Kenneth Kuttler

32.4. THE COVARIATION 871

variables which is Ft∧δ measurable because, for a given ω, when α p becomes larger thant, the sum in 32.11 loses its dependence on p. Thus from pointwise convergence in 32.11,[

Mδ

](t)≡ lim

n→∞∑k≥0

∥∥M(t ∧δ ∧ τ

nk+1)−M (t ∧δ ∧ τ

nk)∥∥2

In case δ = ∞, the above gives an Ft measurable random variable denoted by [M] (t) suchthat

[M] (t)≡ limn→∞

∑k≥0

∥∥M(t ∧ τ

nk+1)−M (t ∧ τ

nk)∥∥2

Now stopping with the stopping time δ , this shows that[Mδ

](t)≡ lim

n→∞∑k≥0

∥∥M(t ∧δ ∧ τ

nk+1)−M (t ∧δ ∧ τ

nk)∥∥2

= [M]δ (t)

That is, the quadratic variation of the stopped local martingale makes sense a.e. and equalsthe stopped quadratic variation of the local martingale.

This has now shown that

∥Mαn (t)∥2− [M]αn (t) = ∥Mαn (t)∥2− [Mαn ] (t)

= Nn (t) , Nn (t) a martingale

and both of the random variables on the left converge pointwise as n→ ∞ to a functionwhich is Ft measurable. Hence so does Nn (t). Of course Nn (t) is likewise a functionof αn ∧ t and so by Proposition 32.3.3 again, it converges pointwise to a Ft measurablefunction called N (t) and N (t) is a continuous local martingale.

It remains to consider the claim about the uniqueness. Suppose then there are twowhich work, [M] , and [M]1. Then [M]− [M]1 equals a local martingale G which is 0 whent = 0. Thus the uniqueness assertion follows from Corollary 32.3.5. ■

Here is a corollary which tells how to manipulate stopping times. It is contained in theabove proposition, but it is worth emphasizing it from a different point of view.

Corollary 32.3.8 In the situation of Proposition 32.3.7 let τ be a stopping time. Then

[Mτ ] = [M]τ .

Proof:

[M]τ (t)+N1 (t) =(∥M∥2

)τ

(t) = ∥Mτ∥2 (t) = [Mτ ] (t)+N2 (t)

where Ni is a local martingale. Therefore,

[M]τ (t)− [Mτ ] (t) = N2 (t)−N1 (t) ,

a local martingale. Therefore, by Corollary 32.3.5, this shows [M]τ (t)− [Mτ ] (t) = 0. ■

32.4 The CovariationDefinition 32.4.1 The covariation of two continuous H valued local martingalesfor H a separable Hilbert space M,N,M (0) = 0 = N (0) , is defined as follows.

[M,N]≡ 14([M+N]− [M−N])

32.4. THE COVARIATION 871variables which is ¥,, measurable because, for a given @, when @, becomes larger thant, the sum in 32.11 loses its dependence on p. Thus from pointwise convergence in 32.11,Me] (t) = lim Y ||M (tA SA th,,) —M (tA SAT)neIn case 5 = ©, the above gives an ¥, measurable random variable denoted by [M] (t) suchthat[mM] (0) = lim YY IM (0 th,1) —M CA B)|)k>0Now stopping with the stopping time 6, this shows thatnoom?| (1) = lim Y ||M (AS Atz,,) —M (ASAT) = (MPO)k>0That is, the quadratic variation of the stopped local martingale makes sense a.e. and equalsthe stopped quadratic variation of the local martingale.This has now shown thatlee (oP — [My = | (|? — [M2] @)= N,(t), N,(t) a martingaleand both of the random variables on the left converge pointwise as n — © to a functionwhich is ¥; measurable. Hence so does N,,(t). Of course N, (t) is likewise a functionof @, At and so by Proposition 32.3.3 again, it converges pointwise to a 7; measurablefunction called N (t) and N (rt) is a continuous local martingale.It remains to consider the claim about the uniqueness. Suppose then there are twowhich work, [M], and [M],. Then [M] — [M], equals a local martingale G which is 0 whent = 0. Thus the uniqueness assertion follows from Corollary 32.3.5.Here is a corollary which tells how to manipulate stopping times. It is contained in theabove proposition, but it is worth emphasizing it from a different point of view.Corollary 32.3.8 In the situation of Proposition 32.3.7 let t be a stopping time. Then[M*] = [My].Proof:Im} (0) +N (2) = (IMI?) @) =P) = (M0) +20)where JN; is a local martingale. Therefore,[M]* (t) — [M*](t) = N2(t)—Mi (1),a local martingale. Therefore, by Corollary 32.3.5, this shows [M]‘ (t) —[M*] (t) =0.32.4 The CovariationDefinition 32.4.1 The covariation of two continuous H valued local martingalesfor H a separable Hilbert space M,N,M (0) =0 = N (0), is defined as follows.[M,N] = -—((M+N]-—|[M—N])Ble