On the Expectation of the First Exit Time of a Nonnegative Markov Process Started at a Quasistationary Distribution

TL;DR

This paper investigates the expected first exit time of a nonnegative Markov process from a bounded region when starting from a quasistationary distribution, providing conditions for its monotonicity with respect to the boundary.

Contribution

It establishes sufficient conditions under which the expected first exit time from a boundary increases as the boundary level A increases for a Markov process starting at a quasistationary distribution.

Findings

E T_A^{Q_A} is increasing in A under certain conditions

Provides theoretical framework for exit time analysis

Applicable to nonnegative Markov processes with stationary transitions

Abstract

Let {M_n}_{n\ge 0}$ be a nonnegative Markov process with stationary transition probabilities. The quasistationary distributions referred to in this note are of the form Q_A(x) = lim_{n\to\infty} P(M_n \le x | M_0 \le A, M_1 \le A, ..., M_n \le A) . Suppose that $M_0$ has distribution $\Qb_A$ and define T_A^{Q_A} = \min\{n | M_n > A, n\ge 1\}, the first time when M_n exceeds A. We provide sufficient conditions for E T_A^{Q_A}$ to be an increasing function of A.