63.2. THE QUADRATIC VARIATION 2121

Lemma 63.2.4 Let X (t) be continuous and adapted to a normal filtration Ft and let η bea stopping time. Then if K is a closed set,

τ ≡ inf{t > η : X (t) ∈ K}

is also a stopping time.

Proof: First consider Y (t) = X (t ∨η)− X (η) . I claim that Y (t) is adapted to Ft .Consider U and open set and [Y (t) ∈U ] . Is it in Ft? We know it is in Ft∨η . It equals

([Y (t) ∈U ]∩ [η ≤ t])∪ ([Y (t) ∈U ]∩ [η > t])

Consider the second of these sets. It equals

([X (η)−X (η) ∈U ]∩ [η > t])

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 63.2.5 Let M (t) be a continuous local martingale for t ∈ [0,T ] having valuesin H a separable Hilbert space adapted to the normal filtration {Ft} such that M (0) = 0.Then there exists a unique continuous, increasing, nonnegative, local submartingale [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 62.7.10 on Page 2085. See alsoLemma 63.2.4. 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