Stochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.

Transcription

1 Ito Calculus I Geneviève Gauthier HEC Montréal

2 Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of the rst important papers was published in 1942).

3 Riemann I Riemann Ito Let f : R! R be a real-valued function. Typically, when we speak of the of a function f, we refer to the Riemann, R b f a (t) dt, which calculates the area under the curve t! f (t) between the bounds a and b.

4 Riemann II Riemann Ito First, we must realize that not all functions are Riemann integrable. Let s explain a little further : in order to construct the Riemann of the function f on the interval [a, b], we divide such an interval into n subintervals of equal length b a n and, for each of them, we determine the greatest and the smallest value taken by the function f.

5 Riemann Ito Riemann III Thus, for every k 2 f1,..., ng, the smallest value taken by the function f on the interval a + (k 1) b a n, a + k b a n is f (n) k inf f (t) b a a + (k 1) t a + k b a n n and the greatest is _ f (n) k sup f (t) b a a + (k 1) n t a + k b n a.

6 Riemann Ito Riemann IV The area under the curve t! f (t) between the bounds a + (k 1) b a n and a + k b a n is therefore greater than the area b a f (n) k of the rectangle of base b a and height f (n) n k, and smaller than the area b a with the same base but with height As a consequence, if we de ne n _ f (n) k _ f (n) k. n of the rectangle I n (f ) n b k=1 n a f (n) k _ n b and I n (f ) n k=1 a _ f (n) k then I n (f ) Z b a f (t) dt _ I n (f ).

7 Riemann V Riemann Ito Illustrations of I n (f ) and _ I n (f )

8 Riemann Ito Riemann VI Riemann integrable functions are those for which the limits _ lim n! I n (f ) and lim n! I n (f ) exist and are equal. We then de ne the Riemann of the function f as Z b a _ f (t) dt lim I n! n (f ) = lim I n (f ). n! It is possible to show that the R b f a (t) dt exists for any function f continuous on the interval [a, b]. Many other functions are Riemann integrable.

9 Riemann VII Riemann Ito However, there exist functions that are not Riemann integrable. For example, let s consider the indicator function I Q : [, 1]! f, 1g which takes the value of 1 if the argument is rational and otherwise. _ I n = 1, which Then, for all natural number n, I n = and implies that the Riemann is not de ned for such a function.

10 Riemann VIII Riemann Ito We will examine a very simple case : the Riemann for boxcar functions, i.e. the functions f : [; )! R which admit the representation f (t) = ci (a,b] (t), a < b, c 2 R where I (a,b] represents the indicator function I (a,b] (t) = 1 if t 2 (a, b] if t /2 (a, b].

11 Riemann IX Riemann Ito If f (t) = ci (a,b] (t) then the Riemann of f is Z t f (s) ds = = Z t 8 < : ci (a,b] (s) ds if t a c (t a) if a < t b c (b a) if t > b.

12 Ito I Ito Let fw t : t g be a standard Brownian motion built on a ltered probability space (Ω, F, F, P), F = ff t : t g. Technical condition. Since we will work with almost sure equalities 1, we require that the set of events which have zero probability to occur are included in the sigma-algebra F, i.e. that the set N = fa 2 F : P (A) = g F. That way, if X is F t measurable and Y = X P almost surely, then we know that Y is F t measurable.

13 Ito Ito II Let fx t : t g be a predictable stochastic. It is tempting to de ne the stochastic of X with respect to W, pathwise, by using a generalization of the Stieltjes : 8ω 2 Ω, Z X (ω) dw (ω). That would be possible if the paths of Brownian motion W had bounded variation, i.e. if they could be expressed as the di erence of two non-decreasing functions. But, since Brownian motion paths do not have bounded variation, it is not possible to use such an approach. We must de ne the stochastic with respect to the Brownian motion as a whole, i.e. as a stochastic in itself, and not pathwise. 1 X = Y P almost surely if the set of ω for which X is di erent from Y has a probability of zero, i.e. P fω 2 Ω : X (ω) 6= Y (ω)g =.

14 Ito I Ito De nition We call X a basic stochastic if X admits the following representation: X t (ω) = C (ω) I (a,b] (t) where a < b 2 R and C is a random variable, F a and square-integrable, i.e. E P C 2 <. measurable Note that such a is adapted to the ltration F. Indeed, 8 < if t a X t = C if a < t b : if b < t which is F measurable therefore F t measurable which is F a measurable therefore F t measurable which is F measurable therefore F t measurable.

