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: where h ( y ) = H ( y, 1) and f ( Z$_{1}$ ) is the final payoff of the asset. 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 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 12 F ( Z$_{t}$, t ) = F ( z$_{0}$, 0) + ∫ t 0 F$_{z}$ ( Z$_{s}$, s ) dZ$_{s}$ H ( Y$_{t}$, t ) = H (0 , 0) + t 0 H$_{y}$ ( Y$_{s}$$_{−}$, s ) dY$_{s}$ + ∫ t 0 H$_{t}$ ( Y$_{s}$ − , s ) ds + 1 2 ∫ t 0 H$_{yy}$ ( Y$_{s}$$_{−}$, s ) d [ Y ]$_{s}$ + ∑ s ≤ t [Δ H ( Y$_{s}$, s ) − H$_{y}$ ( Y$_{s}$$_{−}$, s )Δ Y$_{s}$ ] . Proof. E [ X θ 1 ] = E [∫ 1 0 ( F ( Z$_{s}$, s ) − H ( Y$_{s}$$_{−}$, s )) dθ$_{s}$ + ∫ 1 0 θ$_{s}$$_{−}$dF ( Z$_{s}$, s ) + [ θ, F ( Z, · ) − H ( Y, · )]$_{1}$ ] . h ( Y ∗ 1 ) = H ( Y ∗ 1 , 1) = f ( Z$_{1}$ ) , (3.10) (Sufficiency) For any admissible trading strategy, by using integration by parts for semimartingales ([14], Corollary II.6.2, p. 68), we have 1. The process θ ∗ t is continuous and has bounded variation. 2. The total order, Y ∗ t = θ ∗ t + B 2 $_{t}$, satisfies