Chapter 14
The Itô Integral
The following chapters deal with Stochastic Differential Equations in Finance. References:
1. B. Oksendal, Stochastic Differential Equations, Springer-Verlag,1995
2. J. Hull, Options, Futures and other Derivative Securities, Prentice Hall, 1993.
14.1 Brownian Motion
(See Fig. 13.3.)
Brownian motion, #
$%'&
1.
2.
3. If
.
is given, always
in the background, even when not explicitly mentioned.
! "
, has the following properties:
Technically,
! (*)+,&
-'./
$%102$43
,
is a continuous function of ,
5$687,9:;9=<><><198?
, then the increments
: @
A87B
C<B<C<D
E?
@
A8?#F : are independent,normal, and
! GH
! G
AJILK : M@
AEILK : Q@
AJIN
-OP$='
AEIN
EOAR$S.IK :
@.IT<
14.2 First Variation
QuadraticWvariation
is a measure of volatility. First we will consider first variation,
.
.
function
153
U,V
WP
, of a
154
f(t)
t2
t
T
t1
Figure 14.1: Example function
W.
.
For the function pictured in Fig. 14.1, the first variation over the interval
U,V
7 WD
$4 W/A:
@
W#
EO @
WA.
$
7
W:
EO
@
R
W-AE
J@
WAAJ
W /@
W
is given by:
R
EO
WJ
#L<
$
W/A
O
7
<
Thus, first variation measures the total amount of up and down motion of the path.
The general definition of first variation is as follows:
$
Definition 14.1 (First Variation) Let $S 7
)B 7 : C<B<C< ? 0
9 :
9=<><><19
?
be a partition of
O
, i.e.,
$2<
The mesh of the partition is defined to be
$
"!$#
I K : @ I +<
' ' ?#F :
I&% 7('
? F :
We then define
U,V
7) WD
$
*,+
-(- ./-(- 7
0
I1% 7
W/AEI K : M@
W/AJIN
<
W
Suppose is differentiable.
Then the Mean Value Theorem implies that in each subinterval
3I2
there is a point such that
W.I K :+
@
WJI $=W
2I C.I K :@
.IN
<
JI JI K : O
,
CHAPTER 14. The Itô Integral
155
?#F :
Then
? F :
0
W/AJILK : M@
I&% 7 W/AEIN
0
$
and
2I .IK : @
W
I&% 7
EIN
+
?#F :
U,V
7) * +
- - ./- - WD
$
$
0
7
W-AJ
JIN
<
7
2I JILK : @
W
I&% 7
14.3 Quadratic Variation
W
Definition 14.2 (Quadratic Variation) The quadratic variation of a function on an interval
? F :
is
W
*+
-(- . -(- $
0
7
W AEI K :+
@
I1% 7
W
JIN
R <
W
Remark 14.1 (Quadratic Variation of Differentiable Functions) If is differentiable, then
, because
? F :
0
W JI K :
@
I1% 7 ? F :
W
JIN
0
R
$
W A 2I I&% 7
9
R AEI K :@JIN
R
? F :
<
0
A 2I R AEI K :@JIN
W
I1% 7
and
8W> ? F :
9
$
* +
- - ./- - 7
* +
- - ./- - 7
<
* +
- - ./- W
7
7
0
I&% 7
W
J
R $='<
Theorem 3.44
>2
$
or more precisely,
!(
) &
O
.< /
C;
$
50;$43 <
In particular, the paths of Brownian motion are not differentiable.
2I R JI K :@
JIN
EWB Q$
156
$
)B87#: ><><><-? 0
Proof:
(Outline)
Let be a partition of
A ILK : @
A I . Define the sample quadratic variation
. To simplify notation, set
?#F :
Then
O
0
$
.
IR <
I1% 7
? F :
@
.
0
S$
We want to show that
I1% 7
*,+
-(- ./-(- 7
.IK :
@
IR
@
.
@.I EOJ<
$6<
Consider an individual summand
@
IR
JI K :@
JIN
$
.I K :L
Q@
This has expectation 0, so
For
, the terms
$
!
>
.
