Y to get 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 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. ■ 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. with 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$_{1}$ . 16 Z$_{t}$ = v + ∫ t 0 σ$_{z}$ ( s ) dB 1 $_{s}$, dY$_{s}$ = Z$_{s}$ − Y$_{s}$ Σ$_{z}$( s ) s + σ $_{2}$ds + dB 2 $_{s}$, E [ X ˜ θ 1 ] < V (0 , Z$_{0}$, 0) = E [ X θ 1 ] , E [ X θ 1 ] = V (0 , Z$_{0}$, 0) . Y$_{t}$ = X$_{t}$ − (1 − t ) ( v + ∫ t 0 1 1 − s dX$_{s}$ − ∫ t 0 1 1 − s dB 2 s ) . Lemma 3.1 Consider the process Y$_{t}$ satisfying the stochastic differential equation