IJPAM: Volume 59, No. 3 (2010)

ROBUSTNESS ESTIMATING OF OPTIMAL STOPPING
PROBLEM WITH UNBOUNDED REVENUE
AND COST FUNCTIONS

Elena Zaitseva
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 $X$. Revenue and cost functions are allowed to be unbounded. The stability (robustness) is understood in the sense that an unknown transition probability $p(\cdot\vert x)$, $x \in X,$ is approximated by the known one $\;\tilde p(\cdot\vert x)$, $x \in X$, and the stopping rule $\tilde\tau_*,$ optimal for the process governed by $\tilde p$ is applied to the original process represented by $p$. 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 $\tau_*$ by its approximation $\tilde\tau_*$. The bound is expressed in terms of the weighted total variation distance between the transition probabilities $p$ and $\tilde p$.

Received: January 8, 2010

AMS Subject Classification: 60G40, 90C40

Key Words and Phrases: discrete-time Markov process, stopping time, total expected return, $N$-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