Semiordering of the probabilities of the first passage time for Markov processes

G. I. Kalmykov

Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 735-742

Résumé

Let $\{\xi(t),\ t\in T\}$ bе a real Markov process. Let $c(t)$ be a real function and $\widehat\tau_\xi(s,c(t))$ $(\tau_\xi(s,c(t)))$ denote the first time, after $s$, of the crossing (the contact) of the curve $x=c(t)$. Two real Markov processes $\{\xi_1(t),t\in T\}$ and $\{\xi_2(t),t\in T\}$ with conditional probabilities $\mathbf P_{s,x}^{(1)}\{B\}$ and $\mathbf P_{s,x}^{(2)}\{B\}$ being considered, sufficient conditions for the inequality \begin{gather*} \mathbf P_{s,x}^{(1)}\{\widehat\tau_{\xi_1}(s,a(t))\le\min(t,\widehat\tau_{\xi_1}(s,b(t))\}\le \\ \le\mathbf P_{s,x}^{(2)}\{\widehat\tau_{\xi_2}(s,a(t))\le\min(t,\widehat\tau_{\xi_2}(s,b(t))\} \end{gather*} are obtained. Here $a(t)$ and $b(t)$ are real functions satisfying $a(t)<x<b(t)$. The analogous results are obtained for $\tau_{\xi_1}$ and $\tau_{\xi_2}$.