@
? F :
@
R
!
0
2
$
I1%7
0
? F :
$
9
$
I
A I K :
I1%7
(if
? F :
0
K :
@
@
0
I&% 7
0
.I>
EO $%'<
I% 7
I K :
@
I R
A I K :
@ I R
JI K :
@
JIC
;<
Thus we have
! GH
! C
@
2
$%'
@
2
9
.
.
I @
6.I K :@JIN
R O
IR
? F :
.IB
C<
.I>
EO
JI K :@.IB
@
@
I K :
@
IR
and
AEI K :@
@ I R
.IK :
@
IR
is normal with mean 0 and variance
I K :
I1%7
$%! G
@
IR
0
$
! G
I1%7
+
? F :
$
@
.
are independent, so
@SJI K :@
? F :
!G
.IB
JO R
<(;<
I R O
@
R
, then
!G
)
$
I*$
CHAPTER 14. The Itô Integral
As !
B
,
@
.
2
157
, so
*,+
-(- ./-(- 7
@
.
2
$6<
Remark 14.2 (Differential Representation) We know that
! G
.I K : @
JIN
R
JI K :
@
@JI EO $%<
We showed above that
!
When
IK :
@
I C
AJI K : M@
A ILK :
is small,
.I .
R
I R
@
which we can write informally as
@.IB
JOD$
.I K :
@
.I>
R <
is very small, and we have the approximate equation
JI K : Q@
JI K :
@
JI R
.
.
$
EI K :
@JIT
#L<
14.4 Quadratic Variation as Absolute Volatility
On any time interval
: O
R
, we can sample the Brownian motion at times
:
$
E7*9 :
96<><B< 9
$
8?
R
and compute the squared sample absolute volatility
? F :
3
0
:
I&% 7
>
M@
@
R
AJILK : @
AJIN
R <
This is approximately equal to
3
R
:
@
R
C
:
@
R
: JO$
R
@
:
$
3 <
As we increase the number of sample points, this approximation becomes exact. In other words,
Brownian motion has absolute volatility 1.
Furthermore, consider the equation
> $
$
3
L
<
7
This says that quadratic variation for Brownian motion accumulates at rate 1 at all times along
almost every path.
158
14.5 Construction of the Itô Integral
The integrator is Brownian motion
following properties:
1. 9.$
2.
.
every set in
is
J
9 :
3. For
are independent of
The integrand is
1.
2.
AJ
is
.
is also in -measurable,
9=<><><19
AJ
8?
.
AJ
AJ
+
, and the
,
,
A : M@
, the increments .
.
, with associated filtration
AE
+
R
@
: +C<B<C<
8?1
@
8? F : , where
-measurable
(i.e., is adapted)
is square-integrable:
!G
R .
2<
S
7
We want to define the Itô Integral:
!D.
$
7
Remark 14.3 (Integral w.r.t. a differentiable function) If
we can define
8W $
7
7
WA
'<
W.
is a differentiable function, then
<
This won’t work when the integrator is Brownian motion, because the paths of Brownian motion
are not differentiable.
14.6 Itô integral of an elementary integrand
Let $
)C 7 : ><><><' ? 0
be a partition of
'O
, i.e.,
$SE7*9:29=<><><19
.
Assume that
is constant on each subinterval
elementary process.
The functions Think of J
J
and
A I E?
$2<
.I1JILK : O
(see Fig. 14.2). We call such a
can be interpreted as follows:
as the price per unit share of an asset at time .
an
CHAPTER 14. The Itô Integral
159
δ( t ) = δ( t 1 )
δ( t )= δ( t 3 )
δ( t ) = δ( t )
0
t2
t1
0=t0
t4 = T
t3
δ( t ) = δ( t 2 )
Figure 14.2: An elementary function .
Think of E7#:+C<B<><
AJI>
8?
as the trading dates for the asset.
Think of
as the number of shares of the asset acquired at trading date
.I K :
trading date
.
! .
