63.8. LEVY’S THEOREM 2169

and so passing to the limit as k→ ∞ with the uniform convergence yields

E(∫

(0,a]λ ∧A(s)dA(s)

)= E

(∫(0,a]

λ ∧A− (s)dA(s))

Now let λ → ∞. Then from the monotone convergence theorem,

E(∫

(0,a]A(s)dA(s)

)= E

(∫(0,a]

A− (s)dA(s))

and so for a.e. ω, ∫(0,a]

(A(s)−A− (s))dA(s) = 0.

Thus letting the measure associated with this Lebesgue integral be denoted by µ,

A(s)−A− (s) = 0 µ a.e.

Suppose then that A(s)−A− (s) > 0. Then µ ({s}) = 0 = A(s)−A(s−) , a contradiction.Hence A(s)−A− (s) = 0 for all s. It is already the case that s→ A(s) is right continuous.Therefore, this proves the theorem.

Example 63.7.16 Suppose {M (t)} is a continuous martingale. Assume

supt∈[0,a]

||M (t)||L2(Ω) < ∞

Then {||M (t)||} is a submartingale and so is{||M (t)||2

}. By Example 63.7.11, this is DL.

Then there exists a unique Doob Meyer decomposition,

||M (t)||2 = Y (t)+ ⟨||M (t)||⟩

where Y (t) is a martingale and {⟨||M (t)||⟩} is a submartingale which is continuous, nat-ural, increasing and equal to 0 when t = 0. This submartingale is called the quadraticvariation.

63.8 Levy’s TheoremThis remarkable theorem has to do with when a martingale is a Wiener process. The proofI am giving here follows [44].

Definition 63.8.1 Let W (t) be a stochastic process which has the properties that whenevert1 < t2 < · · ·< tm, the increments {W (ti)−W (ti−1)} are independent and whenever s< t, itfollows W (t)−W (s) is normally distributed with variance t−s and mean 0. Also t→W (t)is Holder continuous with every exponent γ < 1/2 and W (0) = 0. This is called a Wienerprocess.

First here is a lemma.