866 CHAPTER 32. QUADRATIC VARIATION

If 0 ∈U, then it reduces to [η > t] ∈Ft . If 0 /∈U, then it reduces to /0 still in Ft . Nextconsider the first set. It equals

[X (t ∨η)−X (η) ∈U ]∩ [η ≤ t]

= [X (t ∨η)−X (η) ∈U ]∩ [t ∨η ≤ t] ∈Ft

from the definition of Ft∨η . (You know that [X (t ∨η)−X (η) ∈U ] ∈Ft∨η and so whenthis is intersected with [t ∨η ≤ t] one obtains a set in Ft . This is what it means to be inFt∨η .) Now τ is just the first hitting time of Y (t) of the closed set. ■

Proposition 32.3.7 Let M (t) be a continuous local martingale for t ∈ [0,T ] havingvalues in H a separable Hilbert space adapted to the normal filtration {Ft} such thatM (0) = 0. Then there exists a unique continuous, increasing, nonnegative, local sub-martingale [M] (t) called the quadratic variation such that

∥M (t)∥2− [M] (t)

is a real local martingale and [M] (0) = 0. Here t ∈ [0,T ] . If δ is any stopping time[Mδ

]= [M]δ

Proof: First it is necessary to define some stopping times. Define stopping timesτn

0 ≡ ηn0 ≡ 0.

ηnk+1 ≡ inf

{s > η

nk : ∥M (s)−M (ηn

k)∥= 2−n} ,τ

nk ≡ η

nk ∧T

where inf /0 ≡ ∞. These are stopping times by Example 31.3.13 on Page 841. See also theabove Lemma 32.3.6. Then for t > 0 and δ any stopping time, and fixed ω, for some k,

t ∧δ ∈ Ik (ω) , I0 (ω)≡ [τn0 (ω) ,τn

1 (ω)] , Ik (ω)≡ (τnk (ω) ,τn

k+1 (ω)] some k

Here is why. The sequence{

τnk (ω)

}∞

k=1 eventually equals T for all n sufficiently large.This is because if it did not, it would converge, being bounded above by T and then bycontinuity of M,

{M(τn

k (ω))}∞

k=1 would be a Cauchy sequence contrary to the requirementthat ∥∥M

(τ

nk+1 (ω)

)−M (τn

k (ω))∥∥

=∥∥M(η

nk+1 (ω)

)−M (ηn

k (ω))∥∥= 2−n.

Note that if δ is any stopping time, then∥∥M(t ∧δ ∧ τ

nk+1)−M (t ∧δ ∧ τ

nk)∥∥

=∥∥∥Mδ

(t ∧ τ

nk+1)−Mδ (t ∧ τ

nk)∥∥∥≤ 2−n

You can see this is the case by considering the cases, t ∧ δ ≥ τnk+1, t ∧ δ ∈ [τn

k ,τnk+1), and

t ∧ δ < τnk . It is only this approximation property and the fact that the τn

k partition [0,T ]which is important in the following argument.

Now let αn be a localizing sequence such that Mαn is bounded as in Proposition 32.3.3.Thus Mαn (t) ∈ L2 (Ω) and this is all that is needed. In what follows, let δ be a stopping