Then the Itô integral
! AJ
$
87>
In general, if
7 EI
9
9
87>
OJ
E7>
EO
7 EO
%
7 % 7
: @
JI K :
: +
: AJ
@
A
R
: EOJ
@
: EO
A
R
AE
M@
A
R
EO 0
% 7
C
K : M@
EO
AJIC
C
.
@
99
9
JIN
EOJ<
14.7 Properties of the Itô integral of an elementary process
Adaptedness For each
N! .
is
.
Linearity If
!DJ
$
-measurable.
7
then
! AJ
J
+
AE
$
/
$
7
7
/
:
99
:
,
I+F :
! .
$
and held until
can be interpreted as the gain from trading at time ; this gain is given by:
: @
87>
.
@
JI
R
9
R
<
160
s
t
t
.....
t l+1
l
t
t
k
k+1
Figure 14.3: Showing and in different partitions.
and
Martingale
!DJ
!DJ
$
7
<
is a martingale.
We prove the martingale property for the elementary process case.
Theorem 7.45 (Martingale Property)
I F :
!DJ
$
0
% 7
+
K : @
EO
.IB
C
.
@
AJIN
EOE
JI
99
EILK :
is a martingale.
be given. We treat the more
difficult case that: and areI inI different
Proof: Let
I
+-K O
K : O
subintervals, i.e., there are partition points and such that and
(See
Fig. 14.3).
S9
9
Write
JF :
!P.
$
0
% 7
+
I+F :
0
% K :
K : @
+
JO
K : @
A
8O
8K : @
AEIN
+
A
EO
.
@
.I EO
We compute conditional expectations:
JF :
!G
0
%7
! G
B
C
K : Q@
K : @
JF :
E
$
0
% 7
>
5$
A ! GH
$
A K : @
K : @
EO
A
-O @
<
CHAPTER 14. The Itô Integral
I+F :
!G
0
% K :
A
>
K : @
. We show that the third and fourth terms are zero.
!
These first two terms add up to
161
I+F :
.
$
!G
I F :
0
!G
% K :
$
%
0
!G
8K :
A
B
Q ! G
K : Q@
K : 7
%
EO @
.I >
.
@
AJIN
$=! G
Theorem 7.46 (Itô Isometry)
$SJI
Proof: To simplify notation, assume
I
Each
%
7
I
0
! R AJ
$
I
%
7
0
$
%
7
has expectation 0, and different
C
7
%
AJIN
<
K : @
O
R
JIB
8O @
R
0
R 7
are independent.
A
J
, so
0
! AJ
$
!TR AJ
$=! G
!G
.IC
! G
R
<
Since the cross terms have expectation zero,
I
!G
0
!1RBAJ
$
R I
0
$
$
! G
% 7
I
0
$=! G
$=! G
!G
% 7
!G
% 7
I
%
7
7
R RN
0
! G
>
K :
@
R R#
R O
K :
@
.
R !G
162
path of δ
path of δ
4
t2
t1
0=t0
t4 = T
t3
Figure 14.4: Approximating a general process by an elementary process
14.8 Itô integral of a general integrand
Fix
. Let be a process (not necessarily an elementary process) such that
AJ
!G
is
7
.
-measurable,
R .
O
,
S<
Theorem 8.47 There is a sequence of elementary processes
* +
? !G
7
? AJ
@
AJ
R
)
?0 ?
% :
#$=<
Proof: Fig. 14.4 shows the main idea.
In the last section we have defined
2
$
!?
7
?P.
AE
for every . We now define
7
AJ
.
$
*,+
? 7
? .
.
<
such that
, over
O
.
CHAPTER 14. The Itô Integral
163
The only difficulty with this approach is that we need to make sure the above limit exists. Suppose
and are large positive integers. Then
!
! ?
Q@
!
!G
$
!G
(Itô Isometry:)
$
R 9
R
9
R
$=! G
7
7
.
EO
? AE
M@
7
?D.
@
J
R
) !?
which is small. This guarantees that the sequence
J
R
.
.
EO R
? AE
M@
? J
M@
#! G
7
L0 ?
AE
Q@
% :
!G
#
O R
J
7
J
@
7
has a limit.
+<
Here is any adapted, square-integrable process.
Adaptedness. For each ,
! AJ
is
.