15 Ito II Ito In fact, X is more than F adapted, it is F predictable, but the notion of predictable is more delicate to de ne when we work with continuous-time es. Note, nevertheless, that adapted es with continuous path are predictable. Intuitively, if X t represents the number of security shares held at time t, then X t (ω) = C (ω) I (a,b] (t) means that immediately after prices are announced at time a and based on the information available at time a (C being F a measurable) we buy C (ω) security shares which we hold until time b. At that time, we sell them all.

16 Ito III Ito Theorem F adapted es, the paths of which are left-continuous, are F predictable es. In particular, F adapted es with continuous paths are F predictible. (cf. Revuz and Yor)

17 Ito IV Ito De nition The stochastic of X with respect to the Brownian motion is de ned as Z t X s dw s (ω) = C (ω) (W t^b (ω) W t^a (ω)) 8 < if t a = C (ω) (W t (ω) W a (ω)) if a < t b : C (ω) (W b (ω) W a (ω)) if b < t. Note that, for all t, the R t X s dw s is a random variable. n R o t Moreover, X s dw s : t is a stochastic.

18 Ito V Ito Paths of a basic stochastic, and its stochastic with respect to the Brownian motion ,1,2,3,4,5,6,7,8,9 1 t ,1,2,3,4,5,6,7,8,9 1 t Integrale_stochastique.xls

19 VI Ito Ito Note that Z t X s dw s = Z X s I (,t] (s) dw s. Exercise. Verify it!

20 Ito VII Ito Interpretation. If, for example, the Brownian motion W represents the variation of the share present value relative to its initial value (W t = S t S, where S t is the share present value at time t), then R t X sdw s is the present value of the pro t earned by the investor. This example is somewhat lame: since the Brownian motion is not lower bounded, the example implies that it is possible for the share price to take negative values. Oops! Note on the graphs that, over the time period when the investor holds a constant number of security shares, the present value of his or her pro t uctuates. That is only caused by share price uctuations.

21 Ito VIII Ito Theorem Lemma 1. If X is a basic stochastic, then Z t X s dw s : t is a (Ω, F, F, P) martingale. Proof of Lemma 1. Recall that, if M is a martingale and τ a stopping time, then the stopped stochastic M τ = fm t^τ : t g is also a martingale. Since the Brownian motion is a martingale, and τ a and τ b where 8ω 2 Ω, τ a (ω) = a and τ b (ω) = b are stopping times, then the stochastic es W τ a = fw t^a : t g and W τ b = fw t^b : t g are both martingales.

22 Ito Now, let s verify condition (M1). If t a, then Z t E P X u dw u IX Ito = E P [jc (W t^b W t^a )j] = E P [jc (W t W t )j] =.

23 Ito By contrast, if t > a, then Z t E P X u dw u X Ito = E P [jc (W t^b W a )j] = E P [jc j jw t^b W a j] h re P C 2 (W t^b W a ) 2i (see next page) ii = re hc P 2 E h(w P t^b W a ) 2 jf a C being F a mble. = = re P hc 2 E P h(w t^b W a ) 2ii W t^b W a being independent from F a. q (t ^ b a) E P [C 2 ] <

24 Ito XI Ito Justi cation for the inequality: for any random variable Y such that E Y 2 <, we have Var [Y ] = E Y 2 (E [Y ]) 2 which implies that E [Y ] q E [Y 2 ].

25 Ito = With respect to condition (M2), Z t X s dw s 8 < if t a C (W t W a ) if a < t b : C (W b W a ) if b < t. XII Ito F measurable therefore F t measurable F t measurable, F b measurable therefore F t measurable

26 Ito XIII Ito Condition (M3) is also veri ed since 8s, t 2 R, s < t, Z t E P X u dw u jf s = E P [C (W t^b W t^a ) jf s ].

27 Ito XIV Ito But, if s a, then F s F a. Since C is F a measurable, then Z t h i E P X u dw u jf s = E P E P [C (W t^b W t^a ) jf a ] jf s = h E P CE P [(W t^b i W t^a ) jf a ] jf s 2 3 = E P 6 7 4C (W a^b W a^a ) jf s 5 {z } =W a W a = since W τ a and W τ b are martingales, = = Z s X u dw u.

