content_image/306479.jpg content_image/306478.jpg content_image/306477.jpg content_image/306476.jpg
The above assumption is needed for the filtering problem of the market maker to be well defined. 

Assumption 3.2 There exists a t ∗ ∈ [0 , 1) such that 

for any t ≥ t ∗ and σ$_{z}$ ( t ) is continuous on [ t $^{∗}$, 1] . 

Moreover, for all t ∈ [0 , 1] we have 

<img src='content_image/306479.jpg'>

<img src='content_image/306478.jpg'>

This assumption insures that: a) close to the market terminal time, the insider’s signal is more precise than the market maker’s (i.e. E [ ( Z$_{1}$ − E [ Z$_{1}$ |F M t ]) 2 |F M t ] > E [ ( Z$_{1}$ − E [ Z$_{1}$ |F I t ]) 2 |F I t ] ), and b) that the insider’s signal is always at least as precise as the market maker’s. 

Remark 3.1 Notice that Assumptions 2.2, 3.1 and 3.2 guarantee that when condition (3.2) is not satisfied 

and that if Ξ( t ) = ∫ t 0 1+ σ 2 $_{z}$( s ) λ $^{2}$( s ) ds t → 1 −→ ∞ , then 

<img src='content_image/306477.jpg'>

<img src='content_image/306476.jpg'>

Furthermore, assumption 3.2 can be relaxed by replacing (3.4) with condition (3.6). 

## Proof. See Appendix A 

Now we are in the position to state the main result of this section which is summarized in the next theorem. 

Theorem 3.1 Suppose that Assumptions 2.1, 2.2, 3.1 and 3.2 are satisfied. Then the pair ( H $^{∗}$, θ$^{∗}$) , 

