How to build and solve continuous-time heterogeneous agents models in asset pricing? The martingale approach and the finite difference method

This paper serves as a tutorial, offering a step-by-step guide for building and numerically solving a preference-heterogeneous agent model in asset pricing. Using a three-stage framework, we clarify the modeling and solution process through a detailed example. Within this framework, we demonstrate how to apply the finite difference method with implicit and upwind schemes to solve the partial differential equation for stock prices, thereby deriving the optimal portfolio, equilibrium asset prices, and their volatility. Additionally, we explore other contexts where this numerical method can be applied, including models with preference heterogeneity using dynamic programming, external habits, and incomplete markets with income heterogeneity and recursive utility. We also address practical considerations in its implementation. This paper does not cover models that incorporate both aggregate and idiosyncratic risks.