Using Age Structure for a Multi-stage Optimal Control Model with Random Switching Time

Stefan Wrzaczek,Michael Kuhn,Ivan Frankovic

doi:10.1007/s10957-019-01598-5

Abstract

The paper presents a transformation of a multi-stage optimal control model with random switching time to an age-structured optimal control model. Following the mathematical transformation, the advantages of the present approach, as compared to a standard backward approach, are discussed. They relate in particular to a compact and unified representation of the two stages of the model: the applicability of well-known numerical solution methods and the illustration of state and control dynamics. The paper closes with a simple example on a macroeconomic shock, illustrating the workings and advantages of the approach.

Highlights

Optimal control models with a variable time horizon continue to be the object of intensive research interest from both a theoretical and an applied point of view
The decision maker is assumed to know the distribution of the terminal time and can derive the expected objective function
We consider a general model that changes the dynamics and/or the objective function at a random switching time, characterized by a known distribution depending on the state and the control variables

Summary

Introduction

Optimal control models with a variable time horizon continue to be the object of intensive research interest from both a theoretical and an applied point of view. Even a numerical treatment is computationally involved to the point of intractability, as the solution up to the switching time includes an explicit expression of the post-switching value function in terms of the state variables and time In this contribution, we consider a general model that changes the dynamics and/or the objective function at a random switching time, characterized by a known distribution depending on the state and the control variables. All information concerning stage 2 is implicitly included in the post-switch value function By treating both stages simultaneously, the new approach allows one to represent the model and its solution in a unified form that expresses explicitly the links between the two stages, and to characterize in a convenient and intuitive way the mechanisms behind the optimal dynamics of the controls and states.

The Model and its Reformulation as a Deterministic Optimal Control Model

Transformation to an Age-Structured Optimal Control Model

Example

Conclusions

Methods