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This theorem implies that if there are times t$_{i}$ such that Σ$_{z}$( t$_{i}$ ) + σ 2 − ∫ t$_{i}$ 0 w $^{2}$( s ) ds = 0, and the intensity of private information arrival is fast enough at these points (i.e. (4.10) is satisfied), then it is: a) rational for the market maker to change her weighting function at these points and b) it is optimal for the insider to reveal her information at these times. 

Moreover, notice that this equilibrium exists even when assumptions 3.1 and 3.2 insuring exis- tence of Markovian equilibrium are not satisfied. That is, even if Σ$_{z}$( s ) + σ 2 − s < 0 for some s , there is a non Markovian equilibrium as long as there exists a piecewise linear increasing function g , the integral of which is bounding the realized variance of the insider signal (Σ$_{z}$( t ) − σ $^{2}$) from below and satisfies the conditions of the theorem. 

The proof of this theorem relies on linear filtering and deterministic time change. 

Proof. To demonstrate that (( H $^{∗}$, w$^{∗}$) , θ $^{∗}$) is an equilibrium, it is enough to show that 

<img src='content_image/297494.jpg'>

where Y ∗ t = θ ∗ t + B 2 t and 

Indeed, if these two conditions are satisfied, since Σ$_{z}$( t$_{i}$ ) + σ 2 − ∫ t$_{i}$ 0 g $^{2}$( s ) ds = 0, we will have that H $^{∗}$( ξ$_{t}$$_{i}$, t$_{i}$ ) = F ( Z$_{t}$$_{i}$, t$_{i}$ ), therefore H ∗ is an admissible and rational pricing rule. Moreover, if condition (4.12) is satisfied, then θ ∗ is continuous with bounded variation. Therefore it follows from Proposition 4.1 that θ ∗ is optimal if condition (4.13) holds (notice that H ∗ satisfies PDE (4.1)). 

<img src='content_image/297495.jpg'>

Thus, to show that Y ∗ satisfies (4.12) and (4.13) is the next goal. The proof is by induction. 

I) Consider the interval [0 , t$_{1}$ ]. At t = 0 we have Y$_{0}$ = 0, Z$_{0}$ = v and Y$_{t}$ satisfies the following stochastic differential equation on [0 , t$_{1}$ ]: 

<img src='content_image/297496.jpg'>

with dZ$_{t}$ = σ$_{z}$ ( t ) dB 1 $_{t}$. From Theorem 10.3 of [12] (note that due to (4.10), the conditions of 

