ>k I σ>k. Then X n τ X n σ = f k X k+1 X k if n 1, k= 7

8 and, since f k, we see as in Example5 that {X n τ X n σ } n is a submartingale: n E f k X k+1 X k F n = f k X k+1 X k + f n E X n+1 X n F n k= which completes the proof. k= f k X k+1 X k, k= The theorem says that if {X n } is a martingale and τ is a bounded stopping time, then E X τ = E X. In this sense, martingales are fair games. 3 Martingale convergence theorems One advantage of using martingales in analysis of stochastic processes is that they make the proofs of convergences easy. The following argument initiated by Doob is a key to establish the convergence theorems for submartingales. For a sequence of real numbers {x, x 1,, x N } and a < b, set τ = and { τn+1 = min {k τ n ; x k a} for even n τ n+1 = N if {k τ n ; x k a} = φ { τn+1 = min {k τ n ; x k b} for odd n. τ n+1 = N if {k τ n ; x k b} = φ Notice here 1 = τ τ 1 τ τ N = N and, if τ n = N for some n < N, then τ n+1 = τ n+ = = τ N = N. Define the upcrossing number of {x, x 1,, x N } between a, b by { { max k; xτk 1 U = U x, x 1,, x N ; a, b = { a, x τ b} k if k; a, x xτk 1 τ b} k = φ. We illustrate those definitions below. Lemma 13 Doob Let {X, X 1, X,, X N } be a submartingale and U be its upcrossing number between a, b. Then b a E U E X N a X a E X N X +. 8

9 Proof. For simplicity assume N is even. Observe X N X = X τ1 X τ + X τ X τ1 + + X τn X τn 1 b a U + X τ1 X τ + X τ3 X τ + + X τn+1 X τn. Since {τ n } are stopping times, applying Theorem1, we see Thus we have E X τn+1 X τn. E X N X b a E U. Set Y n = X n a. Then it is easy to see that {Y n } is also a submartingale and UX, X 1,, X N ; a, b = UY, Y 1,, Y N ; a, b, we complete the proof. Now the main result is Theorem 14 Suppose {X n } n is a submartingale satisfying E X n + C for all n x + = x with some C >. Then there exists a finite random variable X such that hold. lim X n = X a.s., and E X < n Proof. For a < b, U N ω = U X ω, X 1 ω,, X N ω ; a, b as N. Define Uω = lim N U N ω. Since E X N X + E X + N + E X C + E X, from Lemma19 it follows that However setting A a,b = EU C + E X, hence Uω < a.s.. 1 { ω Ω; lim inf n X n ω < a < b < lim sup n } X n ω, we see U ω = for ω A a,b. Hence we have P A a,b = from 1. Set A = Then P A = and for ω A c, we have A a,b. a<b a,b Q lim inf X n ω = lim sup X n ω X ω [, + ]. n n 9

10 However Fatou s lemma implies Here we have used E X lim inf n E X n C + E X <. E X n = E X + n which completes the proof. As for supermartingales we have E Xn E X n + E X, Theorem 15 Let {F n } <n be a family of σ fields such that F n F n+1 F 1 F, and {X n } <n be a family of random variables adapted to {F n }. Then there exists a random variable X, + ] such that Moreover if then lim n X n = X lim EX n <, n a.s.. E X <, hence X <. Proof. The number of downcrossings D of sequences {X M, X M+1,, X 1, X } between a, b can be defined analogously as the number of upcrossings and we can prove the following estimate b a E D E X M b X b b + E X, because { X n } is a submartinagle and its upcrossing number between b, a is eaqual to the downcrossing number for {X M, X M+1,, X 1, X }. Now the rest of the proof is the same as that of Theorem14 except that X may take +. However if lim EX n < is valid, then n E X n = EX n + EXn EX n + EX = E X lim EX n + EX n <. Corollary 16 Let {F n } <n<+ be a family of σ fields such that F n F n+1, and X be a random variable with finite expectation. Then { E X F+ as n + E X F n E X F as n almost surely and in L 1 Ω, P, where F + = σ F n, F = σ F n. n n 1

11 4 Martingalescontinuous time parameter In this section we discuss martingales with continuous time parameter. {F t } t be a filtration with continuous parameter, that is, F s F t if s t. Let {X t } t be a stochastic processes a family of random variables adapted to {F t }, that is, for each t X t is F t measurable. We can define the notions of martingales, submartingales and supermartingales in this case just like the discerete case. In the following arguments we need to impose the right continuity of {X t } t, that is, P lim X s = X t for every t = 1 s t Throughout this lecture we assume that all stochastic processes are at least right continuous. For a filtration {F t } set Then we have F t+ = s>t F s = n 1 F t+ 1 n F t. Lemma 17 Let {X t } t be a right continuous submartingale with respect to {F t }. Then {X t } t is a submartingale with respect to {F t+ }. Proof. Let t > s and choose n 1 such that t > s + 1 n. Then Fs+ E X t 1 X n s+ 1. n The right hand side converges to X s as n. On the other hand, the left hand side converges to E X t F s+ due to Corollary16, which completes the proof. Let Keeping this Lemma in our minds, from now on we assume filtrations are always right continuous, that is, F t+ = F t for every t. The three main results for martingales with discrete parameter are also valid for martingales with continuous parameter. Theorem 18 Martingale inequalities i Let {X t } t be a right continuous submartingale. Then for any λ > P sup X s λ λ 1 E X t ; sup X s λ λ 1 E X + t λ 1 E X t. s t s t Suppose {X t } t is a right continuous martingale. Then ii P sup X s λ λ p E X t p for p 1, λ >, s t p iii E sup X s p p E X t p for p > 1. s t p 1 11

