Solving optimal stopping problems for Lévy processes in infinite horizon via $A$-transform

We present a method to solve optimal stopping problems in infinite horizon for a L\'evy process when the reward function can be non-monotone. To solve the problem we introduce two new objects. Firstly, we define a random variable $\eta(x)$ which corresponds to the argmax of the reward function. Secondly, we propose a certain integral transform which can be built on any suitable random variable. It turns out that this integral transform constructed from $\eta(x)$ and applied to the reward function produces an easy and straightforward description of the optimal stopping rule. We check the consistency of our method with the existing literature, and further illustrate our results with a new example. The method we propose allows to avoid complicated differential or integro-differential equations which arise if the standard methodology is used.