28 Ito XV Ito If s > a, then C is F s measurable and, as a consequence, Z t E P X u dw u jf s = E P [C (W t^b W t^a ) jf s ] = CE P [(W t^b W t^a ) jf s ] = C (W s^b W s^a ) = Z s X u dw u. The R X s dw s is indeed a martingale.

29 Ito I We will need a few results later on, that we shall now start to develop. Since the Ito for basic es is a martingale, then Z t Z E P X s dw s = E P X s dw s =. The next result will allow us to calculate variances and covariances.

30 Ito II Theorem Lemma 2. If X and Y are basic es, then for all t, Z t Z t Z t E P X s Y s ds = E P X s dw s Y s dw s.

31 Ito III Proof of Lemma 2. It will be su cient to show that Z Z Z E P X s Y s ds = E P X s dw s Y s dw s since being basic es, are also basic es. X = CI (a,b] and Y = eci (ea, eb] X I (,t] = CI (a^t,b^t], Y I (,t] = eci (ea^t, eb^t] et XY I (,t] = C eci ((a_ea)^t,(b^eb)^t] (1)

32 Ito IV Thus, if equation (1) is veri ed, we can use it to establish the third equality in the following calculation: Z t E P X s Y s ds Z = E P X s Y s I (,t] (s) ds Z = E P X s I (,t] (s) Y s I (,t] (s) ds Z Z = E P X s I (,t] (s) dw s Y s I (,t] (s) dw s Z t Z t = E P X s dw s Y s dw s.

33 Ito So, let s establish Equation (1) E P Z X s Y s ds V Z Z = E P X s dw s Y s dw s. Three cases must be treated separately: i (i) (a, b] \ ea, eb =?, i (ii) (a, b] = ea, eb, i i (iii) (a, b] 6= ea, eb and (a, b] \ ea, eb 6=?.

34 Ito VI Proof of (i). i Since (a, b] \ ea, eb =?, we can assume, without loss of generality, that a < b ea < eb. Since X s Y s = C eci (a,b] (s) I (ea, eb] (s) =, then E P R X sy s ds =.

35 Ito VII Now, since C, ec and (W b W a ) are F ea measurable, Z Z E P X s dw s Y s dw s = E P h C (W b W a ) ec W e b W ea i h h = E P E P ii C (W b W a ) ec Wb e W ea jfea 2 3 = E P 6 4C ec (W b W a ) E P 7 Wb e W ea jf ea 5 {z } = = E P Z X s Y s ds =W ea W ea = thus establishing Equation (1) in this particular case.

36 Ito Then, VIII Proof of (ii). Let s now assume that (a, b] = E P Z X s Y s ds = E P Z = E P C ec C eci (a,b] dt Z I (a,b] dt i = E P h C ec (b a) = (b a) E P h C ec i. i ea, eb.

37 Ito On the other hand, IX Z Z E P X s dw s Y s dw s h i = E P C (W b W a ) ec (W b W a ) h = E P C ec h ii E P (W b W a ) 2 jf a since C and ec are F a = E P h C ec E P h (W b W a ) 2ii mbles since h W b W a is independent of F a i = E P C ec (b a) since W b W a is N (, b a) h i Z = (b a) E P C ec = E P X s Y s ds thus establishing Equation (1) for this other particular case.

38 Ito Proof of (iii). We have that (a, b] 6= i (a, b] \ ea, eb 6=?. X Basic i ea, eb and a = a _ ea and b = b ^ eb. On the one hand, Z Z E P X s Y s ds = E P CI (a,b] eci (ea, eb] dt Z = E P C eci (a,b ]dt Z = E C P ec I (a,b ]dt h i = E P C ec (b a ) h i = (b a ) E P C ec.

39 Ito XI In what follows, in case there were intervals of the form (α, β] where α β, then we de ne (α, β] =?. On the other hand, since Z X s dw s = C (W b W a ) = C (W b W b + W b W a + W a W a ) = C (W b W b ) + C (W b W a ) + C (W a W a ) Z Z Z = C I (b,b] dw s + C I (a,b ] dw s + C I (a,a ] dw s, (2)

40 Ito Z Y s dw s = ec Wb e W ea XII = ec W e b W b + W b W a + W a W ea = ec Wb e W b + ec (W b W a ) + ec (W a Z Z Z W ea ) = ec I (b,eb] dw s + ec I (a,b ] dw s + ec I (ea,a ] dw s (3)

41 Ito and XIII (a, a ] \ (ea, a ] =?, (a, a ] \ (a, b ] =?, (a, a ] \ i (a, b ] \ (ea, a ] =?, (a, b ] \ b, eb =?, (b, b] \ (ea, a ] =?, (b, b] \ (a, b ] =?, (b, b] \ i b, eb =?, i b, eb =?, we can use the results obtained in points (i) and (ii) in order to complete the proof: using the expressions from lines (2) and

42 Ito (3), XIV Z Z E P X s dw s Y s dw s Z = E CI Z P (a,b ]dw s eci (a,b ]dw s h i = (b a ) E P C ec Z = E P X s Y s ds. The proof of Lemma 2 is now complete.

