4.3Ito’s Integral for General

Intrgrands徐健勳

Review• Construction of the Integral

T

0 (t) dW(t)

1

Assume that ( ) is constant in t on each

subinterval ,

Such a process ( ) is a simple processj j

t

t t

t

Tttt n 100

tudWutI

0)()()(

1

1

10

In general, if , then

( ) ( ) ( ) ( ) ( ) ( ) ( )

k k

k

j j j k kj

t t t

I t t W t W t t W t W t

Ttt n

• Martingale

• Ito Isometry

• Quadratic Variation

2

2

0.

tEI t E u du

2

0,

tI I t u du

4.3 Ito Integral for General Integrands• In this section, the Ito Integral

for integrands that allowed to vary

continuously with time and jump• Assume • is adapted to the filtration

• Also assume

T

0 (t) dW(t)

)(t

)(t

0 ),( ttF

T

0

2 )(E dtt

)(t

• In order to define , we approximate

by simple process.

• it is possible to choose a sequence of simple

processes such that as these processes

converge to the continuously varying .

• By “converge”, we mean that

)(tT

0 (t) dW(t)

)(tn

n

)(t

2

0lim 0

T

nnE t t dt

Theorem 4.3.1 Let T be a positive constant and let

be adapted to the filtration F(t) and be an adapted

stochastic process that satisfies Then

has the following properties.

(1)(Continuity) As a function of the upper limit of integration t,

the paths of I(t) are continuous.

(2)(Adaptivity)For each t, I(t) is F(t)-measurable.

,0 ,t t T

2

0.

TE t dt

0 0

lim ,0t t

nnI t u dW u u dW u t T

(3)(Linearity) If and

then

furthermore, for every constant c,

(4)(Martingale) I(t) is a martingale.

(5)(Ito isometry)

(6)(Quadratic variation)

0

tI t u dW u

0

,t

J t u dW u

0;

tI t J t u u dW u

0

tcI t c u dW u

2

2

0.

tEI t E u du

2

0,

tI I t u du

• Example 4.3.2 Computing We choose a large

integer n and approximate the integrand by the

simple process

0( ) ( ).

TW t dW t

t W t

0 0 0

2

1 1

n

TW if tn

T T TW if tn n nt

n T n TW if t T

n n

• As shown in Figure 4.3.2. Then

• By definition,

• To simplify notation, we denote

0|)()(|lim 2

0

dttWtET

nn

0 0

1

0

lim

1 lim

T T

nn

n

n j

W t dW t t dW t

j TjT jTW W Wn n n

WjjTWn

21 1 1 1

2 21 1 1

0 0 0 0

1 12 2

11 0 0

1 1 12 2 2

10 0 0

1 1 12 2 2

1 1 2 2

1 1 1 2 2 2

n n n n

j j j j j jj j j j

n n n

k j j jk j j

n n n

n k j j jk j j

W W W W W W

W W W W

W W W W W

1 12 2

10 0

12

10

1 2

1 2

n n

n j j jj j

n

n j j jj

W W W W

W W W W

• We conclude that

• In the original notation, this is

21 1

21 1

0 0

1 12 2

n n

j J j n j jj j

W W W W W W

1

0

21

2

0

1

11 1 2 2

n

jj

n

j

j TjT jTW W Wn n n

j T jTW T W Wn n

• As , we get

• Compare with ordinary calculus. If g is a differentiable

function with g(0)=0, then

n

2

0

2

1 1 ,2 21 1 2 2

TW t dW t W T W W T

W T T

T Tt

tTgtgtdgtg

0

2

0

2 )(21)(

21)()( |

• If we evaluated

then we would not have gotten ,this term.

In other words,

1

0

112lim

n

n j

j T j T jTW W Wn n n

T21

2

0

12

TW t dW t W T

n

jTWn

TjWn

TjW

n

jn

)1(21

lim1

0

To simplify notation, we denote

and21

21

jWn

TjW

jWnjTW

1

0

21

0 21

1

0

2

21

1

0

2

21 2

121)(

21 n

jj

n

jjj

n

j j

n

jjj

WWWWWW

1

0

2

21

1

0 211

1

0

21

1

0

2

211 2

121)(

21 n

j j

n

j jj

n

jj

n

j jj WWWWWW

---①

---②

Then, ①-②

1

0

1

0 21

1

0 211

21

1

0

2

21

21 n

jj

n

j j

n

j jjj

n

jj WWWWWW

1

0

1

01

1

0 21

222

21

21

21 n

j

n

jjj

n

j jnjj WWWWWW

We conclude that

1

0

1

0

22

211

2

21

1

01

21 2

1)(21lim)(

21limlim

n

j

n

jnjjnjjn

n

jjjjn

WWWWWWWW

22)(

1

0

1

0 21

1

0

2

21

Tn

TWWVarWWEn

j

n

jjj

n

jjj

02

442

43

4243

4)()(

2)()(

2

222

21

0

1

02

2

1

02

21

0

2

21

1

0

4

21

1

0

2

2

21

1

0

2

21

nasn

Tn

Tn

Tn

T

nT

nT

nT

nT

nTWWE

nTWWE

nTWWEWWVar

n

j

n

j

n

j

n

jjj

n

jjj

n

jjj

n

jjj

We conclude that and

1

0

2

21 4

1)(lim21 n

jjjn

TWW

1

0

2

211 4

1)(lim21 n

j jjnTWW

1

0

1

0

22

211

2

21

1

01

21 2

1)(21lim)(

21limlim

n

j

n

jnjjnjjn

n

jjjjn

WWWWWWWW

)(21)(

21

41

41)1(2

1

lim 221

0

TWTWTTnjTW

nTjW

n

TjW

n

jn

, So

2

0

12

TW t dW t W T We have

• is called the

Stratonovich integral.

• Stratonovich integral is inappropriate for finance.

• In finance, the integrand represents a position in an

asset and the integrator represents the price of that

asset.

2

0

12

TW t dW t W T

• The upper limit of integrand T is arbitrary, then

• By Theorem4.3.1

• At t = 0, this martingale is 0 and its expectation is 0.

• At t > 0, if the term is not present and EW2(t) = t, it is not

martingale.

2

0

1 1 , 02 2

TW u dW u W T t t

12

t