Here is my problem (I'll present it in terms of two maximizations, for simplicity):
Given two cost functions, solve simultaneously
$V_1(y_1) = {min_{u_1 \in \mathcal{A}(y)}} J(y_1, u_1)$
$V_2(y_1) = {min_{u_1 \in \mathcal{A}(y)}} J(y_1, u_1)$
$st$ $dy_{ti} = a_t(y_i, u_i, x)dt + b_t(y_i, u_i, x)dW_t$ $\forall$ $i \in 1, 2$
$where$ $f(y1, y2, u1, u2, x) = 0$
That is, solve two dynamic programming problems where each problem is a function of the state. However, the evolution of these state variables is determined in a non-linear way by the choice. $x$ should adjust to simultaneously solve the system.
This produces a system of HJB equations, whose boundary conditions are more or less clear, but which are coupled in a strange way. Does anyone know of this sort of problem? Are there standard methods to approach this?
My only intuition at this time is to search the space of possible values of $x$ numerically by brute force. My idea is to choose $x$, solving the HJBs, check the constraint. I could do this for a set of points or use some search algorithm (eg MCMC)... but it seems very costly. Any thoughts?