62.3 The Covariation
Definition 62.3.1 The covariation of two continuous H valued local martingales for H
a separable Hilbert space M,N,M
= 0 =
, is defined as follows.
Lemma 62.3.2 The following hold for the covariation.
Proof: From the definition of covariation,
is a local martingale. Thus
is equal to the difference of two
increasing continuous adapted processes and it also equals a local martingale. By
, this process must equal 0. Now consider the second claim.
is a local martingale. ■
Corollary 62.3.3 Let M,N be two continuous local martingales, M
in Proposition 62.2.5. Then
is of bounded variation and
is a local martingale. Also for τ a stopping time,
In addition to this,
is bilinear and symmetric.
is the difference of increasing functions, it is of bounded
which equals a local martingale from the definition of
. It remains
to verify the claim about the stopping time. Using Corollary
The really interesting part is the next equality. This will involve Corollary 62.2.3
the difference of two increasing adapted processes. Also, this equals
is a local martingale. Let
localizing sequence for both M
Such a localizing sequence is of the form
τnM ∧ τnN
where these are localizing sequences for the indicated local submartingale.
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
measurable and so by the Doob optional sampling
This shows the claim is true.
Now from 62.3.11 and Corollary 62.3.3,
Now consider the next claim that
. From the definition, it
By the first part of the corollary which ensures
is of bounded variation, the left
side is the difference of two increasing adapted processes and so by Corollary
again, the left side equals 0. Thus from the above,
Finally consider the claim that
is bilinear. From the definition, letting
valued local martingales,
The left side can be written as the difference of two increasing functions thanks to
of bounded variation and so by Lemma
it equals 0.
symmetric from the definition.