[Paper Review] On the One-Dimensional Optimal Switching Problem

Erhan Bayraktar, Masahiko Egami|ArXiv.org|Jul 1, 2007

Capital Investment and Risk Analysis20 references4 citations

TL;DR

This paper presents a direct solution to the one-dimensional optimal switching problem for diffusive processes using dynamic programming and excessive function theory, establishing the value function's smooth fit and concavity properties without assuming Hölder continuity of the reward function. The key contribution is an explicit characterization of the optimal switching policy and value function via sequential approximation and Markov process theory.

ABSTRACT

We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.

Motivation & Objective

• To develop a direct, non-verification-based method for solving optimal switching problems in one-dimensional diffusions.
• To characterize the value function as the solution of coupled optimal stopping problems via dynamic programming.
• To prove the smooth fit principle and concavity properties of the value function using excessive function theory.
• To construct an explicit solution for non-degenerate cases using the excessive characterization of optimal stopping value functions.
• To generalize existing results by avoiding restrictive assumptions such as Hölder or Lipschitz continuity on the reward function.

Proposed method

• Apply a sequential approximation method to prove the dynamic programming principle without requiring Hölder continuity of the reward function.
• Characterize the value function as the solution to two coupled optimal stopping problems using the essential supremum and Markov process properties.
• Use the excessive function characterization (from Dayanik and Karatzas) to explicitly construct the value function in non-degenerate cases.
• Establish sufficient conditions under which the optimal switching problem reduces to a standard optimal stopping problem.
• Employ the strong Markov property and continuity arguments to prove the continuity and regularity of the value function iteratively.
• Use induction to verify that the value function sequence converges and maintains continuity and boundedness under the given growth conditions.

Experimental results

Research questions

• RQ1 Can the optimal switching problem in one-dimensional diffusions be solved directly without relying on verification arguments?
• RQ2 What conditions ensure the smooth fit principle and concavity of the value function in the absence of Hölder continuity?
• RQ3 How can the dynamic programming principle be rigorously established under minimal regularity assumptions on the reward function?
• RQ4 Under what conditions does the optimal switching problem reduce to a standard optimal stopping problem?
• RQ5 Can an explicit solution be constructed for non-degenerate cases using excessive function theory?

Key findings

• The dynamic programming principle for optimal switching problems is rigorously established via sequential approximation, without assuming Hölder continuity of the reward function.
• The value function is shown to satisfy the smooth fit principle and exhibit concavity after an appropriate transformation, using properties of excessive functions.
• The continuation regions in the optimal switching problem are not necessarily connected, as demonstrated by a counterexample in the paper.
• An explicit solution is constructed for non-degenerate cases using the excessive characterization of optimal stopping value functions.
• The value function sequence converges uniformly, and each iterate remains continuous under the given growth and continuity conditions.
• The solution method is general and applies to all one-dimensional diffusions, including mean-reverting processes like the Ornstein-Uhlenbeck process.