12 Proof. Let Q be the set of all rational numbers and F n be finite subsets of Q [, T ] such that F n F n+1, max F n = t for every n 1 and F n = Q [, T ]. Since {X t } t Fn is a submartingale with discrete parameter, we can apply Theorem8 and Corollary9. The right continuity of {X t } implies n=1 max X s sup X s as n, s F n s [,T ] which shows i. ii and iii can be proved similarly. To prove the optional stopping theorem we give a remark on stopping times. The notion of stopping times can be defined similarly, that is, a non-negative random variable τ is called an {F t } stopping time if {τ t} F t for every t. The right continuity of {F t } implies this definition is equivalent to because We give examples. {τ < t} F t for every t >, {τ < t} = { τ t 1 n} Ft n=1 {τ t} = { }. τ < t + 1 n Ft+ n=1 Example 19 Let {X t } t be a right continuous stochastic process adapted to {F t }. For A R set { inf {t ; Xt A} if { } = φ τ A = if { } = φ. If A is an open set or a closed set, then τ A becomes an {F t } stopping time. Proof. Assume first A is open. Then {τ A < t} = Suppose A is closed. Then where A n = {τ A t} = r Q, r<t {X r A} F t. {τ An t} F t, n=1 { x R ; dist x, A > 1 } n open set in R. For a general Borel set A, the measurability of τ A is a highly non-trivial problem, and we need extra properties for {X t } t and the completion of probability spaces. 1

13 Lemma i Let σ, τ be stopping times. Then so are σ τ, σ τ, σ + τ. ii Let {σ n } n 1 be a sequence of stopping times which are decreasing or increasing. Then so is lim σ n. n Proof. We left the proof for the readers. Theorem 1 Optional stopping theorem Let {X t } t be a right continuous submartingale and σ, τ be stopping times such that τ σ. Assume E sup t T X t < for each T >. 3 Then {X t τ X t σ } becomes a submartingale. In particular, if τ is bounded, we have E X τ E X σ. Proof. For fixed n 1, set τ = k + 1 n if σ = k + 1 n if and Then n τ, n σ are k n τ < k + 1 k n n σ < k + 1, n F k = F. k { } n Fk stopping times, because { n τ k} = {τ < k } n F k n = F k. { } Since Xk = X is a submartingale with respect to k Fk, we see from n Theorem1 for k > l E Xk n τ X k n σ F l X l n τ X l n σ. Now assume k 1 t < k n inequality implies E n, l 1 X k n τ X k n σ I A E s < l n and let A F n s F l. Then the above X l n τ X l n σ I A, On the other hand, since τ τ, σ σ and k l t, n s as n, the right n continuity of {X t } shows X k n τ X t τ, X l n τ X s τ, X k n σ X t σ, X l n σ X s σ as n. X However, since X k, n τ k n σ sup u t+1 X u, which is integrable due to the assumption 3, the dominated convergence theorem proves E X t τ X t σ I A E X s τ X s σ I A for A F s. 13

14 { } Remark Since a more detailed argument shows that X k n τ, X k n σ are uniformly integrable, we see that the condition 3 is unnecessary in general. If {X t } t is a right contin- Theorem 3 Martingale convergence theorem uous submartingale satisfying E X + t C for any t, then there exists a finite random variable X such that Proof. For fixed n 1, set X t X as t a.s. and E X <. T n = { } k ; k =, 1,,, T = n T n. Then T is countable and dense in [,. For N 1 and a < b, let UN n a, b be the upcrossing number of {X t } t Tn [,N] between a, b. Then from Lemma13, we have n=1 b a E U n Na, b E X + N + E X C + E X. Since as n, N, UN n a, b is increasing, let Ua, b be its limit. Then we see which implies E Ua, b C + E X b a Ua, b < a.s.. <, Varying a < b among all rational numbers, we see that with probability 1 lim X t [, + ] exists a.s.. t t T The right continuity of {X t } implies On the other hand, we have lim t t T X t = lim t X t. E X t = E X + t EXt EX + E X + t EX + C, hence Fatou s lemma shows E X EX + C <. 14

15 5 Brownian motion A typical and the most important martingale with continuous parameter is Brownian motion. In this section we introduce this process and study its basic properties. Brownian motion was found by an English botanist Brown in 188 as a random motion of some components of pollen. In 195 Einstein used this process to study the existence of molecules without knowing the previous discovery by Brown. Brownian motion was first defined mathematically as a continuous stochastic process by Wiener in 193, and this process is called sometimes Wiener process. Lévy studied properties of paths of Brownian motions from very original points of view. Kolmogorov tried to develop a dynamical theory of Markov processes and pointed out that Brownian motion could play a basic role for its purpose. Ito initiated a random dynamical theory based on Brownian motion in 194 and started a rigorous treatment of stochastic differential equations. A stochastic process {B t } t on a probability space Ω, F, P is called a Brownian motion if it satisfies 1 Each sample {B t } t is continuous as a function of t. For any sequence { t t 1 t n }, the random variables { Bt1 B t, B t B t1,, B tn B t} are independent. 3 The distribution of B t B s is N, t s, that is, a Gaussian distribution with expectation and variance t s. These three properties determine uniquely the distribution of a Brownian motion as a stochastic process. These three properties are not independent. Actually owing to the central limit theorem the properties 1 and imply that the distribution of B t B s becomes a Gaussian. The existence of a stochastic process satisfying the three properties is not trivial, and we had to wait Wiener. He constructed a Brownian motion as a Fourier series with random coefficients. Now a more intuitive introduction of the process is possible as a limit of a simple random walk. We give here its short explanation. Let {Y n } n 1 be independent random varibales taking values {±1} with equal probability 1/. Set { Y1 + X X n = + + X n if n 1, if n =. Define a continuous stochastic process {X t } t by X t = t n X n+1 + n + 1 t X n if n t n + 1. Since EX t = and E X t = t n + n if n t n + 1, we see Xmt E = mt n + n if n mt n + 1. m m Denoting t = n + ε, we have m Xmt E = mε + n m m. 15

