Strong Markov property

At an almost surely finite stopping time, a Markov process restarts from its stopped state with the same transition law and independently of the history conditional on that state.

Let $X$ switch between $\pm 1$ at rate $q$ and be killed into zero at rate $\lambda$. To exceed a new level it must cross it in state $+1$. Restarting there by the Strong Markov property gives the multiplicative identity ${\psi }_{+}(c+d)={\psi }_{+}(c){\psi }_{+}(d)$, hence an exponential tail. First-step conditioning gives ${\psi }_{-}(c)={\int }_0^{\infty }q{e}^{-(q+\lambda )u}{\psi }_{+}(c+u)du$ and ${\psi }_{+}(c)=e^{-(q+\lambda )c}+{\int }_0^{c}q{e}^{-(q+\lambda )u}{\psi }_{-}(c-u)du$. Substituting the exponential determines its rate as $\sqrt{\lambda (2q+\lambda )}$.

Let $a={\mathbb{P}}_i(T_j<T_i)$ and $b={\mathbb{P}}_j(T_i<T_j)$. Before returning to $i$, the chain visits $j$ zero times with probability $1-a$ and, for $k\ge 1$, exactly $k$ times with probabilityWhen $b>0$, the expected number of visits is $a/b$.