21 - Characterisation of Mean Hitting Times

Characterisation of Mean Hitting Times

Let $I$ be the state space then

Vector of Mean Hitting Times

$$k^A=(k_i^A,i\in I)$$

is the minimal non-negative solution to

$$k_i^A=\{\begin{array}{ll}0& \text{ if }i\in A\\ 1+{\sum }_j{p}_{ij}{k}_j^A& \text{ if }i\notin A\end{array}$$

Proof

Condition on first jump so for $i\notin A$

$$\begin{array}{rl}{k}_i^A& ={\mathbb{E}}_i[H^A]=\sum _j{\mathbb{E}}_i[H^A\mid X_1=j]{\mathbb{P}}_i(X_1=j)\\ & =\sum _j{p}_{ij}(1+k_j^A)\\ & =1+\sum _j{p}_{ij}{k}_j^A\end{array}$$

Minimality can be shown using similar idea to Characterisation of Hitting Probabilities