Explicit solutions to some optimal variance stopping problems

Abstract

The stopping problem with variance as the optimality criterion is introduced. Due to the variance criterion, smooth fit cannot be applied directly. The problem is solved by embedding it into tractable auxiliary optimal stopping problems, where smooth fit is used to obtain explicit, optimal solutions. Optimal strategies are presented in closed form for several examples. A characteristic feature is that the optimal stopping boundaries depend on the initial value of the gain process, i.e. the state space of the gain process does not split into one continuation set and one stopping set.

Keywords:

• variance criterion
• quadratic optimal stopping
• geometric Brownian motion
• Jacobi diffusion
• square-root process
• variance stopping for a given mean

2000 Mathematics Subject Classification::

• Primary 60G40
• Secondary 90C20

Notes

1. http://www.math.ku.dk/ ∼jesper

Additional information

Notes on contributors

Jesper Lund Pedersen

1 1. http://www.math.ku.dk/ ∼jesper