2470 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

Lemma 73.4.1 In Situation 73.2.1 the following formula holds for a.e. ω for 0 < s < twhere M (t) ≡

∫ t0 Z (u)dW (u) which has values in W. In the following, ⟨·, ·⟩ denotes the

duality pairing between V,V ′.

⟨BX (t) ,X (t)⟩= ⟨BX (s) ,X (s)⟩+

+2∫ t

s⟨Y (u) ,X (t)⟩du+ ⟨B(M (t)−M (s)) ,M (t)−M (s)⟩

−⟨BX (t)−BX (s)− (M (t)−M (s)) ,X (t)−X (s)− (M (t)−M (s))⟩

+2⟨BX (s) ,M (t)−M (s)⟩ (73.4.6)

Also for t > 0

⟨BX (t) ,X (t)⟩= ⟨BX0,X0⟩+2∫ t

0⟨Y (u) ,X (t)⟩du+2⟨BX0,M (t)⟩+

⟨BM (t) ,M (t)⟩−⟨BX (t)−BX0−BM (t) ,X (t)−X0−M (t)⟩ (73.4.7)

Proof: From the formula which is assumed to hold,

BX (t) = BX0 +∫ t

0Y (u)du+BM (t)

BX (s) = BX0 +∫ s

0Y (u)du+BM (s)

ThenBM (t)−BM (s)+

∫ t

sY (u)du = BX (t)−BX (s)

It follows that⟨B(M (t)−M (s)) ,M (t)−M (s)⟩−

⟨BX (t)−BX (s)− (M (t)−M (s)) ,X (t)−X (s)− (M (t)−M (s))⟩

+2⟨BX (s) ,M (t)−M (s)⟩

= ⟨B(M (t)−M (s)) ,M (t)−M (s)⟩−⟨BX (t)−BX (s) ,X (t)−X (s)⟩+2⟨BX (t)−BX (s) ,M (t)−M (s)⟩

−⟨B(M (t)−M (s)) ,M (t)−M (s)⟩+2⟨BX (s) ,M (t)−M (s)⟩

Some terms cancel and this yields

=−⟨BX (t)−BX (s) ,X (t)−X (s)⟩+2⟨BX (t) ,M (t)−M (s)⟩

=−⟨BX (t)−BX (s) ,X (t)−X (s)⟩+2⟨B(M (t)−M (s)) ,X (t)⟩

= −⟨B(X (t)−X (s)) ,X (t)−X (s)⟩

+2⟨

BX (t)−BX (s)−∫ t

sY (u)du,X (t)

⟩