16 However, ε 1, therefore as m it holds that m Xmt E t. m Then the central limit theorem tells us that the distribution of {X mt / m} converges to N, t as m. If we take a closer look at vector valued random variables { Xmt1 X mt m, X mt X mt1,, X } mt n X mt m, m we can show that their distributions converge to independent Gaussian distibutions N, t 1 t, N, t t 1,, N, t n t. To construct a Brownian motion as a limit of {X mt / m} this argument is not sufficient and we have to discuss the convergence of the distributions µ m of {X mt / m} induced on the space C [, = {w; w is a continuous function on [, }. Donsker established this convergence and the limit µ is a probability measure on C [, governing the probability law of the Brownian motion. Although its sample path is continuous, it is known that any sample path is differentiable at no points of [,, which makes it difficult to construct a stochastic analysis based on Brownian motions. To readers who get interested in a more detailed explanation of the construction of Brownian motions, we recomend them to look suitable text books. A Brownian motion in d-dimensional space R d d dim.b.m. in short can be defined similarly. Let { B t = } Bt 1, Bt,, Bt d be an t Rd valued stochastic process satisfying 1 Each sample {B t } t is continuous as a function of t. For any sequence { t t 1 t n }, the random variables { Bt1 B t, B t B t1,, B tn B t} are independent. 3 The distribution of B t B s is N, t s I, that is, a Gaussian distribution with expectation and variance matrix t s I. From this definition we can easily see that each { } Bt i { {B } 1, { } t B t t,, { } } B d t t are independent. t Set F t = σ {B s ; s t}. We remark Proposition 4 For a d dim.b.m.{b t } t, { B i t are martingales with respect to {F t }. t is a 1-dim.B.M. and { } }t and BtB i j t δ ij t t 16

17 Proof. For t s, E B i t F s = E B i t B i s F s + E B i s F s = E B i t B i s + B i s = B i s. We have used here the fact that B i t B i s is independent of F s. On the other hand, EBtB i j t F s B i = E t Bs i B j t Bs j F s + E BsB i j t Bs j F s + E BtB i s j F s = E Bt i Bs i B j t Bs j + BsEB i j t Bs j + BsE j Bt i F s = δ ij t s + BsB i s. j A Brownian motion has a lot of symmetries. reveals a part of these symmetries. The following proposition Proposition 5 Let {B t } t be a d-dim.b.m. starting from. Then i For a fixed c >, {B ct / c} t is a d-dim.b.m. starting from. ii For a fixed t, {B t+t B t } t is a d-dim.b.m. starting from. iii {tb t 1} t is a d-dim.b.m. starting from. Proof. Only iii is not trivial to prove. To show this, we have only to look the property for ˆB t = tb t 1. To see this, we consider a family of random vectors {ˆB ˆB } t, t1,, ˆB tn. It is easy to see that their joint distribution is a Gaussian in R n+1d with expectation. If we could show that the distributions of ˆB ˆB t, t1,, ˆB tn, B t,b t1,, B tn are equal, then we have the property } for {ˆB t. However Gaussian distributions in multi-dimensional space are t determined by their expectations and covariances, we have only to compute for t s and 1-dim.B.M. E ˆBt ˆBs = tse B t 1B s 1 = ts t 1 s 1 = s = E B t B s. 6 Stochastic integral To define an integral based on a function Y on the real line, usually the function has to be of bounded variation. However, as we see in Brownian motions, martingales does not have paths with bounded variation. Therefore the ordinary Lebesgue-Stieltjes integral can not be applied to an integration with respect to martingales. Fortunately martingales have an extended version of variation, which will be seen in this section, and we make use of this property for the definition of the stochastic integral. 17

18 6.1 Variational processes Let { = t < t 1 < t < < t n = T } be a partition of [, T ] and denote it by Π. For p 1 and a function {Y t } on [, T ] set V p Π, Y = Yti+1 Y ti p, V1 Y = sup V 1 Π, Y. Π i= A function {Y t } t is called of bounded variation on [, T ] if V 1 Y <. Now, denoting Π = max {t i+1 t i ; i n 1}, we have for p > V p+1 Π, Y V 1 Π, Y sup s t T t s Π Y t Y s p V 1 Y sup Y t Y s p. s t T t s Π Therefore, if {Y t } t is continuous and of bounded variation, then V p+1 Π, Y as Π. What we would like to show in this section is the existence of non-decreasing, continuous function A t for square integrable martingale M such that V Π n, M A T if Π n. Then a decomposition: M t = N t + A t 4 is valid, where {N t } t is a continuous martingale. The {A t } t is called the variational process associated with {M t } t. This process is crucial to define stochastic integrals based on martingales. Since { } Mt is a submartingale,usually this process is introduced through the Doob-Meyer decomposition of submartingales. However in this lecture we employ another approach to take a shortcut. Fix a right continuous filtration {F t } t throughout this section. A stochastic process {f t } t adapted to the given filtration is called a stepfunction if there exists an increasing sequence {t n } n and random variables {φ n } n such that { = t < t 1 < t < < t n <, t n as n, f t = φ i if t t i, t i+1 ], where φ i is measurable w.r.t. F ti In this section we assume the boundedness for stepfunctions, that is, there exists a constant C such that f t ω C for all t and ω Ω. For a later purpose we denote the set of all bounded stepfunctions by L = L F. Let {M t } t be a continuous martingale which is square integrable, that is, EMt < for every t and satisfies M =. We define a stochastic integral I t f of a stepfunction based on this martingale by I f = and k 1 I t f = φ i i + φ k t if t k < t t k+1, 5 i= where i = M ti+1 M ti, t = M t M tk. Since there are many ways to express a stepfunction, it should be proved that the above integral does not depend on the expressions. We leave its proof to the readers. 18

