4.3 ito’s integral for general intrgrands 徐健勳. review construction of the integral
TRANSCRIPT
4.3Ito’s Integral for General
Intrgrands徐健勳
Review
• Construction of the IntegralT
0 (t) dW(t)
1
Assume that ( ) is constant in t on each
subinterval ,
Such a process ( ) is a simple process
j 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
nn
E 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
nn
I 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 t
nT T T
W 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
nj
W t dW t t dW t
j TjT jTW W W
n n n
Wj
jTW
n
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 1
2 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 1
2 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 W
n n n
j T jTW T W W
n 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 )(2
1)(
2
1)()( |
• If we evaluated
then we would not have gotten ,this term.
In other words,
1
0
112
limn
nj
j Tj T jT
W W Wn n n
T2
1
2
0
1
2
TW t dW t W T
n
jTW
n
TjW
n
TjW
n
jn
)1(21
lim1
0
To simplify notation, we denote
and
21
21
jWn
TjW
jWn
jTW
1
0
21
0 2
1
1
0
2
2
1
1
0
2
2
1 2
1
2
1)(
2
1 n
jj
n
jj
j
n
jj
n
jj
jWWWWWW
1
0
2
2
1
1
0 2
11
1
0
21
1
0
2
2
11 2
1
2
1)(
2
1 n
jj
n
jj
j
n
jj
n
jj
j WWWWWW
---①
---②
Then, ①-②
1
0
1
0 2
1
1
0 2
112
1
1
0
2
2
1
2
1 n
jj
n
jj
n
jj
jj
n
jj WWWWWW
1
0
1
01
1
0 2
1222
2
1
2
1
2
1 n
j
n
jjj
n
jj
njj WWWWWW
We conclude that
1
0
1
0
22
2
112
2
1
1
01
2
1 2
1)(
2
1lim)(
2
1limlim
n
j
n
jn
jj
nj
jn
n
jjj
jnWWWWWWWW
22)(
1
0
1
0 2
1
1
0
2
2
1
T
n
TWWVarWWE
n
j
n
jj
j
n
jj
j
02
44
2
4
3
4243
4)()(
2)()(
2
222
21
0
1
02
2
1
02
21
0
2
2
1
1
0
4
2
1
1
0
2
2
2
1
1
0
2
2
1
nasn
T
n
T
n
T
n
T
n
T
n
T
n
T
n
T
n
TWWE
n
TWWE
n
TWWEWWVar
n
j
n
j
n
j
n
jj
j
n
jj
j
n
jj
j
n
jj
j
We conclude that and
1
0
2
2
1 4
1)(lim
2
1 n
jj
jnTWW
1
0
2
2
11 4
1)(lim
2
1 n
jj
jn
TWW
1
0
1
0
22
2
112
2
1
1
01
2
1 2
1)(
2
1lim)(
2
1limlim
n
j
n
jn
jj
nj
jn
n
jjj
jnWWWWWWWW
)(2
1)(
2
1
4
1
4
1)1(21
lim 221
0
TWTWTTn
jTW
n
TjW
n
TjW
n
jn
, So
2
0
1
2
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
1
2
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, 0
2 2
TW u dW u W T t t
1
2t