Local times, optimal stopping and semimartingales. (English) Zbl 0773.60031

The author considers a martingale \(X\) such that \(X\) and its Snell envelope \(S\) are continuous and in \(H^ 1\). He gives a maximal characterization of \(S\) in terms of a stochastic differential equation involving the local time of \(S-X\) at zero. This result is applied to the optimal stopping problem for continuous functions of diffusions, yielding sufficient conditions for the smooth pasting condition to hold. Finally, the intuitive assertion that \(S\) is “a martingale on the go-region and equal to \(X\) on the stop-region” is shown to be incorrect by a counterexample.


60G40 Stopping times; optimal stopping problems; gambling theory
60G07 General theory of stochastic processes
60H20 Stochastic integral equations
60G44 Martingales with continuous parameter
60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
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