Optimal price function: application of calculus of variations The problem, I am trying to solve is based on the paper by Rochet and Vila 1994 (see literature below). In fact, it is a variant of the seminal paper of Kyle 1985 in the finance/economics literature. Despite the problem being framed in the finance literature, the mathematical problem associated with it is far more general and is based on calculus of variations. Let me briefly introduce the problem:


Let $V \sim N(0, \sigma^2_V)$ denote the random asset value, $U\sim N(0, \sigma^2_U)$ the random liquidity trader's demand, $X$ the order by the strategic trader, and $Y=X+U$ the total order flow
The strategic trader observes the realization of the asset value $V=v$, the exogenous liquidity trades $U=u$, and submits an order $x(u,v)$ that maximizes his trading profits: $\pi(x,u,v)=x(v-P(y))$
The market maker observes total order flow $Y=y$ and sets the trading price equal to the conditional expected value of the asset, i.e. $P(y)=E[V|y]$.
A market equilibrium is characterized by a trading strategy $x(u,v)$ and a pricing rule $P(y)$ such that: i) Given the pricing rule, for every $(u,v)$, the trading strategy $x(u,v)$ maximizes the informed trader's profit $\pi$ and ii) given the trading strategy, the pricing rule $P(y)$ sets the price equal to the expected value of $V$ conditional on $Y=y$.


Now the authors show that this problem is equivalent to the following optimization problem $\min_{P(\cdot)} E [ \max_x (v-P(x+u))x]$, that is, the problem can be solved by the calculus of variations. They state the answer to be $P(y)=\frac{\sigma_V}{\sigma_U}y$ which is true, once we "guess and verify" the solution. However, I want to find the solution from scratch, that is, without "guess and verify".

My problems arise as the function over which the functional of $P$ is defined is itself an optimal value. We know that the insider maximizes $x(v-P(x+u))$ or equivalently $(y-u)(v-P(y))$. From the first-order condition, we can calculate the optimal demand $y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$ which results in an optimal profit of $\pi(y^*,u,v)=\frac{(v-P(y^*))^2}{P'(y^*)}$. 

Therefore, the market maker's optimization problem can be rewritten as $$\min_{P(\cdot)} E [ \max_x (v-P(x+u))x]= \int_{ \infty}^{\infty} \int_{-\infty}^{\infty} \pi(y^*,u,v) f_V(v) f_U(u) du dv = \int_{ \infty}^{\infty} \int_{-\infty}^{\infty} \frac{(v-P(y^*))^2}{P'(y^*)} f_V(v) f_U(u) du dv  $$ with $f_V$ and $f_U$ being the density functions of $V$ and $U$ respectively.

From the fact that $\pi^*$ is the maximum of the informed trader's objective function, we can use the envelope theorem to simplify: i) $\frac{d \pi(y^*)}{d u}=\frac{\partial \pi(y,u,v)}{\partial u}|_{y=y^*}=-(v-P(y^*))$ and  ii) $\frac{d \pi(y^*)}{d v}=(y^*-u)$.

This is where I am stuck. My problem is the fact that the argument $y^*(u,v)$ of the function we are searching for $P(\cdot)$ is itself the solution of a maximization problem. Especially, I do not know how to calculate the integral with respect to $v$ and $u$.

Any comments are highly appreciated. Thank you!

Literature:


Rochet and Vila 1994, "Insider Trading wihtout Normality", The Review of Economic Studites, Vol. 61, No. 1, pp. 131-152
Kyle 1985, "Continuous Auctions and Insider Trading", Econometrica, Vol. 52, No. 6, pp 1315-1335
Not really an answer, but too long for comment. 

The $P$ in your 
$$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$$ 
expression from the insider's problem and the $P$ in the expression 
$$
\min_{P(\cdot)} \cdots
$$ 
from the market maker's problem should not be the same. That is not the definition of rational expectation equilibrium in this context.

The insider takes the pricing rule as given, per definition of rational expectation equilibrium. This means insider has optimal demand 
$$
y(u,v, P_1(\cdot))^*= \frac{v-P_1(y^*)}{P_1'(y^*)} +u
$$ for any pricing rule $P_1(\cdot)$. Then market maker minimizes 
$$
E[ ( y(u,v, P_1(\cdot))^* - u) \cdot ( v - P_2(y(u,v, P_1(\cdot))^*) )  ]
$$ over $P_2$---taking $P_1$ as fixed. It's a simultaneous game.
If the minimizing $P_2$ coincides with $P_1$, then $P= P_1 = P_2$ is an equilibrium pricing rule. Doing this in the one-period Kyle setting recovers Kyle's lambda, whereas your calculation would not.

The optimization problem you're solving is therefore not the correct one.

The insider optimizes with respect to a conjectured pricing rule $P_1$. In equilibrium, the market maker confirms the insider's conjecture, making the latter's expectation rational.

Your calculation assumes that the market maker acts after the insider. This is not the case in Kyle and I don't believe it's the case in Rochet and Vila either.