19 Lemma 6 {I t f} t is a continuous martingale satisfying for t > s E I t f F s = Is f + E φ i i + φ k t F s, 6 k 1 i=l where we assume t l = s without loss of generality. Proof. The continuity is clear from the definition. A computation E I t f F s = E I s f F s + E k 1 φ i i + φ k t F s = I s f, shows the martingale property of {I t f}. Now we calculate the conditional expectation of the square: E k 1 I t f F s = E I s f + φ i i + φ k t i=l F s k 1 k 1 = I s f + E φ i i + φ k t F s + I s fe φ i i + φ k t F s i=l i=l + E φ i φ i i j + φ i φ k i t F s l i<j k 1 because for i < j we see and which proves 6. = I s f + E l i k 1 k 1 i=l φ i i + φ k t, i=l E φ i φ i i j F s = E E φ i φ i i j Ftj Fs = E φ i φ i i E j Ftj Fs =, E φ i φ k i t F s = E E φ i φ k i M t M tk F tk F s = E φ i φ k i E M t M tk F tk F s =, Since we have to discuss convergence of sequences of continuous martingales, let M T = M T F be the set of all continuous square integrable martingales on [, T ] and introduce a norm of M = {M t } t T M T M = M,T = E M T. Lemma 7 With this norm the space M T becomes a Hilbert space. 19

20 Proof. For M = {M t } t T M T, we see M t = E M T F t if t T. Therefore if we have a Cauchy sequence {M n } n 1 M T, whose {M n T } converges to an f L Ω, P in L norm, then a martingale defined by M t = E f F t becomes a limit of {M n } n 1 in M T if we could show the continuity of {M t}. To see this we apply ii of Theorem18 to a martingale M n M m, that is, P sup Mt n Mt m λ λ E MT n MT m 7 t T for any λ >. Since {MT n} n 1 is a Cauchy sequence, for any k 1 we can choose a subsequence {n k } k 1 such that M n E k+1 T M n k T 1 8 k. Set A k = { ω Ω; sup n M k+1 t ω M n k t ω } 1 t T k, A = n 1 A k. k n Then 7 implies P A k k for every k 1, thus P A P A k P A k 1 as n, k k n k n k n hence we see P A =. If ω / A, then there exists a K 1 such that sup n M k+1 t ω M n k t ω 1 t T k for any k K. For k > l, we have for every t [, T ] M n k t ω M n l t ω l i k 1 M n i+1 t ω M ni t ω l i k 1 1 i as k, l, hence {M n k t ω} t T becomes a Cauchy sequence with sup-norm and converges uniformly on [, T ] to a continuous function {M tω} t T. Since Mt n = E MT n F t, we see MT n MT m F t M n t M m t E = E Mt n Mt m E MT n MT m as n, m, thus for every t [, T ], {M n t } becomes also a Cauchy sequence converging to M t, which concludes M t = M t. The Lemma below indicates that continuous martingales can not be functions of bounded variation unless they are constants.

21 Lemma 8 Suppose {X t } t is an adapted continuous process and has a decomposition X t = M t + A t with a continuous martingale {M t } and a bounded variation process {A t } with A =. Under the condition that E V 1 A <, this decomposition is unipue. Proof. Suppose we have another such decomposition {M t, A t} of {X t } t and set Y t M t M t = A t A t, {Y t } t is a continuous martingale with bounded variation. Here, first we assume {Y t } t is bounded by C. Then observing V Π, Y CV 1 Y, V 1 Y V 1 A + V 1 A L 1, we have by the dominated convergence theorem E V Π, Y as Π. 8 On the other hand, from the martingale property for t > s it follows that E Y t Y s = E Yt Y t Y s + Ys = E Y t Ys, hence we see E V Π, Y = E Y ti+1 Y ti = E i= i= Y t i+1 Y t i = EY T. This together with 8 shows EYT =, which is nothing but Y t = for every t from the martingale property of {Y t }. If {Y t } t is not bounded, we have only to stop {Y t } by τ c = inf {t ; Y t > c}, then {Y t τc } becomes a martingale, because it satisfies 3: E sup t T Y t E V 1 Y E V 1 A + V 1 A <. Applying the above argument to {Y t τc }, we see Y t τc =. The rest of the proof is clear. Now, for {M t } t M T and a fixed partition Π = { = t < t 1 < t < < t n = T } of [, T ] set { f Π t = M ti if t i < t t i+1, i =, 1,,, n 1 and f Π =, N Π t = I t f Π and k 1 A Π t = M Mti+1 t i if t k t < t k+1, k =, 1,,, n 1 i=. A Π T = M Mti+1 t i i= 1

