A Probabilistic Solution to High-Dimensional Continuous-Time Macro and Finance Models

Abstract

This paper introduces the probabilistic formulation of continuous-time economic models: forward stochastic differential equations (SDE) govern the dynamics of backward-looking variables, and backward SDEs capture that of forward-looking variables. Deep learning streamlines the search for the probabilistic solution, which is less sensitive to the "curse of dimensionality." The paper proposes a straightforward algorithm and assesses its accuracy by considering a multiple-country model with an explicit solution under symmetric states. Combining with the finite volume method, the algorithm can obtain global dynamics of heterogeneous-agent models with aggregate shocks, in which agents consider the distribution of individual states as a state variable.

Keywords: backward stochastic differential equation, deep reinforcement learning, the curse of dimensionality, heterogeneous-agent continuous-time model, finite volume method

JEL Classification: C63, E44, F40, E27, E37