2170 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Lemma 63.8.2 Let {X (t)} be a real martingale adapted to the filtration Ft for t ∈ [a,b]

some interval such that for all t ∈ [a,b] ,E(

X (t)2)< ∞. Then

{X (t)2− t

}is also a mar-

tingale if and only if whenever s < t,

E((X (t)−X (s))2 |Fs

)= t− s.

Proof: Suppose first{

X (t)2− t}

is a real martingale. Then since {X (t)} is a martin-gale,

E((X (t)−X (s))2 |Fs

)= E

(X (t)2−2X (t)X (s)+X (s)2 |Fs

)= E

(X (t)2 |Fs

)−2E (X (t)X (s) |Fs)+X (s)2

= E(

X (t)2 |Fs

)−2X (s)E (X (t) |Fs)+X (s)2

= E(

X (t)2 |Fs

)−2X (s)2 +X (s)2

= E(

X (t)2− t|Fs

)+ t−X (s)2

= X (s)2− s+ t−X (s)2 = t− s

Next suppose E((X (t)−X (s))2 |Fs

)= t− s. Then since {X (t)} is a martingale,

t− s = E(

X (t)2−X (s)2 |Fs

)= E

(X (t)2− t|Fs

)+ t−X (s)2

and so0 = E

(X (t)2− t|Fs

)−(

X (s)2− s)

which proves the converse.

Theorem 63.8.3 Suppose {X (t)} is a real stochastic process which satisfies all the con-ditions of a real Wiener process except the requirement that it be continuous. Then both{X (t)} and

{X (t)2− t

}are martingales.

Proof: First define the filtration to be

Ft ≡ σ (X (s)−X (r) : r ≤ s≤ t) .

Claim: If A ∈Fs, then∫Ω

XA (X (t)−X (s))dP = P(A)∫

Ω

(X (t)−X (s))dP.

Proof of claim: Let G denote those sets of Fs for which the above formula holds.Then it is clear that G is closed with respect to countable unions of disjoint sets and