67.9. SOME REPRESENTATION THEOREMS 2309

which proves uniqueness.With the above major result, here is another interesting representation theorem. Recall

that if you have an Ft adapted function f and f ∈ L2 (Ω× [0,T ] ;Rn) , then∫ t

0 fT dW is amartingale. The next theorem is sort of a converse. It starts with a Gt martingale andrepresents it as an Itô integral. In this theorem, Gt continues to be the filtration determinedby n dimensional Wiener process.

Theorem 67.9.5 Let M be an Gt martingale and suppose M (t) ∈ L2 (Ω) for all t ≥ 0.Then there exists a unique stochastic process, g(s,ω) such that g is Gt adapted and inL2 (Ω× [0, t]) for each t > 0, and for all t ≥ 0,

M (t) = E (M (0))+∫ t

0gT dW

Proof: First suppose f is an adapted function of the sort that g is. Then the followingclaim is the first step in the proof.

Claim: Let t1 < t2. Then

E(∫ t2

t1fT dW|Gt1

)= 0

Proof of claim: This follows from the fact that the Ito integral is a martingale adaptedto Gt . Hence the above reduces to

E(∫ t2

0fT dW−

∫ t1

0fT dW|Gt1

)=∫ t1

0fT dW−

∫ t1

0fT dW = 0.

Now to prove the theorem, it follows from Theorem 67.9.4 and the assumption that Mis a martingale that for t > 0 there exists ft ∈ L2 (Ω× [0,T ] ;Rn) such that

M (t) = E (M (t))+∫ t

0ft (s, ·)T dW

= E (M (0))+∫ t

0ft (s, ·)T dW.

Now let t1 < t2. Then since M is a martingale and so is the Ito integral,

M (t1) = E (M (t2) |Gt1) = E(

E (M (0))+∫ t2

0ft2 (s, ·)T dW|Gt1

)

= E (M (0))+E(∫ t1

0ft2 (s, ·)T dW

)Thus

M (t1) = E (M (0))+∫ t1

0ft2 (s, ·)T dW = E (M (0))+

∫ t1

0ft1 (s, ·)T dW

and so0 =

∫ t1

0ft1 (s, ·)T dW−

∫ t1

0ft2 (s, ·)T dW