On undiscounted non‐linear optimal multiple stopping

We study and formulate an undiscounted non‐linear optimal multiple stopping problem, with an application to the valuation of the perpetual American‐style discretely monitored Asian options. When the reward process is continuous, we follow a vector‐valued approach. Under the right‐continuity of this process, the problem can be reduced to a sequence of ordinary optimal stopping problems. In the Markovian case, we characterise the value function of the problem in terms of excessive functions. Finally, in case of a regular diffusion, we provide an optimal sequence of stopping times. The results are illustrated by some examples, where the value function of the problem is given explicitly.