62.5 The Quadratic Variation And Stochastic Integration
Let ℱ_{t} be a normal filtration and let
{M (t)}
be a continuous local martingale adapted
to ℱ_{t} having values in U a separable real Hilbert space.
Definition 62.5.1Let ℱ_{t}be a normal filtration and let
n∑−1
f (t) ≡ fkX(tk,tk+1](t)
k=0
where
{tk}
_{k=0}^{n}is a partition of
[0,T]
and each f_{k}is ℱ_{tk}measurable, f_{k}M^{∗}∈ L^{2}
(Ω )
where
M ∗ (ω ) ≡ sup ||M (t)(ω)||
t∈[0,T]
Such a function is called an elementary function.Also let
{M (t)}
be a local martingaleadapted to ℱ_{t}which has values in a separable real Hilbert space U such that M
(0)
= 0.For such an elementary real valued function define
∫ t n∑−1
fdM ≡ fk(M (t∧ tk+1) − M (t∧tk)).
0 k=0
Then with this definition, here is a wonderful lemma.
Lemma 62.5.2For f an elementary function as above,
{ ∫ }
t0 fdM
is a continuouslocal martingale and
( )
||||∫ t ||||2 ∫ ∫ t 2
E |||| fdM |||| = f (s) d[M ](s)dP. (62.5.13)
0 U Ω 0
(62.5.13)
If N is another continuous local martingale adapted to ℱ_{t}and both f,g are elementaryfunctions such that for each k,
f M ∗,g N ∗ ∈ L2 (Ω ),
k k
then
( (∫ t ∫ t ) ) ∫ ∫ t
E fdM, gdN = f gd[M, N ] (62.5.14)
0 0 U Ω 0
(62.5.14)
and both sides make sense.
Proof: Let
{τl}
be a localizing sequence for M such that M^{τl} is a bounded
martingale. Then from the definition, for each ω
( )
∫ t ∫ t τl ∫ t τl
0 fdM = lli→m∞ 0 fdM = lli→m∞ 0 fdM
and it is clear that
{ }
∫ tf dM τl
0
is a martingale because it is just the sum of
some martingales. Thus
{τl}
is a localizing sequence for ∫_{0}^{t}fdM. It is also
clear ∫_{0}^{t}fdM is continuous because it is a finite sum of continuous random
variables.
Next consider the formula which is really a version of the Ito isometry. There is no
loss of generality in assuming the mesh points are the same for the two elementary
functions because if not, one can simply add in points to make this happen. It suffices to
consider 62.5.14 because the other formula is a special case. To begin with, let
{τl}
be a
localizing sequence which makes both M^{τl} and N^{τl} into bounded martingales. Consider
the stopped process.
(( ∫ t ∫ t ) )
E fdM τl, gdN τl
0 0 U
( (
n∑−1 τl τl
= E fk (M (t∧ tk+1)− M (t∧ tk)) ,
n−1 k=0 ))
∑ g (N τl (t∧ t ) − Nτl (t∧t ))
k=0 k k+1 k
Now consider the claim about the measures. It was just shown that
[(M + N)− (M +N )s] ≤ 2([M − M s]+[N − N s])
and from Corollary 62.3.3 this implies that for t > s
[M + N ](t)− [M + N ](s∧ t)
= [M + N ](t)− [M + N ]s(t)
s s
= [M + N − (M + N )](t)
= [M − M s + (N − N s)](t)
≤ 2[M − M s](t)+ 2 [N − N s](t)
≤ 2([M ](t)− [M ](s)) + 2([N ](t)− [N](s))
Thus
ν ([s,t]) ≤ 2(ν ([s,t]) +ν ([s,t]))
M+N M N
By regularity of the measures, this continues to hold with any Borel set F in place of
[s,t]
. ■
Theorem 62.5.5The integral is well defined and has a continuous version which is alocal martingale.Furthermore it satisfies the Ito isometry,
(||∫ t ||2) ∫ ∫ t
E |||| f dM |||| = f (s)2 d[M ](s)dP
|| 0 ||U Ω 0
Let the norm on G_{N}∩G_{M}be the maximum of the norms on G_{N}and G_{M}and denote byℰ_{N}and ℰ_{M}the elementary functions corresponding to the martingales N and Mrespectively. Define G_{NM}as the closure in G_{N}∩G_{M}of ℰ_{N}∩ℰ_{M}. Then forf,g ∈G_{NM},
( (∫ ∫ )) ∫ ∫
t t t
E 0 fdM, 0 gdN = Ω 0 fgd[M, N ] (62.5.20)
(62.5.20)
Proof: It is clear the definition is well defined because if
{fn}
and
{gn}
are two
sequences of elementary functions converging to f in L^{2}
( 2 )
Ω;L ([0,T],ν(⋅))
and if
∫_{0}^{1t}fdM is the integral which comes from