content_image/324550.jpg content_image/324549.jpg content_image/324547.jpg
Thus, this result gives the characterization of the optimal insider’s trading strategy in a slightly more general form than stated in Theorem 3.1. This is summarized in the following proposition. 

Proposition 3.1 Suppose that Assumptions 2.1, 2.2, 3.1 and 3.2 are satisfied. Then, given an admissible pricing rule H ∈ H satisfying the partial differential equation (PDE) (3.7), an admissible trading strategy θ ∗ ∈ A ( H ) is optimal for the insider if and only if: 

1. The process θ ∗ t is continuous and has bounded variation. 

2. The total order, Y ∗ t = θ ∗ t + B 2 $_{t}$, satisfies 

where h ( y ) = H ( y, 1) and f ( Z$_{1}$ ) is the final payoff of the asset. 

## Proof. 

<img src='content_image/324550.jpg'>

(Sufficiency) For any admissible trading strategy, by using integration by parts for semimartingales ([14], Corollary II.6.2, p. 68), we have 

<img src='content_image/324549.jpg'>

By applying Itˆ o formula for semimartingales ([14], Theorem II.6.33, p. 81) to H ( y, t ) and F ( z, t ), and using the fact that F ( Z$_{t}$, t ) is a true martingale, we get 

<img src='content_image/324547.jpg'>

Since Y$_{t}$ = θ$_{t}$ + B 2 $_{t}$, we have that [ Y ]$_{t}$ = t + 〈 θ $^{c}$〉 $_{t}$+ 2 〈 θ $^{c}$, B $^{2}$〉 $_{t}$+ ∑ s ≤ $_{t}$(Δ θ$_{s}$ ) $^{2}$. Therefore, using 

