Chapter 33

Quadratic Variation and Stochastic In-tegration

Let Ft be a normal filtration and let {M (t)} be a continuous local martingale adapted toFt having values in U a separable real Hilbert space.

Definition 33.0.1 Let Ft be a normal filtration and let

f (t)≡mn−1

∑k=0

fkX(tk,tk+1] (t)

where {tk}mnk=0 is a partition of [0,T ] and each fk is Ftk measurable, fkM∗ ∈ L2 (Ω) where

M∗ (ω)≡ supt∈[0,T ]

∥M (t)(ω)∥

Such a function is called an elementary function. Also let {M (t)} be a continuous localmartingale adapted to Ft which has values in a separable real Hilbert space U such thatM (0) = 0. For such an elementary real valued function, define∫ t

0f dM ≡

mn−1

∑k=0

fk (M (t ∧ tk+1)−M (t ∧ tk)) . (33.1)

Since the t→Ft is increasing, this definition is well defined. Also the set of elementaryfunctions is a vector space.

Then with this definition, here is a wonderful lemma.

Lemma 33.0.2 For f an elementary function as above,{∫ t

0 f dM}

is a continuous localmartingale and

E

(∥∥∥∥∫ t

0f dM

∥∥∥∥2

U

)=∫

Ω

∫ t

0f (s)2 d [M] (s)dP. (33.2)

If N is another continuous local martingale adapted to Ft and both f ,g are elementaryfunctions such that for each k,

fkM∗,gkN∗ ∈ L2 (Ω) ,

then

E((∫ t

0f dM,

∫ t

0gdN

)U

)=∫

Ω

∫ t

0f gd [M,N] (33.3)

and both sides make sense.

Proof: Let {τ l} be a localizing sequence for M such that Mτ l is a bounded martingale.Then from the definition, for each ω∫ t

0f dM = lim

l→∞

∫ t

0f dMτ l = lim

l→∞

(∫ t

0f dM

)τ l

and it is clear that{∫ t

0 f dMτ l}

is a martingale because it is just the sum of some martin-gales. Thus {τ l} is a localizing sequence for

∫ t0 f dM. It is also clear

∫ t0 f dM is continuous

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