content_image/296800.jpg content_image/296798.jpg content_image/296799.jpg content_image/296797.jpg content_image/296796.jpg
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. 

## Proof. 

(Sufficiency) As in the proof of Proposition 3.1, for any admissible trading strategy we have 

<img src='content_image/296800.jpg'>

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 

<img src='content_image/296798.jpg'>

<img src='content_image/296799.jpg'>

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 

<img src='content_image/296797.jpg'>

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 

<img src='content_image/296796.jpg'>