43 Ito I Ito De nition We call X a simple stochastic if X is a nite sum of basic es : X t (ω) = n C i (ω) I (ai,b i ] (t). i=1 Since such a is a sum of F predictable es, then it is itself predictable with respect to the ltration F.

44 Ito II Ito De nition The stochastic of X with respect to the Brownian motion is de ned as the sum of the stochastic s of the basic es which constitute X : Z t X s dw s = n Z t C i I (ai,b i ] (s) dw s. i=1

45 Ito III Ito Paths of a simple stochastic and its stochastic with respect to the Brownian motion ,1,2,3,4,5,6,7,8,9 1 4 t ,1,2,3,4,5,6,7,8, t The simple is of the form C 1 I ( 1 1, 6 1 ] + C 2I ( 1 2, 3 4 ]

46 Ito IV Ito Integrale_stochastique.xls

47 Ito V Ito First, we must ensure that such a de nition contains no inconsistencies, i.e. if the simple stochastic X admits at least two representations in terms of basic stochastic es, say n i=1 C i I (ai,b i ] and m j=1 e C j I (eaj,eb j], then the is de ned in a unique manner: n Z t m Z t C i I (ai,b i ] (s) dw s = ec j I i=1 j=1 (eaj,eb j] (s) dw s. Exercise. Prove that the de nition of the stochastic for simple es does not depend on the representation chosen.

48 Ito VI Ito Theorem Lemma n 3. If X is a simple stochastic, then R o t X sdw s : t is a (Ω, F, F, P) martingale. Proof of Lemma 3. The stochastic Z t X s dw s = n Z t C i I (ai,b i ] (s) dw s i=1 of the simple X = n i=1 C i I (ai,b i ] is the sum of the stochastic s of the basic es it is made up of. Since a nite sum of martingales is also a martingale, the results comes from the fact that stochastic s of basic es are martingales (Lemma 1).

49 Ito VII Ito The next result is rather technical and the proof can be found in the Appendix. It will useful to us later on. Theorem Lemma 4. If X is a simple, then for all t, Z t " Z t # 2 E P Xs 2 ds = E P X s dw s.

50 Ito VIII Ito That lemma is, besides, very useful to calculate the variance of a stochastic. Indeed, Z t Var P X s dw s " Z t # 2 = E P X s dw s = E P Z t Z t Xs 2 ds = E P X 2 s ds. 1 Z t EP X s dw s C A {z } = 2 It is possible to extend this calculation to other es X and toestablish in a similar manner a method to calculate the covariance between two stochastic s (see the Appendix).

51 Ito es I Ito We would like to extend the class of es for which the stochastic with respect to the Brownian motion can be de ned. We will choose the class of F predictable es for which there exists a sequence of simple es approaching them. Theorem F adapted es, the paths of which are left-continuous are F predictable es. In particular, F adapted es with continuous paths are F predictible. Let X be ann F predictable o for which there exists a sequence X (n) : n 2 N of simple es converging to X when n increases to in nity.

52 Ito es II Ito You can t speak of convergence without speaking of distance. Indeed, how does one measure whether a sequence of stochastic es approaches some other? What is the distance between two stochastic es? To answer such a question, we must de ne the space of stochastic es on which we work as well as the norm we put on that space. Recall that kk A : A! [, ) is a norm on the space A if (i) kx k A =, X = ; (ii) 8X 2 A and 8a 2 R, kax k A = jaj kx k A ; (iii) 8X, Y 2 A, kx + Y k A kx k A + ky k A (triangle inequality).

53 Ito Let A = X es III Ito X is af predictable such that E P R X 2 t dt <. Exercise, optional and rather di cult! The function kk A : A! [, ) de ned as kx k A = s E P Z Xt 2 dt is a norm on the space A.

