2472 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE
Proof: Consider the formula in Lemma 73.4.1.
⟨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)⟩
−⟨B(X (t)−X (s)− (M (t)−M (s))) ,X (t)−X (s)− (M (t)−M (s))⟩+2⟨BX (s) ,M (t)−M (s)⟩ (73.4.9)
Now let t j denote a point of Pk from Lemma 73.1.1. Then for t j > 0,X (t j) is just the valueof X at t j but when t = 0, the definition of X (0) in this step function is X (0)≡ 0. Thus
m−1
∑j=1
⟨BX(t j+1
),X(t j+1
)⟩−⟨BX (t j) ,X (t j)
⟩+⟨BX (t1) ,X (t1)⟩−⟨BX0,X0⟩
= ⟨BX (tm) ,X (tm)⟩−⟨BX0,X0⟩
Using the formula in Lemma 73.4.1, for t = tm this yields
⟨BX (tm) ,X (tm)⟩−⟨BX0,X0⟩= 2m−1
∑j=1
∫ t j+1
t j
⟨Y (u) ,X rk (u)⟩du+
+2m−1
∑j=1
⟨B∫ t j+1
t j
Z (u)dW,X (t j)
⟩
+m−1
∑j=1
⟨B(M(t j+1
)−M (t j)
),M(t j+1
)−M (t j)
⟩
−m−1
∑j=1
⟨B(X(t j+1
)−X (t j)−
(M(t j+1
)−M (t j)
)),
X(t j+1
)−X (t j)−
(M(t j+1
)−M (t j)
)⟩+2∫ t1
0⟨Y (u) ,X (t1)⟩du+2
⟨BX0,
∫ t1
0Z (u)dW
⟩+ ⟨BM (t1) ,M (t1)⟩
−⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩ (73.4.10)
First consider
2∫ t1
0⟨Y (u) ,X (t1)⟩du+2
⟨BX0,
∫ t1
0Z (u)dW
⟩+ ⟨BM (t1) ,M (t1)⟩ .