content_image/310809.jpg content_image/310808.jpg content_image/310807.jpg content_image/310806.jpg content_image/310805.jpg
Y to get 

<img src='content_image/310809.jpg'>

Notice that Y$_{t}$ is continuous, therefore θ$_{t}$ has bounded variation almost surely. Moreover, H ( Y$_{1}$, 1) = f ( Z$_{1}$ ) almost surely, hence this choice of θ gives 

<img src='content_image/310808.jpg'>

Since, by the sufficiency proof, we have that for any ˜ θ$_{t}$ which is either not continuous or does not satisfy equation (3.10) 

we know that any such ˜ θ$_{t}$ is not optimal. 

<img src='content_image/310807.jpg'>

■ 

From this characterization result, it follows that the θ ∗ t given by (3.9) is an optimal insider trading strategy given an admissible pricing rule H ∗ satisfying (3.7) and (3.8). Establishing the rationality of the pricing rule H $^{∗}$, on the other hand, is not so direct. Therefore, to set up the stage for proving that the ( H $^{∗}$, θ$^{∗}$) given in Theorem 3.1 is indeed an equilibrium, we first need to demonstrate the following lemma. 

Lemma 3.1 Consider the process Y$_{t}$ satisfying the stochastic differential equation 

with 

<img src='content_image/310806.jpg'>

<img src='content_image/310805.jpg'>

where B 1 t and B 2 t are two independent standard Brownian motions, v is N (0 , σ ) independent of F B $^{1}$,B2 1 and Σ$_{z}$( t ) = ∫ t 0 σ 2 $_{z}$( s ) ds . Suppose that σ , σ$_{z}$ ( t ) and Σ$_{z}$( t ) satisfy Assumptions 2.1, 2.2, 3.1 and 3.2. Then, on the filtration ¯ F Y t , the process Y$_{t}$ is a standard Brownian motion and Y$_{1}$ = Z$_{1}$ . 

