Kenneth Kuttler

872 CHAPTER 32. QUADRATIC VARIATION

Lemma 32.4.2 The following hold for the covariation.

[M] = [M,M]

[M,N] = local martingale+14

(∥M+N∥2−∥M−N∥2

)= (M,N)+ local martingale.

Proof: From the definition of covariation,

[M] = ∥M∥2−N1

[M,M] =14([M+M]− [M−M]) =

(∥M+M∥2−N2

)= ∥M∥2− 1

4N2

where Ni is a local martingale. Thus [M]− [M,M] is equal to the difference of two in-creasing continuous adapted processes and it also equals a local martingale. By Corollary32.3.5, this process must equal 0. Now consider the second claim.

[M,N] =14([M+N]− [M−N]) =

(∥M+N∥2−∥M−N∥2 +N

)= (M,N)+

14N

where N is a local martingale. ■

Corollary 32.4.3 Let M,N be two continuous local martingales,

M (0) = N (0) = 0,

as in Proposition 32.3.7. Then [M,N] is of bounded variation and

(M,N)H − [M,N]

is a local martingale. Also for τ a stopping time,

[M,N]τ = [Mτ ,Nτ ] = [Mτ ,N] = [M,Nτ ] .

In addition to this,[M−Mτ ] = [M]− [Mτ ]≤ [M]

and alsoM,N→ [M,N]

is bilinear and symmetric.

Proof: Since [M,N] is the difference of increasing functions, it is of bounded variation.

(M,N)H − [M,N] =

(M,N)H︷︸︸︷14

(∥M+N∥2−∥M−N∥2

)

−

[M,N]︷︸︸︷14([M+N]− [M−N])

872 CHAPTER 32. QUADRATIC VARIATIONLemma 32.4.2 The following hold for the covariation.[(M] = |M,M]1[M,N] = local martingale + 4 (i+? — IM —NI\’)= (M,N) + local martingale.Proof: From the definition of covariation,[M] = ||M\|*—.%MM) = 2 ((M+M|—(M—M)) =< (\Im+m\?—.15)= MP 5%where -% is a local martingale. Thus [M]—[M,M] is equal to the difference of two in-creasing continuous adapted processes and it also equals a local martingale. By Corollary32.3.5, this process must equal 0. Now consider the second claim.1(MN) = (M+N|-[M—N)) =; (M+? - MNP +7)= (M,N)+ wawhere .¥% is a local martingale. HiCorollary 32.4.3 Let M,N be two continuous local martingales,M (0) = N(0) =0,as in Proposition 32.3.7. Then [M,N] is of bounded variation and(M,N) xq — [M,N]is a local martingale. Also for Tt a stopping time,[M,N]° = [M*,N*| =[M*,N] =[M,N‘}.In addition to this,[M —M*] = [M] —[M"] < [M|and alsois bilinear and symmetric.Proof: Since [M,N] is the difference of increasing functions, it is of bounded variation.(MLN) 4(MN), — [M,N] = 5 (\IM+N|? IM —NIP)[M,N]—7((M +N] —[M —N))