| Abstract: |
| We derive an explicit asymptotic expansion for the profit function in a class
of long-term average stochastic impulse control problems for one-dimensional
diffusions with two revenue streams: intervention rewards and state-dependent
subsidies. Optimality results characterize the long-term average optimal value $F^*$
as the maximum of a nonlinear function on $\mathbb{R}^2$ and the optimal
threshold policy as its argmax. When the diffusion`s left boundary is an entrance point, the profit function $J^*(T,x)$, representing the expected accumulated optimal profit up to time $T$, satisfies
$J^*(T,x) \approx F^* T + G(x) - \langle G, \nu^* \rangle$, where $\nu^*$ is the stationary distribution of the optimally controlled system and $G$ the relative value function. The correction term $G(x) - \langle G, \nu^* \rangle$ can also be expressed using the first two moments of the profit contributions and the average cycle time of the optimal process. This interpretable representation aids applications and
reveals connections to cumulative renewal processes. The results are joint work with R. Stockbridge and C. Zhu and are based on Long-Term Average Impulse and Singular Control of a Growth Model with Two Revenue Sources (2026a, https://arxiv.org/pdf/2601,09646.pdf) and Long-Term Average Impulse Control with Mean Field Interactions (2026b, https://arxiv.org/pdf/2505.11345.pdf). |
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