22 Then it holds that M t = N Π t + A Π t + M t M tk if t k t < t k+1. 9 Lemma 9 Let {M t } t M T and suppose {M t} t is bounded, that is, there exists a C > such that M t ω C holds for every t and ω Ω. Then there exists a continuous and non-decreasing process {A t } t T such that holds if Π n. E sup t T A Πn t A t 1 Proof. From 6 it follows that E Nt Π 4C E A Π t + M t M tk = 4C E Mt 4C 4, E Nt Π Nt Π E sup s t f s Π fs A Π Π Π t + M t M t k, 11 1 where Π Π is the partition of [, T ] generated by Π, Π. We easily see from 9,11 Thus 1 implies E E A Π t + M t M tk E M 4 t + E N Π t 1C 4. E Nt Π Nt Π sup fs Π fs Π 4 E At Π Π + M t M t k s t 1C E sup fs Π fs Π s t Since { } fs Πn converges to {Ms } uniformly on [, T ] and sup s t f Π n 4 s fs Πm 4 C 4, the bounded convergence theorem shows the right hand side of 13 converges to as n, m. Then } Lemma7 concludes that there exists a {N t } t M T {N such that Πn converges to {N t } t on [, T ]. Therefore { A Πn t } t also converges to a continuous stochastic process {A t} t, because t the reminder term M t M tk clearly converges to. Now we can show the decomposition 4.

23 Theorem 3 For {M t } t M T there exist a unique martingale {N t} t and a non-decreasing stochastic process {A t } t with A = such that 4 holds. Proof. The uniqueness has been proved in Lemma8. The existence of the decomposition is already proved in case M is bounded. If {M t } t is not bounded, we truncate it by τ c = inf {t ; M t > c}, From iii of Theorem18, we see E sup M s E s T sup Ms s T 4E M T, 14 which assures 3 of Theorem1. Then {M t τc } becomes a bounded continuous martingale and by the previous argument we have a decomposition If c > c, then τ c > τ c, hence M t τ c = N c t + A c t. N c t τ c + A c t τ c = N c t + A c t. Then the uniqueness of the decomposition implies N c t τ c = N c t, A c t τ c = A c t, therefore we define N t = N c t, A t = A c t if t τ c. The property 14 shows E M t τ c 4E M t, and hence E A t τc + E M = E M t τc 4E M t. Letting c, we see A t τc A t from τ c, hence E A t 4E Mt <, and N t L 1. We have to check the martingale property of {N t }. To verify this we have only to see the L 1 convergence of Nt c N t for every fixed t. However this is clear from E Nt c N t E Mt τ c Mt + E A t τc A t = E M t τc Mt + E A t EA t τc. M t τc Mt converges to a.s. as c and is dominated by an integrable sup s T Ms. Therefore the first term converges to. The second term tends to, because {A t } is non-decreasing. For a given {M t } M T the non-decreasing stochastic process {A t} t in Theorem3 is called the variational process of {M t } t and is denoted by M t. It is convenient to define a variational process for a product of two martingales. Making use of an identity M t N t = M t + N t M t N t, 4 for two such processes {M t, N t } t we introduce M, N t = M + N t M N t. 4 { M, N t } can be understood as the unique continuous process of bounded variation which makes M t N t M, N t a martingale. This remark implies the following 3

24 Lemma 31 i, t satisfies the property of inner products, that is,, t is bilinear and M, M t. ii For any stopping time τ, Let M τ t = M t τ, N τ t = N t τ. Then M τ, N τ t = M, N t τ. From Proposition4 we easily see Example 3 Let { B t = } Bt 1, Bt,, Bt d t be a d dim. B.M. Then B i, B j t = δ ij t. 6. Stochastic integral If a process {Y t } is of bounded variation that is, V 1 Y <, an integral of suitable functions based on the process is possible, which is called as the Lebesgue-Stiltjes integral. In this section we define a stochastic integral based on continuous martingales, which was initiated by Ito. As we have seen in the last section, martingales are not of bounded variation but of bounded variation of the second order in a weak sence. In this case we have to discuss the integral keeping in mind the quadratic structure of martingales developed in the last section. For stepfunctions we have already defined a stochastic integral?? based on martingales. We try to extend the integral to wider class of random functions. Throughout this section, we fix a large T > and define the integral on [, T ]. For M M T and a function f = ft, ω on [, T ] Ω which is measurable with respect to B[, T ] F, we define T f = E ft, ω d M t, and denote the set of all such functions f satisfying f < by L Ω, P, M. Clearly this space is a Hilbert space and the space of all stepfunctions L restricted on [, T ] is contained in L Ω, P, M. Define L M = the closure of L in L Ω, P, M. 15 The following Lemma tells us a typical function belonging to L M. Lemma 33 Suppose a function f = ft, ω is left-continuous for every fixed ω Ω and f t, is F t measurable for each t. If f L Ω, P, M, then f L M. Proof. First suppose f is bounded. For each n 1, introduce { f, ω, if t = f n t, ω = f k, ω, if t ] k n, k+1. n, k =, 1,, n Then f n L and the left-continuity implies f n t, ω ft, ω for any fixed t, ω [, T ] Ω. Therefore the bounded convergence theorem shows f n f, hence f L M. If f is not bounded, we prepare ϕ N x = N, if x N x, if x < N N, if x N, 4