54 Ito es IV Ito n o It is said that the sequence X (n) : n 2 N A converges to X 2 A when n increases to in nity if and only if lim X (n) n! s Z X = lim E P A n! =. X (n) t X t 2 dt

55 Ito es V Ito It is tempting to de ne the stochastic of X with respect to the Brownian motion as the limit of the stochastic s of the simple es, i.e. Z t Z t X s dw s = lim X s (n) dw s. n!

56 Ito es VI Ito The latter equation raises two issues, one of which being whether such a limit exists. For example, let s assume we work on the space of strictly positive numbers A fx 2 R jx > g and we study the sequence 1 n : n 2 N. That sequence does not converge 1 in the space A since lim n! n = /2 A. Of course, that sequence converges in space R. R t The question is: does lim n! X s (n) dw s exist in the space in which we work? The second issue is that this equation implies that, the greater n, the closer R t X s (n) dw s is to R t X s dw s. But, what is the distance between two stochastic s?

57 Ito es VII Ito Recall that, in Lemma 3, we have established that, for all n, the R X s (n) dw s of a simple is a martingale. Moreover, it is possible to establish that E " Z X (n) s dw s 2 # <.

58 Ito es VIII Ito As a consequence, the sequence used to approach R X s dw s is contained in the space ( ) M is a martingale M = M q such that sup t E P [Mt 2. ] < Exercise, optional and laborious! The function kk M : M! [, ) de ned as is a norm on M. r kmk M = sup t E P [M 2 t ]

59 Ito Thus, we say that the sequence es IX Ito n o M (n) : n 2 N of martingales belonging to the normed space (M, kk M ) converges to M if and only if M 2 M and lim M (n) n! s M = lim sup E P M (n) t M n! t =. M t 2

60 Ito es X Ito The idea behind the construction of R t X s dw s is that, on the one hand, we have the space (A, kk A ) of the es for which we construct the stochastic and, on the other hand, we have the space (M, kk M ) containing the stochastic s of the es in the rst space. But, we have chosen the norms in such a way that, if the X is a simple, then kx k A = R X s dw s M (Ref.: Lemma 5 in the Appendix).

61 Ito es XI Ito This n implies that, o if the sequence of simple es X (n) : n 2 N is a Cauchy sequence in the rst space, i.e. X (n) X (m) m,n! A!, n R o t then the sequence X s (n) dw s : n 2 N of stochastic s is a Cauchy sequence in the second space: Z t Z X s (n) dw s X s (m) M m,n! dw s!. The sequence n o M (n) : n 2 N is a Cauchy sequence on the normed space (M, kk M ) if lim M (n) M (m) M =. n,m!

62 Ito es XII Ito n o If M (n) : n 2 N is a Cauchy sequence, we cannot, in general, state that such a sequence converges, since we are not certain that the limit-point belongs to the set M.

63 Ito es XIII Ito The only thing left to verify is that n the limit-point of the R o t sequence of stochastic s dw s : n 2 N is X s (n) indeed an element of M (ref.: Lemma 6 in the Appendix). We will then be able to de ne the stochastic of the predictable X = lim n! X (n) as R t lim n! X s (n) dw s.

64 Ito Predictible We have de ned the stochastic with respect to the (Ω, F, F, P) Brownian motion for all X F predictable es satisfying the condition Z E P Xt 2 dt <. For each of these es, we n have established that the R o t family of stochastic s X sdw s : t is a (Ω, F, F, P) martingale.

65 Ito I Predictible It is possible to extend the class of es for which the stochastic can be de ned. It involves local martingales as well as semi-martingales. We refer to the book by Richard Durrett those who would like to know more. Let s however mention that it is possible, for such es, n that the family of stochastic s R o t X sdw s : t is no longer a martingale.

66 Ito II Predictible Here is a result allowing us to verify whether the stochastic with which we work is a martingale : Theorem Lemma. h If X is a F predictible such that R i E P t X s 2 ds < then R s X sdw s : s t is a (Ω, F, F, P) martingale (ref. Revuz and Yor, Corollary 1.25, page 124).

67 Ito III Predictible It is possible to de ne the stochastic with respect to other stochastic es than the Brownian motion by using the same construction as the one we have just provided.

