A Direct Method for Solving Optimal Switching Problems of One-Dimensional Diffusions

In this paper, we propose a direct solution method for optimal switching problems of one-dimensional diffusions. This method is free from conjectures about the form of the value function and switching strategies, or does not require the proof of optimality through quasi-variational inequalities. The direct method uses a general theory of optimal stopping problems for one-dimensional diffusions and characterizes the value function as sets of the smallest linear majorants in their respective transformed spaces.

💡 Research Summary

The paper introduces a novel “direct” method for solving optimal switching problems (OSPs) driven by one‑dimensional diffusion processes. Traditional approaches to OSPs rely heavily on conjecturing the shape of the value function and the optimal switching strategy, then verifying optimality through quasi‑variational inequalities (QVIs) or other variational techniques. Such methods often require sophisticated functional‑analytic arguments and can be cumbersome when the underlying diffusion has state‑dependent drift or volatility. In contrast, the authors propose a framework that bypasses any a‑priori guesswork and eliminates the need for QVI analysis by exploiting the well‑established theory of optimal stopping for one‑dimensional diffusions.

The model is set up as follows. Let (X_t) be a regular diffusion on an interval (I\subset\mathbb{R}) with infinitesimal generator (\mathcal{L}i) when the system is in mode (i\in{1,\dots,N}). Each mode is associated with an instantaneous reward function (f_i(x)) and a discount rate (\rho>0). Switching from mode (i) to mode (j) incurs a possibly state‑dependent cost (K{ij}(x)). A control consists of a sequence of stopping times ({\tau_k}) at which the controller decides to switch modes, together with the target mode for each switch. The objective is to maximize the expected discounted cumulative reward over an infinite horizon.

The key insight is to treat each mode separately as an optimal stopping problem. For a fixed mode (i), the authors introduce the scale function (s_i) and speed measure (m_i) of the diffusion (\mathcal{L}_i). Two fundamental solutions (\varphi_i) (increasing) and (\psi_i) (decreasing) of (\mathcal{L}_i u=0) are constructed, and the transformation
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