4.3 Ito's Integral for General
Integrands
報告者：張議文
古豐上
In order to make Ito isometry exist
• In general, it is possible to choose a sequence
n  t  of simple processes such that as n  
these processes converge to the continuously
varying   t  .
• By “converge”, we mean that
2
lim E  n  t     t  dt  0
T
n
0
Example 4.3.2
T
• Computing0 W (t )dW (t ). We choose a large
integer n and approximate the integrand
  t   W  t  by the simple process
T

if 0  t 
 W  0  0
n

T
2T
 W T 
if  t 
 

n
n
n
n t   


n  1 T

   n  1 T 
t T
 if
W 
n
n

 

T
0
W (t )dW (t ).
 g  t  dg  t    g  t  g   t  dt
T
T
0
0
1 2 T 1 2
 g  t   g T 
0
2
2
By ordinary calculus. If g is a differentiable
function with g  0  0, then
By definition

T
0
W  t dW  t   lim   n  t  dW  t 
T
n  0
 jT     j  1 T
 lim  W 
W


n 
n
 n   
j 0
n 1

 jT  
 W 

 n  

Lebesgue Integral → Riemann Integral
Let
 jT 
Wj  W 

n


and W0  W  0  0
n 1
2
1
1 n 1 2 n 1
1 n 1 2
W j 1  W j    W j 1   W j  W j 1   W j


2 j 0
2 j 0
2 j 0
j 0
1 n 2 n 1
1 n 1 2
  Wk   W j  W j 1   W j
2 k 1
2 j 0
j 0
1 2 1 n 1 2 n 1
1 n 1 2
 Wn   Wk   W j  W j 1   W j
2
2 k 0
2 j 0
j 0
1 2 n 1 2 n 1
 Wn   W j   W j  W j 1
2
j 0
j 0
1 2 n 1
 Wn   W j W j  W j 1 
2
j 0
Conclude that
n 1
n 1
1 2 1
W j WJ 1  W j   Wn   W j 1  W j 

2
2 j 0
j 0
2
In the original notation
 jT     j  1 T
Wj 

 W 
n
 n   
j 0
n 1

 jT  
 W 
 
 n  

1 2
1    j  1 T
W T    W 
2
2 j 0  
n
n 1

 jT  
 W 

 n  

2
Letting n   ,
1 2
1
0 W  t  dW  t   2 W T   2 W ,W  T 
1 2
1
 W T   T
2
2
T
• Usually, evaluating the integrand at the lefthand endpoint of the subinterval.
• If evaluating the integrand at the midpoint,

1
  j  2 T
n 1

lim  W  
n 
n

j 0



    j  1 T
 W 
n
  


then ( see Exercise 4.4 )

T
0
W t 
1 2
dW  t   W T 
2

 jT  
 W 

 n  

•

T
0
W t 
1 2
dW  t   W T  is called the
2
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.
• The difference of the Stratonovich integral and
the Ito integrand is sensitive. Stratonovich
integral is less sensitive than Ito integrand.
• The upper limit of integrand T is arbitrary, then
• By Theorem4.3.1
• At t = 0, Stratonovich integral and Ito integral
martingale are 0 and their expectation are 0.
1
• At t > 0, if the term  2 t is not present and
EW2(t) = t, Stratonovich integral does not
follow martingale property.
The End