Linearity. If
!DJ
$
-measurable.
7
then
! AJ
J
and
!D.
!D.
Itô Isometry.
!G
AE
$
$
7
/
7
/
7
!DJ
$
Martingale.
Continuity.
+
+<
is a martingale.
is a continuous function of the upper limit of integration .
! R .
$6! G
7
R
.
Example 14.1 () Consider the Itô integral
We approximate the integrand as shown in Fig. 14.5
.
R
14.9 Properties of the (general) Itô integral
!DJ
$
164
2T/4
T/4
Figure 14.5: Approximating the integrand By definition,
)(+*
$#&%&'
We compute
F
!
0
, - .
?/1
,DE :
C(+*
B#&%&'
,
!
0
F
F
, - D
F
F
6
F
F
6
F
F
F
, E :
D
F
:
43
5/
17698
3+:
5/;1
<3
,>=
GH
0
.
!"
To simplify notation, we denote
so
O
, over
,.- 0/21
with
if
!
0
if
if
T
3T/4
3A@
!
0
,
,D- !
0
, - D
I
F
:
,.E :
,
0
,
, E .
F
!
0
6
,
, - D
,.E IJ- ,D- !
!
G
0
0
F
, ,DE , :
,D- ,D- !
0
,
,.E :
,
,D- 6
F
,
F
0
,D- ,
F
CHAPTER 14. The Itô Integral
G!
0
Therefore,
,D- 165
,
,DE :
,
F
F
F
:
G!
0
,DE :
, - .
or equivalently
0
,.- 5/21
Let
(
3
?/
1 6<8
34:
/1
3
F
F
F
:
F
,
!
0
5/
,D- @
1 698
3
3
/ 1
F
and use the definition of quadratic variation to get
:
Remark 14.4 (Reason for the
term) If
R
F
W
7
8W
W
F :
F
is differentiable with
M$
:
$
R
$
R
:
W/ W7
W#
/$6
, then
W R
W R
2
+<
7
In contrast, for Brownian motion, we have
7
:
$
;
@
R
R R
:
<
:
The extra term
there, because
R
comes from the nonzero quadratic variation of Brownian motion. It has to be
!G
but
7
$6
!G
(Itô integral is a martingale)
:
2
Q$
R
R :
;<
R
14.10 Quadratic variation of an Itô integral
Theorem 10.48 (Quadratic variation of Itô integral) Let
!DJ
$
Then
! C.
$
7
7
RN
+<
<
166
This holds
even if is not an elementary process.! The quadratic
variation formula says that at each
R time , the instantaneous absolute volatility of is
J.
This is the absolute volatility of the
) in the Brownian motion. Informally,
Brownian motion scaled by the size of the position (i.e.
we can write the quadratic variation formula in differential form as follows:
T! .
!D.
$
Compare this with
.
R .
#L<
#L<
.
$
)>-7 : ><><><8?'0
$
Proof:
anelementary
process ). Let be the partition for , i.e.,
JI*9
9.IK :
/$68?
JIB
(For
for
. To simplify notation, assume
. We have
? F :
8! >E
$
Let us compute
8! CJI
8! BJI>
K :
M@
0
$
. Let
JI2$
7*9
)
:
K : M@
!
$
8! BJI
0
$
@ @@@'@T
It follows that
0
I&% 7
?#F :
0
I&% 7
$
@8@ @'@'@ @ K :+
@
K : @
0
% 7
!D
R#
@
@
EIN
7
RB <
EO EO R
K : @
R JI>
>.IK :
R AEIN
>.I K :
? F :
8! BJ
$
F :
R .I>
!D
% 7
be a partition
JI K : <
EO1<
0
$
JI>
F :
$
.IC
so
: @
:>><><><'
9=<><>< 9
$
E! CAJILK
7B
Then
!D
8! >.I
8! C.I K : Q@
I1%7
.I +<
JO R
.
$
Chapter 15
Itô’s Formula
15.1 Itô’s formula for one Brownian motion
W
W/
J
We want a rule.
to “differentiate” expressions of the form , where
is a differentiable
function. If were also differentiable, then the ordinary chain rule would give
W
J
M$=W
AJ
J
+
which could be written in differential notation as
EW/
.
