{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Boltzmann mean field game solver\n", "\n", "We consider the time-dependent BMFG with diffusion on a bounded domain $\\Omega = [0,\\bar{z}]$. The dynamics are given by\n", "\\begin{align*}\n", "&\\partial_t f(z,t) - \\frac{\\sigma^2}{2} \\partial_{zz} f(z,t) = f(z,t) \\int_0^z \\alpha(S(y,t)) f(y,t) dy - \\alpha(S(z,t)) f(z,t) \\int_z^{\\bar{z}} f(y,t) dy\\\\\n", "&\\partial_t V(z,t) - r V(z,t) + \\frac{\\sigma^2}{2} \\partial_{zz} V(z,t) =\n", "(1-S(z,t)) z + \\alpha(S(z,t)) \\int_{z}^{\\infty} \\left[V(y,t) - V(z,t) \\right] f(y,t) dy\\\\\n", "&S(z,t) = \\text{argmin}_{s\\in\\mathcal{S}} \\left[(1-S(z,t)) z + \\alpha(S(z,t)) \\int_{z}^{\\infty} \\left[V(y,t) - V(z,t) \\right] f(y,t) dy\\right]\n", "\\end{align*}\n", "with initial condition $f(z,0) = f_0(z)$ and terminal condition $V(z,T) = 0$. \n", "\n", "We solve the system using a fixed point iteration:\n", "