the insider to reveal her information not only at the market terminal time, but also in the interim times whenever the market maker changes her weighting function. (Sufficiency) As in the proof of Proposition 3.1, for any admissible trading strategy we have On the other hand, consider the functions where ξ ∗ $_{i}$( z ) is the solution of H ( ξ ∗ $_{i}$( z ) , t$_{i}$ ) = F ( z, t$_{i}$ ). For t ≤ t$_{i}$ let These functions are well defined (it is easy to check that E [ | J $^{i}$( ∫ t$_{i}$ 0 w ( s ) dB 2 $_{s}$, Z$_{t}$$_{i}$ ) | ] < ∞ ) and satisfy the partial differential equation with terminal condition V $^{i}$( ξ, z, t$_{i}$ ) = J $^{i}$( ξ, z ). Therefore, V $^{i}$( ξ, z, t$_{i}$ ) ≥ V ( ξ ∗ $_{i}$( z ) , z, t$_{i}$ ) = 0 for any fixed z and any ξ = ξ ∗ $_{i}$( z ). Moreover, since H ( ξ, t ) is a nondecreasing continuous function of ξ , and due to the definition of an admissible pricing rule, we have 21 V i $_{ξ}$( ξ, z, t ) + F ( z, t ) − H ( ξ, t ) = 0 . V i t ( ξ, z, t ) + w $^{2}$( s ) 2 V i $_{ξξ}$( ξ, z, t ) + σ 2 $_{z}$( t ) 2 V i $_{zz}$( ξ, z, t ) = 0 V $^{i}$( ξ, z, t ) = E [ J i ( ξ + ∫ t$_{i}$ t w ( s ) dB 2 $_{s}$, z + ∫ t$_{i}$ t σ$_{z}$ ( s ) dB 1 s )] . J $^{i}$( ξ, z ) = ∫ ξ $^{∗}$( z ) ξ ( F ( z, t$_{i}$ ) − H ( x, t$_{i}$ )) dx, E [ X θ 1 ] = E [∫ 1 0 ( F ( Z$_{s}$, s ) − H ( ξ$_{s}$$_{−}$, s )) dθ$_{s}$ + ∫ 1 0 θ$_{s}$$_{−}$dF ( Z$_{s}$, s ) + ∫ 1 0 F$_{z}$ ( Z$_{s}$, s ) d [ θ $^{c}$, Z ]$_{s}$ − ∫ 1 0 H$_{ξ}$ ( ξ$_{s}$$_{−}$, s ) w ( s ) d [ θ $^{c}$]$_{s}$ − ∫ 1 0 H$_{ξ}$ ( ξ$_{s}$$_{−}$, s ) w ( s ) d [ θ $^{c}$, B $^{2}$] s − ∑ s ≤ 1 Δ H ( ξ$_{s}$, s )Δ θ$_{s}$   . Proof.