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

Jacka, S. D.

Ann. Probab. 21, No. 1, 329-339 (1993).

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.

Reviewer: W.Stadje (Osnabrück)

MSC:

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

Keywords:

semimartingale; local time; Snell envelope; optimal stopping problem