The above assumption is needed for the filtering problem of the market maker to be well defined. for any t ≥ t ∗ and σ$_{z}$ ( t ) is continuous on [ t $^{∗}$, 1] . Moreover, for all t ∈ [0 , 1] we have 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 Furthermore, assumption 3.2 can be relaxed by replacing (3.4) with condition (3.6). Now we are in the position to state the main result of this section which is summarized in the next theorem. 10 Proof. See Appendix A Assumption 3.2 There exists a t ∗ ∈ [0 , 1) such that Theorem 3.1 Suppose that Assumptions 2.1, 2.2, 3.1 and 3.2 are satisfied. Then the pair ( H $^{∗}$, θ$^{∗}$) , lim t → $_{1}$λ $^{2}$( t )Ξ( t ) log log (Ξ( t )) = 0 . (3.6) λ ( t ) = exp { − ∫ t 0 1 Σ$_{z}$( s ) + σ 2 − s ds } t → 1 −→ 0 , Σ$_{z}$( t ) − t + σ 2 ≥ 0 . (3.5) 1 − t > ∫ 1 t σ 2 $_{z}$( s ) ds (3.4)