In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. The set-up and methodology are sufficiently general to allow the analysis of problems with both finite-time and infinite-time horizon. Finally we apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case.
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