$=W
$=W8
J
J
AE
J
.
However, is not differentiable, and in particular has nonzero quadratic variation, so the correct
formula has an extra term, namely,
8W
AJ
$6W-
.
AE
:
W R
#
<
.
This is Itô’s formula in differential form. Integrating this, we obtain Itô’s formula in integral form:
W
AJ
@
W
#
$
7
7
W8
:
W 8
7
R
<
Remark 15.1 (Differential vs. Integral Forms) The mathematically meaningful form of Itô’s formula is Itô’s formula in integral form:
W
AJ
@
W
#
$
7
W8
167
R
:
7
W 8
<
168
This is because we have solid definitions for both integrals appearing on the right-hand side. The
first,
W
7
is an Itô integral, defined in the previous chapter. The second,
W 8
7
is a Riemann integral, the type used in freshman calculus.
For paper and pencil computations, the more convenient form of Itô’s rule is Itô’s formula in differential form:
:
8W
J
$6W
.
.
R
W
.
<
8W
J
+
.
There is an intuitive meaning but no solid definition for the terms
and appearing
in this formula. This formula becomes mathematically respectable only after we integrate it.
15.2 Derivation of Itô’s formula
Consider
Let
I W
:
$
I K :
R
R
, so that
W8
$
W 8 $
3#<
be numbers. Taylor’s formula implies
W ILK : @
ILK :
W IN
$
IN
WA
@
:
IC
R
ILK :
In this case, Taylor’s formula to second order is exact because
W
IB
RW(8
@
IC
+<
is a quadratic function.
general case, the above equation is only approximate, and the error is of the order of
InI the
. The total error will have limit zero in the last step of the following argument.
Fix
W
:
$
R ? F :
@
W
W
I&% 7
be a partition of
0
I&% 7
JI K : /@
W/
.IB
EO
JILK : @
JIN
EOBW8
$
JIN
JILK : @
:
JI ?#F :
0
I % 7 &
. Using Taylor’s formula, we write:
R
R ?#F :
$
'O
:
@
R
0
$
)>E7# : ><><C< E?0
$
and let .I JO
R
:
R
? F :
0
I% 7
JILK : @
JIN
EO R W(A
?#F :
0
I&% 7
ILK : @
.IK : M@
.I JO R <
.IB
CHAPTER 15. Itô’s Formula
We let 169
to obtain
W
2
@
W
#
$
7
$
W8
7
:
/
C 2
R
:
W(A
7
R
:
This is Itô’s formula in integral form for the special case
W :
$
R <
R
15.3 Geometric Brownian motion
Definition 15.1 (Geometric Brownian Motion) Geometric Brownian motion is
J
$
where
and
)#
.
/
:
@
R
R
*
are constant.
Define
W/A
$
- )#
:
@
R
R
so
.
$=W
Then
W
$
.
+<
:
@
R
R
*$
W W ,$
W W
R W<
According to Itô’s formula,
-WL
J
Q$
$=W
$
$
J
# W
:
@
.
R
:
R
R W
W
J
:
W .
R
R W
#
/
Thus, Geometric Brownian motion in differential form is
J
Q$
.
J
.
+
and Geometric Brownian motion in integral form is
.
$
- 7
7
<
<
170
15.4 Quadratic variation of geometric Brownian motion
In the integral form of Geometric Brownian motion,
J
$
7
the Riemann integral
AJ
M$
is differentiable with U
U
J
.
$
7
7
. This term has zero quadratic variation. The Itô integral
.
$
7
/
is not differentiable. It has quadratic variation
Thus the quadratic variation of
we write
.
AJ
$
7
R
R
<
is given by the quadratic variation of
J
$
AJ
J
J
R
$
. In differential notation,
R AJ
R
15.5 Volatility of Geometric Brownian motion
9
:
Fix
volatility of
9
R
on
. Let
: O
R
3
R
@
$
)>87#B<C<B<E?'0
O
R
. The squared absolute sample
is
? F :
0
Q: &
I % 7
JI K :
@
3
.I>
EO R
R
( : be a partition of
@
R
Q:
R
R
: R
:
As R
, the above approximation becomes exact. In other words, the instantaneous relative
R
volatility of is . This is usually called simply the volatility of .
