Semimartingale properties of the lower Snell envelope in optimal stopping under model uncertainty. (English) Zbl 1380.60049

Treviño Aguilar, Erick

Braz. J. Probab. Stat. 31, No. 1, 194-213 (2017).

Summary: Optimal stopping under model uncertainty is a recent topic under research. The classical approach to characterize the solution of optimal stopping is based on the Snell envelope which can be seen as the value process as time runs. The analogous concept under model uncertainty is the so-called lower Snell envelope and in this paper, we investigate its structural properties. We give conditions under which it is a semimartingale with respect to one of the underlying probability measures and show how to identify the finite variation process by a limiting procedure. An example illustrates that without our conditions, the semimartingale property does not hold in general.

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
91G20	Derivative securities (option pricing, hedging, etc.)

Keywords:

model uncertainty; optimal stopping; robustness; semimartingales; Snell envelopes