25 and set f N = ϕ N f. Then f N L M and it is easy to see that f N f as N, hence f L M. Now we look back 6. Since Mt = N t + M t, we have E ft i i Fs = f ti E i Fs = f ti M ti+1 M ti. Therefore we see E I t f k 1 F s Is f = E ft i i + ft k t F s i=l = E f u ω d M u F s s 16 In particular, setting s = and taking expectation, we have E I t f = E f u ω d M u. 17 Notice I f M T and M T is a Hilbert space Lemma7. For f L M, choose a sequence of {f n } L such that f f n as n. Then 17 shows E I T f n I T f m = E T f n,u ω f m,u ω d M u = f n f m, hence {I f n } n 1 converges in M T, and its limit is denoted by I f = I M f if necessary. This I M f is called a stochastic integral based on a martingale M. Now suppose M, N M T and f L M, g L N. Then I M f and I N g are in M T. The next problem is to compute their, t. For this purpose, first we assume f, g L. Then a similar calculation as 6,15 shows E It M fit N k 1 g F s I M s fis N g = E f ti g ti i i + f tk g tk t t F s i=l = E f u ωg u ωd M, N u F s with i = N t i+1 N ti, t = N t N tk. Therefore we have s I M f, I N g = f t u ωg u ωd M, N u 18 Since the convergence of the right hand side of 18 is not trivial for general f L M, g L N, we have to examine it. From 18 it follows that for f, g L T f u ωg u ωd M, N u I M f T I N g T T T = f u ω d M u g u ω d N u. 19 5

26 Let hs be a non-random stepfunction. Then for any bounded variation function vt, we see T T sup hudvu = d v u, h 1 where h = sup { hu ; u [, T ]} and v u is the total variation on [, u] of v. Let h 1, h be two non-random stepfunctions. Then in 19 replacing f, g by h 1 f, h g respectively, we have T T T f u ωg u ω d M, N u f u ω d M u g u ω d N u. Now it is easy to see that holds for any f L M, g L N. Summing up these argument, we obtain Theorem 34 Suppose M, N M T and 18 are valid. and f L M, g L N. Then We remark the following Lemma 35 Let M M T, f L M and.τ be a stopping time. Then I M t τ f = I M τ t f τ, where M τ t = M t τ, f τ t = I [,τ] tf t. Proof. For f L, the statement is clearly valid and then pass to the limit. In the calculation notice ii of Lemma Localization of stochastic integral It is not convenient that the stochastic integral is defined only for square integrable martingales. In this section we try to extend the notion of martingales with the help of the optional stopping theorem. Set { } M = M {Mt } loc,t = t T ; there exists stopping times {τ n } n 1 such that τ n τ n+1 and M τn = {M t τn } t T M, T and L loc M { f = {ft } = t T ; there exists stopping times {τ n } n 1 such that τ n τ n+1 and f τn = { I [,τn]tf t } t T L M }. An element of M loc,t is called a local martingale. For M M loc,t, f L loc M, IMτn t f τn is defined as stochastic integrals introduced in Section5.. If n > m, then from Lemma35 we have I M τm t Therefore we can define f τm = I M τn τm t f τn τm = I M τn t τ m f τn. I M t f = I M τn t f τn if t τ n. 6

27 It is clear that I M f M loc,t. For M, N M loc,t, we define M, N t = M τn, N τn t if t τ n. This definition can be seen well-defined through ii of Lemma31. It is easy to see that I M f, I N g t = f s g s d M, N s, 1 for M, N M loc,t and f, g L loc M. 6.4 Ito s formula If A t is a continuous function with bounded variation of the first order and f is a differentiable function whose derivative f is continuous, then the following formula is valid. fa t fa = f A s da s. This can be seen as follows. Let Π = { = t < t 1 < t < < t n = t}. Set fy fx = f xy x + ε 1 x, yy x. Then for a large enough C such that A s [ C, C] for any s [, t] Thus ρ 1 ε sup { ε 1 x, y ; y x ε, x, y [ C, C] } as ε. fa t fa = fa fatk+1 t k Then as Π, k= = f A tk A tk+1 A tk + ε 1 Atk, A tk+1 Atk+1 A tk k= k= f A tk t A tk+1 A tk f A s da s, k= and ε 1 Atk, A tk+1 Atk+1 A tk ε1 Atk, A tk+1 Atk+1 A tk k= k= ρ 1 Π V 1 A. If the process A t is replaced by a martingale M t, the formula corresponding to is no longer valid, and we have to taking accout of the next term of the Taylor expansion of f. 7

28 Theorem 36 Let M i = { M i t } M loc,t and { A i t} be adapted continuous processes with bounded variation for 1 i d. Then for f = fx 1, x,, x d which has continuous derivatives up to the second order with respect to any variables, we have fx t fx = d i=! f xi X s dx i s + 1 d i,j=! f xi,x j X s d M i, M j s, 3 where X t = Xt 1, Xt,, Xt d and X i t = Mt i + A i t. Proof. For simplicity we assume d = 1. We assume first M, A are bounded by C. From the assumption on f it follows that with fy fx = f xy x + 1 f xy x + ε x, y y x, ρ ε sup { ε x, y ; y x ε, x, y C } as ε. Then for Π = { = t < t 1 < t < < t n = t} we see fx t fx = fx fxtk+1 t k = f X tk X tk+1 X tk k= + 1 f X tk X tk+1 X tk + ε Xtk, X Xtk+1 tk+1 X tk k= I Π 1 + I Π + I Π 3. k= k= Since f X t is continuous and bounded, it is clear that I1 Π = f X tk M tk+1 M tk + f X tk A tk+1 A tk k= f X s dm s + k= f X s da s = f X s dx s, as Π. Here the first term converges in L Ω, P from Theorem34. To treat I3 Π, set ε Π, X = sup s,t T, s t Π X t X s. Then However and I 3 Π ρ ε Π, X X tk+1 X tk. 4 k= Xtk+1 X tk Atk+1 A tk + Mtk+1 M tk k= k= k= Atk+1 A tk ε Π, A V1 A as Π. k= 8

