Topics In Analysis

62.3 The Covariation

Proof: From the definition of covariation,

2 [M ] = ||M || − N1

where N_i is a local martingale. Thus − 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. where N is a local martingale. ■

Proof: Since

is the difference of increasing functions, it is of bounded variation.

which equals a local martingale from the definition of and . It remains to verify the claim about the stopping time. Using Corollary 62.2.6

1 [M,N ]τ =- ([M + N]− [M − N ])τ 4

The really interesting part is the next equality. This will involve Corollary 62.2.3.

[M, N ]τ − [M τ,N ] = [M τ,N τ]− [M τ,N ]

the difference of two increasing adapted processes. Also, this equals

τ τ τ local martingale − (M ,N )+ (M ,N )

Claim: 

− = is a local martingale. Let σ_n be a localizing sequence for both M and M. Such a localizing sequence is of the form τ_n^M ∧ τ_n^N 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 N^σ_n and M^σ_n 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 M^τ

is M which is ℱ_τ 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 M^τ

is ℱ_τ measurable,

This shows the claim is true.

Now from 62.3.11 and Corollary 62.3.3,

[M, N]τ − [M τ,N] = 0.

Similarly

[M, N ]τ − [M, N τ] = 0

Now consider the next claim that

= −. From the definition, it follows

[M − M τ]− ([M ]+ [M τ]− 2 [M, M τ])

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 62.2.3 again, the left side equals 0. Thus from the above,

Finally consider the claim that

is bilinear. From the definition, letting M₁,M₂,N be H valued local martingales,

Hence

[aM + bM ,N ]− (a [M ,N]+ b[M ,N]) = local martingale. 1 2 1 2

The left side can be written as the difference of two increasing functions thanks to

of bounded variation and so by Lemma 62.1.5 it equals 0. is obviously symmetric from the definition. ■