This paper studies recursive exchange economies with short sales. Agents maximize discounted expected utility. The asset structure is general and includes real securities, infinite-lived stocks, options, and other derivatives. The main result shows the existence of a competitive equilibrium...

We use supervised machine learning to approximate the expectations typically contained in the optimality conditions of an economic model in the spirit of the parameterized expectations algorithm (PEA) with stochastic simulation. When the set of state variables is generated by a stochastic...