29 The second term is nothing but M tk+1 M tk = A Π t k= see9. Then from Lemma9 it follows that A Πn t M t in L Ω, P as Π n. Hence, by choosing a suitable subsequence {n k }, we see A Πn k t M t uniformly on [, T ] a.s.. Then 4 shows I Πn k 3. As for I Π, only the term J Π f X tk M tk+1 M tk k= has a non-trivial limit. However we have As we have seen that A Πn k t J Π = f X s da Π s. M t uniformly on [, T ] a.s., hence J Πn k f X s d M s. Consequently we have proved 3 if d = 1 and M is bounded. For a general M M τn τc loc,t, we have only to truncate it, namely Mt = M t τn τ c. Set τ = τ n τ c. Then 3 is valid for M τ, hence from ii of Lemma31 and the definitions in Section5.3 it follows that fx t τ fx = f X s τ dx s τ + 1 f X s τ d M τ s = τ f X s dx s + 1 τ f X s d M s. Since τ = τ n τ c as n, c we complete the proof if d = 1. A process X t = M t + A t with a continuous local martingale M and a continuous adapted process with bounded variation is called a semi-martingale. Ito s formula is a chain rule in stochastic analysis and it can be understood more intuitively if we write 3 in a differential form: dfx t = d f xi X t dxt i + 1 i=! d f xi,x j X t d M i, M j. t i,j=! The variational process is expressed sometimes as d M, N t = dm t dn t, and if A, A are continuous processes with bounded variation, we set da t dm t = dm t da t = da t da t =. Then the Ito s formula can be written as dfx t = d f xi X t dxt i + 1 i=! d f xi,x j X t dx i dx j t. i,j=! 9

30 7 Applications Ito s formula has many applications, however here we give two of them. For other applications see the exercises. 7.1 Martingale characterization of Brownian motions Lévy pointed out that Brownian motions can be characterized by martingales and it led Stroock-Varadhan to characterization of various stochastic processes through martingales, which is now called the martingale problem. Theorem 37 Let M i M loc,t with M i, M j t = δ ijt and M i = for i = 1,,, d. Then M t = Mt 1, Mt,, Mt d is a d dimensional Brownian motion. Proof. For ξ R d apply Ito s formula to e iξ Mt : e iξ Mt = 1 + i = 1 + i d ξ k e iξ Ms dms k 1 k=1 d ξ k ξ l e iξ Ms d M k, M l s k,l=1 d ξ k e iξ Ms dms k 1 ξ e iξ Ms ds. k=1 Thus we have for t s E e iξ Mt F s = e iξ M s 1 ξ E e iξ Mu F s du. Regarding this relationship as an integral equation with unknown E e iξ Mt F s, we obtain E e iξ Mt F s = e iξ M s e 1 t s ξ, which is equivalent to E e iξ Mt Ms F s = e 1 t s ξ. This shows that M t M s is independent of F s and M t M s is distributed as a Gaussian distribution of expectation and variance matrix t s I, which concludes that M t is a d dim.b.m.. s Corollary 38 Let B t be a d dim.b.m. and g ij t, ω 1 i,j d be an orthogonal matrix for every t and ω Ω. Assume g ij L B. Then M t = M 1 t, Mt,, Mt d with Mt i = becomes a d dim.b.m.. d j=1 g ij s, ωdb j s 3

31 7. Brownian functionals and stochastic integral Let B t = B 1 t, B t,, B d t be a d dim.b.m. starting from and set F t = σ {B s ; s t} = σ {B s B u ; u s t}. In this section we show that any F T measurable square integrable random variable can be expressed by a stochastic integral based on {B t }. For simplicity assume d = 1. Set m linear combination of e iξ kb rk B rk 1 F = k=1. with ξ k R, r r 1 r m T Lemma 39 F is dense in L Ω, F T, P. Proof. Let { Y = f Br1 B r,, B rm B rm 1 ; f is a Borel measurable F 1 = function on R m with E f Br1 B r,, B rm B rm 1 < Then { F F 1 and F 1 is dense in L Ω, F T, P. Hence we have only to prove that linear combination of e iξ x with ξ R m} is dense in L R m, N, Λ, where N, Λ is a Gaussian distribution with expectation and covariance matrix r 1 r r r r m r m 1 However this is clear by the Fourier transform. }. Theorem 4 For any X L Ω, F T, P such that X = EX + T there exists a unique f L B f s, ω db s. 5 Proof. Let S be the set of all X which is represented as 5. 1 X = e iξbt Br S, r t T. Due to Ito s formula we have X = e 1 ξ t r + T r I [r,t siξe iξbs Br e 1 ξ s t db s, hence setting f s, ω = I [r,t siξe iξbs Br e 1 ξ s t, we obtain e iξbt Br S. Suppose X k S with bounded f k for k = 1,,, n and f k s, ωf l s, ω = for k l. Then X 1 X X n S. Let X k t = EX k + f k s, ωdb s. 31

