Definition 62.3.1The covariation of two continuous H valued local martingales for Ha separable Hilbert space M,N,M
(0)
= 0 = N
(0)
,is defined as follows.
[M,N ] ≡ 1 ([M + N]− [M − N ])
4
Lemma 62.3.2The following hold for the covariation.
[M ] = [M, M ]
1 ( 2 2)
[M, N ] = local martingale+ 4 ||M + N ||− ||M − N||
= (M, N )+ local martingale.
Proof: From the definition of covariation,
2
[M ] = ||M || − N1
( )
[M, M ] = 1 ([M + M ]− [M − M ]) = 1 ||M + M ||2 − N2
4 4
= ||M ||2 − 1N2
4
where Ni is a local martingale. Thus
[M ]
−
[M, M ]
is equal to the difference of two
increasing continuous adapted processes and it also equals a local martingale. By
Corollary 62.2.3, this process must equal 0. Now consider the second claim.
1 1( 2 2 )
[M,N ] = 4 ([M + N ]− [M − N ]) = 4 ||M + N|| − ||M − N || + N
1
= (M,N )+ 4 N
where N is a local martingale. ■
Corollary 62.3.3Let M,N be two continuous local martingales, M
the difference of two increasing adapted processes. Also, this equals
τ τ τ
local martingale − (M ,N )+ (M ,N )
Claim:
(M τ,N)
−
(M τ,N τ)
=
(M τ,N − N τ)
is a local martingale. Let σn be a
localizing sequence for both M and M. Such a localizing sequence is of the form
τnM∧ τnN where these are localizing sequences for the indicated local submartingale.
Then obviously,
τ τ τ σn σ ∧τ σ σ ∧τ σ ∧τ
(− (M ,N )+ (M ,N )) = − (M n ,N n)+ (M n ,N n )
where Nσn and Mσn are martingales. To save notation, denote these by M and
N respectively. Now use Lemma 62.1.1. Let σ be a stopping time with two
values.
E ((M τ (σ),N (σ)− Nτ (σ ))) = E (E ((M τ (σ ),N (σ) − N τ (σ))|ℱτ))
Now Mτ
(σ)
is M
(σ ∧τ)
which is ℱτ measurable and so by the Doob optional sampling
theorem,
= E (M τ (σ),E (N (σ) − Nτ (σ) |ℱτ))
= E (M τ (σ),N (σ∧ τ)− N (τ ∧ σ)) = 0
while
E ((M τ (t),N (t)− N τ (t))) = E (E ((M τ (t),N (t) − Nτ (t))|ℱ τ))
Since Mτ
(t)
is ℱτ measurable,
= E ((M τ (t),E (N (t)− N τ (t)|ℱ τ)))
= E ((M τ (t),E (N (t ∧τ) − N (t∧ τ)))) = 0
2 ( 2 2 )
= ||M − M τ|| − ||M || + ||M τ|| − 2(M, M τ) + local martingale
= local martingale.
By the first part of the corollary which ensures
[M,M τ]
is of bounded variation, the left
side is the difference of two increasing adapted processes and so by Corollary 62.2.3
again, the left side equals 0. Thus from the above,