15.6 First derivation of the Black-Scholes formula
7
Wealth of
an investor. An investor begins with nonrandom initial wealth
and at each time ,
.
shares of stock. Stock is modelled by a geometric Brownian motion:
holds
J
Q$
AJ
J
J
<
CHAPTER 15. Itô’s Formula
171
.
can be random, but must be adapted. The investor finances his investing by borrowing or
lending at interest rate .
Let
J
denote the wealth of the investor at time . Then
AJ
J
Q$
$
AJ
$
.
.
J
J
.
@
J
.
EO #
.
J
AE
EO
@
Risk premium
Value of an option. Consider an European option which pays
.
$
the value of this option
at
time
if
the
stock
price
is
O
is
option at each time
.
+<
L
AJ
$
$
$
$
:
R
:
R
R
A hedging portfolio
starts with some initial wealth
A
J
time tracks
. We saw above that
J
$
L
AJ
To
ensure that
coefficients, we obtain the
Equating the
#
J
Q$4
@
R
7
:
O
J
+<
R
# R
R
O
and invests so that the wealth
at time . Let
denote
. In other words, the value of the
J
AE
at each
<
for all , we equate coefficients in their differentials. Equating the
-hedging rule:
J
coefficients, we obtain:
J
JO) J
2
The differential of this value is
AE
J
@
:
R
$
R
R
$
.
<
@
<
, and we are seeking to cause to agree with . Making these substitutions,
(where and
) which simplifies to
In conclusion, we should let be the solution to the Black-Scholes partial differential equation
satisfying the terminal condition
If an investor starts with and uses the hedge , then he will have
.
for all , and in particular, But we have set
we obtain
$
A
$
E
$
A R
R
7
L
J
$
A -
:
2
J
$
$
R
@
.
:
R
R
:
R
R
R
$
$
R
<
$
+<
#
.
$
2
Q$
.
172
15.7 Mean and variance of the Cox-Ingersoll-Ross process
The Cox-Ingersoll-Ross model for interest rates is
where
and
.
$
J
:
.
.
R
@
7
R .
J
AE
+
W
. This is
R AJ
.
AE
.
AJ
R J
@
$
W .
#
@
$
The mean of
J
7
J
, where
W $
<
R
. We obtain
AJ
$6W8
$
-#
We apply Itô’s formula to compute
R AJ
$EW/
AJ
are positive constants. In integral form, this equation is
E
$
@
#Q@
.
J
R J
#
.
.
.
#
@
J
J
.
R
.
#
R
AE
. The integral form of the CIR equation is
-#
E
$
7
@
7
<
Taking expectations and remembering that the expectation of an Itô integral is zero, we obtain
Differentiation yields
which implies that
Integration yields
We solve for
!G
.
$
!G
J
$
!G
! G
J
!G
.
J
Q@
Q$
, then
!G
J
Q$
!G
J
$
* +
!G
$
J
$
<
J
+
3
.
<
!G
/
.
$
#
/@
, then
!G
F for every . If
7
@
AJ
!G
.
$
!G
$
@
7
$
P@
:
!G
If
$
@
<
3>
+<
<
exhibits mean reversion:
CHAPTER 15. Itô’s Formula
.
Variance of
173
. The integral form of the equation derived earlier for
R .
$
=
R
/
R R .
$
Differentiation yields
!G
which implies that
@
R J
$
R !G
R AJ
$
R
R
R
R
!
AJ
R J
@S ! G
$=! G
$
R
R
!G
is
7
!G
7
R
<
<
R J
+
R J
!G
R ! G
3
R
3
/
R .
.