32 Then Ito s formula implies thus X 1 tx t = X 1 X + = EX 1 EX + X 1 sdx s + X sdx 1 s + X 1 sf s + X sf 1 s db s + d X 1, X s f 1 sf sds, T X 1 X = EX 1 EX + X 1 sf s + X sf 1 s db s S. Inductively we can show X 1 X X n S. 3 Suppose X n S and X n X in L Ω, P. Then X S. This follows immediately from Lemma7. Summing up the above argument and Lemma39 conclude the theorem. For higher dimensional Brownian motions also the corresponding theorem is valid, that is, Theorem 41 Let Ω, F T, P be a probability space generated by a d dim.b.m. {B t }. Then for any X L Ω, F T, P,there exist unique f i L B i i = 1,,, d such that X = EX + d i=1 T f i s, ω db i s. This theorem was first proved by K.Ito by applying the Wiener-Ito expansion and is now used to Black-Sholes theory to show the completeness of markets. Corollary 4 Let Ω, F T, P be a probability space generated by a d dim.b.m. {B t }. Then F t+ = F t for every t. Proof. Let X L Ω, F t+, P. Then for any n 1, X L Ω, F t+1/n, P, and applying Theorem41, we have X = EX + However the uniqueness implies d i=1 +1/n f n i s, ω db i s. fi n s, ω = fi m s, ω if s [, t + 1 ] and n m, n hence we see X = EX + d i=1 f 1 i s, ω db i s. 3

33 Corollary 43 Let Ω, F T, P be a probability space generated by a d dim. B.M. {B t }. Any square integrable martingale M = {M t } on this probability space can be represented by a stochastic integral based on the Brownian motion, and hence is automatically continuous. Proof. Since M T L Ω, F T, P, Theorem41 implies Therefore we see M T = EM T + d i=1 T M t = E M T F t = EM T + f i s, ω db i s. d i=1 f i s, ω db i s. 8 Stochastic differential equations Ito constructed stochastic integral based on Brownian motions in order to define rigorously ordinary differential equations whose coefficients are derivatives of Brownian motions. His equatios are now called as stochastic differential equations SDE and are used to describe various random phenomena. This section is devoted to a brief introduction to the theory of SDE. Let {a ij t, x, b i t, x} 1 i d,1 j d be functions on [, R d satisfying certain smoothness conditions. A SDE is dxt i = a ij t, X t db j t + b i t, X t dt, 1 i d. d j=1 More precisely this equation has to be understood in an integral form: d Xt i = x i + j=1 a ij s, X s db j s + b i s, X s ds, 1 i d. 6 If a ij t, x =, then the above equation becomes an ordinary differential equation. We prove here the existence and uniqueness of the equation 6. Assume {a ij t, x, b i t, x} 1 i d,1 j d are continuous functions [, T ] R d satisfying a ij t, x a ij t, y + b i t, x b i t, y K x y 7 with some K > and for any x, y R d, t [, T ]. Set { L c = X t = Xtω i } d ; X i=1 t is continuous,adapted and satisfying X T <, where X t = E sup X s. s t 33

34 Theorem 44 Under the condition 7 the equation 6 has a unique solution for any initial condition x i 1 i d R d. Proof. For {X t } L c define Since d Φ X i t = x i + j=1 a ij s, X s db j s + b i s, X s ds, 1 i d. a ij t, x + b i t, x K x + a ij t, + b i t, K x + K 1 8 with K 1 = max { a ij t, + b i t, ; t T, 1 i d, 1 j d }, we see from iii of Theorem18 and 8 E sup Φ X i t t T x d i + E sup a ij s, X s dbs j t + E sup b i s, X s ds t T j=1 t T d T T x i + 8E a ij s, X s ds + T E b i s, X s ds j=1 x i + 16K d T E sup X t + 4T E sup X t + K t T t T with some other constant K. Therefore Φ X L c. Similarly from 7 we have for X, Y L c Φ X Φ Y t K 3 with some constant K 3. Therefore setting X Y s ds 9 X t = x i, X n t = Φ X t for n 1, we see which implies Thus X n+1 X n t K 3 X n+1 X n K 3t n t n! X n X ds for n 1, s X n+1 X n T <. n= X 1 X ds. s Since L c is complete as we have seen in Lemma7, there exists an X L c such that X n X in L c. This X satisfies X =Φ X, hence 6. The uniqueness 34

35 follows from a truncation argument. Let {X t, X t} be two solutions of 6 and set τ c = inf {t ; X t X t c}. Then we see hence Xt τc E X Xs τc t τ c K 3 E X s τ c ds, which implies the uniqueness. Xt τc E X t τ c =, 8.1 SDE and partial differential equations Let {B t } be a d dim.b.m. and ut, x be a smooth bounded function on [, R d satisfying u t = 1 u, u, x = fx. 3 Then applying Ito s formula to M t = ut t, B t + x, we see dm t = u t T t, B t + xdt + u xi T t, B t + x dbt i i,j d 1 i d u xi,x j T t, B t + x db i tdb j t = u t T t, B t + xdt + 1 i d + 1 u T t, B t + x dt = u xi T t, B t + x dbt, i 1 i d u xi T t, B t + x db i t hence {M t } turns out to be a martingale. Therefore we have EM T = EM = ut, x = EfB T + x. This shows the uniqueness of the bounded solutions for the equation 3 and ut, x = EfB t + x satisfies the equation 3. This argument can be extended to the solutions of SDE whose coefficients do not depend on t. Under the condition of Theorem44 on the coefficients, set L = 1 1 i,j d σ ij x + b i x, x i x j x i 1 i d where σ ij x = 1 k d a ik xa jk x. 35