68 Ito I DURRETT, Richard (1996). Calculus, A Practical, CRC Press, New York. KARATZAS, Ioannis and SHREVE, Steven E. (1988). Brownian Motion and Calculus, Springer-Verlag, New York. KARLIN, Samuel and TAYLOR, Howard M. (1975). A First Course in Processes, second edition, Academic Press, New York. LAMBERTON, Damien and LAPEYRE, Bernard (1991). au calcul stochastique appliqué à la nance, Éllipses, Paris. REVUZ, Daniel and YOR, Marc (1991). Continuous Martingale and Brownian Motion, Springer-Verlag, New York.

69 Lemma 4 I Ito Lemma 4 Lemma 5 Lemma 6 Comments Here is a second result similar to Lemma 2, but for simple es. Theorem Lemma 4. If X is a simple, then for all t, Z t " Z t # 2 E P Xs 2 ds = E P X s dw s.

70 Ito Lemma 4 Lemma 5 Lemma 6 Comments Lemma 4 II Proof of Lemma 4. Let X = n i=1 C i I (ai,b i ] be a simple. " Z t # 2 E P X s dw s = E P 2 4 n Z t i=1! 3 2 C i I (ai,b i ] (s) dw s 5 n n Z t Z t = E P C i I (ai,b i ] (s) dw s C j I (aj,b j ] (s) dw s i=1 j=1 n n Z t = E P C i I (ai,b i ] (s) C j I (aj,b j ] (s) ds by Lemma 2. i=1 j=1 " Z # t = E P n n C i I (ai,b i ] (s) C j I (aj,b j ] (s) ds i=1 j=1 2 3 Z t = E P n 2 Z t 4 C i I (ai,b i ] (s)! ds5 = E P Xs 2 ds. i=1

71 Lemma 5 I Ito Lemma 4 Lemma 5 Lemma 6 Comments Theorem Lemma 5. For any simple Y, ky k A = R Y s dw s M. Proof of Lemma 5. Note that 8ω 2 Ω, the function t! R h t Y R i s 2 (ω) ds as well as the function t!e P t Y s 2 ds

72 Ito Lemma 4 Lemma 5 Lemma 6 Comments are non-decreasing. As a consequence, Z ky k 2 A = EP Ys 2 ds = E P Z t = lim E P Y 2 t! s ds lim t! Lemma 5 II Z t Ys 2 ds by monotone convergence theorem. Z t = sup E P Ys 2 ds t " Z t # 2 = sup E P Y s dw s by Lemma 4. = t Z Y s dw s 2 M by the very de nition of kk M.

73 Ito Lemma 4 Lemma 5 Lemma 6 Comments Lemma 5 III From n the latter result, it follows that the sequence R o (n) X s dw s : n 2 N of stochastic s of simple es with respect to the Brownian motion is a Cauchy sequence since Z Z lim n,m! X s (n) dw s X s (m) M dw s = lim X (m) A = X (n) n,m! where the rst equality is a direct consequence from Lemma 5, X (n) and X (m) being simple stochastic es, and the second n equalityo comes from the fact that the sequence X (n) : n 2 N is a Cauchy sequence since it converges to X.

74 Ito Lemma 4 Lemma 5 Lemma 6 Comments Theorem Lemma 6 I Lemma 6. The normed space (M, kk M ) is complete a (Durrett, 1996, Theorem 4.6, page 57). a A normed space is said to be complete if any Cauchy sequence converges, i.e. the limit-point of any Cauchy sequence belongs to the space. n R o (n) Since X s dw s : n 2 N is a Cauchy sequence on the complete space (M, kk M ) then such a sequence converges, thus R establishing the existence of what we have called Xs dw s, and its limit-point belongs to (M, kk M ), which implies R that the stochastic of X with respect to W, Xs dw s, is a martingale, even for some adapted es which are not simple es.

75 Comments I Ito Lemma 4 Lemma 5 Lemma 6 Comments It is possible to prove that the de nition of the stochastic, for the predictable stochastic es X 2 A which own a sequence of simple es approaching them, n does notodepend n on the sequence chosen, i.e. if X (n) : n 2 N and X e (n) : n 2 No are such two sequences of simple es, that X lim (n) n! X = and lim n! ex (n) X =, A A then Z lim n! Z X s (n) dw s = lim n! ex (n) dw s.

76 Comments II Ito Lemma 4 Lemma 5 Lemma 6 Comments Last comment: for any predictable stochastic X n 2 (A, kk A ), o there exists such a sequence X (n) : n 2 N of simple es that lim X (n) n! X =. A (Durrett, 1996, Lemma 4.5, page 57)