+<
and integrating the last equation, after considerable
Q@
@
@
R AJ
!G
/ /
J
@
R J
R
7
R ! G
$
!G
!G
7
/ Using the formula already derived for
algebra we obtain
R $
R J
R ! G
%
R
7
Taking expectations, we obtain
!G
R
F
R
R
F R
/
@
#
3
F
R
<
J
R
Q@
3
/
F R
R
/
@
#
3
F
R
<
15.8 Multidimensional Brownian Motion
Definition 15.2 ( -dimensional Brownian Motion) A -dimensional Brownian Motion is a process
:
.
$
.
><B<C<'
.
with the following properties:
Each If
$
IE
is a one-dimensional
Brownian motion;
, then the processes
AJ
and J
are independent.
Associated with a -dimensional Brownian motion, we have a filtration
For each , the random vector For each 9
:
?
96<B<><19
.
.
is
AE
-measurable;
, the vector increments
are independent of
J
: Q@
J
+B<C<B<
8?T
@
8? F : ) AE
0
such that
174
15.9 Cross-variations of Brownian motions
Because each component is a one-dimensional
Brownian motion, we have the informal equation
.
#L<
J
$
However, we have:
Theorem 9.49 If
$
$
,
)B87#><B<C< 8?0
Proof: Let of and on
'O
.
J
$=
'O
be a partition of
. For
$
to be
? F :
.
0
I1% 7 $
JI K :
@
JIB
8O1
, define the sample cross variation
.I K :
@
JIN
EO#<
The increments appearing on the right-hand side of the above equation are all independent of one
another and all have mean zero. Therefore,
!G
!
We compute
? F :
.
0
I
R
JI .IK : Q@
? F :
. First note that
0
I&% 7 $
R
. $6<
.
A K : @
JON
JILK : @
K : @
R
.IN
JO <>
.IK : M@
.IC
EO1
AJILK : @
.I>
8O
All the increments appearing in the sum of cross terms are independent of one another and have
mean zero. Therefore,
!
C
. $%! G
.
0
I1% 7 $%! G
.I K : Q@
But expectation
.IK :
EIN
EO R
@.I !
As R
? F :
JI K :
@
.I K :L
M@
and . It follows that
JIN
EO R JI>
EO R
, we have
. $
!
B
AEI K : @
JIB
EO R <
? F :
0
AJI K :@JI R 9
I1% 7
. are independent of one another, and each has
? F :
>
0
I1% 7
JI K :@
.IB
$
!G
, so . converges to the constant
.
< ;<
$=
.
CHAPTER 15. Itô’s Formula
175
15.10 Multi-dimensional Itô formula
To keep the notation as simple as possible, we write the Itô formula for two processes driven by a
two-dimensional Brownian motion. The formula generalizes to any number of processes driven by
a Brownian motion of any number (not necessarily the same number) of dimensions.
Let
and
be processes of the form
J
$
7
-#
.
$
7
7
7
::
:
R
:
:
7
:
7
R
RR
+
R
R
+<
Such processes, consisting of a nonrandom initial condition, plus
a Riemann
integral,
plus one or
more Itô integrals, are called semimartingales. The integrands
and
can be any
adapted processes. The adaptedness of the integrands guarantees that and are also adapted. In
differential notation, we write
$
Given these two semimartingales
$
N:R :
$
$
$
$
$
WL
::
R
R :
:
R
:
R
:
R
:
:
RR
R
$
W
In integral form, with
is
W
$
7
where
W
J
+
W
7
$=W
:: W
W
+
R
: W
:
R
O
W
#
+
W
<
R
W
R
AE
:
.
and
:
R
R
be semimartingales. Then we
RR
W
W
O <
as decribed earlier and with all the variables filled in, this equation
W
W
and
AJ
Q@
>
R
1
1
RR
R
R
R
Let
be a function of three variables, and let
have the corresponding Itô formula:
:
8WL
R
: N:R R R R
7 R
R
RR
:
R
:
R R
:
:
:
::
:
:/
, the quadratic and cross variations are:
:
R
R R
RR
::
::
R
::
#
: : :R R :R : and
$
$
, for
R :
:
7
W
R
:
:R
:
) 3# T0
R
W
,
::
$
RR
W
:
R
O
:
R
R
W
RR
:
R
R
, and $
.
R :
R W
RR
O