一维扩散奇异控制问题与约束控制问题的渐近关系
On asymptotic relations between singular and constrained control problems of one-dimensional diffusions
摘要：
本文研究了具有折扣目标和遍历目标的一维奇异和约束控制问题之间的渐近关系。通过约束控制问题，我们的意思是控制只允许在独立的泊松到达时间。我们表明，当潜在的扩散是经常性的，折扣问题的解决方案收敛于阿贝尔意义下的遍历对应。此外，我们还证明了当Poisson率趋于无穷大时，约束问题的解收敛到奇异问题的解。我们用漂移的布朗运动和Ornstein-Uhlenbeck过程来说明结果。
英文摘要：
We study the asymptotic relations between certain singular and constrained control problems for one-dimensional diffusions with both discounted and ergodic objectives. By constrained control problems we mean that controlling is allowed only at independent Poisson arrival times. We show that when the underlying diffusion is recurrent, the solutions of the discounted problems converge in Abelian sense to those of their ergodic counterparts. Moreover, we show that the solutions of the constrained problems converge to those of their singular counterparts when the Poisson rate tends to infinity. We illustrate the results with drifted Brownian motion and Ornstein-Uhlenbeck process.
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