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In this short letter, we present an explicit upper bound for the optimal value of a bidimensional optimal stopping problem Eis a geometric Brownian motion coupled with an arbitrary diffusion process y(.), θ (., .) and c(.) are given positive, continuous functions and β > 0 is a fixed constant. The present result is derived from a corresponding Lagrangian dual problem, and using a recent result of Makasu (Seq Anal 27:435-440, 2008). Examples are given to illustrate our main result.

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A stochastic Gronwall lemma is proved in Scheutzow (2013) in the case when the exponent p lies in the interval 0 < p < 1. In this paper, we extend the lemma to the entire interval 0 < p < ∞. We construct simple examples to illustrate the present result.