# IJPAM: Volume 59, No. 3 (2010)

**ROBUSTNESS ESTIMATING OF OPTIMAL STOPPING**

PROBLEM WITH UNBOUNDED REVENUE

AND COST FUNCTIONS

PROBLEM WITH UNBOUNDED REVENUE

AND COST FUNCTIONS

Department of Mathematics

Instituto Tecnológico Autónomo de México

Rio Hondo # 124, Col. Tizapan San Angel

C.P. 01080, Mexico D.F., MEXICO

e-mail: [email protected]

**Abstract.**We study the stability of the optimal
stopping problem for a discrete-time Markov process on a general
space state . Revenue and cost functions are allowed to be
unbounded. The stability (robustness) is understood in the sense
that an unknown transition probability , is
approximated by the known one
, , and
the stopping rule optimal for the process governed
by is applied to the original process represented by .
The criteria of stopping rule optimization is the total expected
return. We give an upper bound for the decrease of the return due to
the replacement of the unknown optimal stopping rule by its
approximation . The bound is expressed in terms of the
weighted total variation distance between the
transition probabilities and .

**Received: **January 8, 2010

**AMS Subject Classification: **60G40, 90C40

**Key Words and Phrases: **discrete-time Markov process, stopping time, total expected return, -contractive operator, stability index, weighted total variation norm

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 59

**Issue:** 3