Hi I am a student in electrical engineering. Currently I am facing a difficult problem solving a coupled integral-differential partial equations arising from mean field game. The problem is similar as in http://mfglabs.com/wp-content/uploads/2012/12/cfe.pdf
The Bellman cost function is defined as
$u(t,R) = \max_{p(s)\geq0,s\geq t} E\left[\int_t^{\infty}-\left[p(s)g_1 - g_2(I(s)+N_0)\right]^2e^{-r(s-t)}ds\right]$ where $g_1,g_2,N_0 > 0$ are constant and $R \geq 0$
$dR = -pdt + \lambda dW$ where $W$ is a Wiener process
$I(t) =\frac{d}{dt}\int Rm(t,R)dR $ where $m(t,R)$ is the probability distribution of $R$ at time $t$
For mean field game, I assume $I(t)$ and $p(t)$ are independent. I applied the HJB equation and the Fokker Planck equation to get a system of partial equations as :
$\partial_tu(t,R) +\lambda^2\partial^2_{RR}u(t,R) -ru(t,R)+ \max_{p\geq 0}\{-(pg_1 - g_2(I+N_0))^2-p\partial_{R} u(t,R)\}= 0 $
$\partial_t m(t,R) - \partial_R [p(t,R)m(t,R)] = \frac{\lambda^2}{2} \partial^2_{RR}[m(t,R)] $
$\int m(t,R)dR = 1 $
$I$ is calculated as above. The initial condition is known as $m(0,R) = f(R)$ where $f$ are known distribution
Solving the Hamiltonian I can get the optimal control $p$. However when I replace $p$ back into the HJB this becomes a system of integral-partial differential equations that I can not solve. The paper suggests using eductive method to find the answer however using Google search returns no result. I want to ask if someone know any directions or (Matlab) methods to approximately solve these equations in order to find $m$ and $u$ ? Thank you very much.