Optimal stopping with discontinuous reward

Mathematics – Probability

Scientific paper

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Scientific paper

We study, for any stopping time $S$, the optimal stopping time problem $v(S)= \esssup_{\theta \geq S} E[\phi(\theta)|\F_S]$, where the reward is given by a family $\{\phi(\theta),\TO\}$ {\em of positive random variables} indexed by stopping times. We solve the problem under the weakest assumptions in terms of the regularity of the reward. More precisely, the reward family $\{\phi(\theta),\TO\}$ is supposed to satisfy some compatibility conditions and to be upper-semicontinuous along stopping times in expectation. We give several properties of the value function. We show the existence of an optimal stopping time. Also, we obtain a characterization of the minimal and the maximal optimal stopping times.

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