An equivalent approximation approach for the Hamilton‐Jacobi‐Bellman equations in intertemporal decision problems

Abstract

In this paper, we develop an equivalent approximation approach to obtain the Hamilton-Jacobi-Bellman (HJB) equations of intertemporal decision problems under uncertainty. An implicit discount function is adopted in the derivation of the rate of change of value function for the HJB equations to provide the broadest coverage of time preferences. Meanwhile, we provide an illustrative example for application by revisiting the intertemporal consumption and portfolio decisions problem. Furthermore, we verify the validity of our approach in robustness tests by incorporating exponential, nonconstant, and stochastic hyperbolic discounting functions, respectively. The HJB equations and their resulting decision rules in the finite and infinite planning horizons have confirmed that our approach provides an efficient and reliable alternative to acquire the HJB equations in the intertemporal economic issues under uncertainty. KEYWORDS equivalent approximation, HJB equation, intertemporal decision, time inconsistency 1976