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\subsubsection*{Examples}

Simple random walk on $\ZZ_+$.
\[ P(0, 1) = 1 \]
\[ P(i, i + 1) = p = 1 - P(i, i -
1) \quad i \ge 1 \]
Want to find $h_i = \PP_i(T_0 < \infty)$.
\[ h_0 = 1 \quad h_i = p h_{i + 1} + q h_{i - 1} \]
\begin{itemize}
    \item $p \neq q$. Then
        \[ h_i = a + b \left( \frac{q}{p}
        \right)^i = a + (1 - a) \left( \frac{q}{p}
        \right)^i \]
        $i = 0$, $a + b = 1$. Assume $q > p$: to
        get non-negative and minimal solution need
        to take $a = 1$. So $h_i = 1$ for all $i
        \ge 1$. If instead we have $q < p$, then
        $a = 0$ implies $h_i = \left( \frac{q}{p}
        \right)^i$ for $i \ge 1$.
    \item If $p = q = \half$. General solution
        $h_i = a + bi$, $h_0 = 1$ implies $a = 1$.
        For minimality need to take $b = 0$. So
        $h_i = 1$ for all $i \ge 1$.
\end{itemize}

\myskip
Birth and death chains.
\[ P(0, 0) = 1, \quad P(i, i + 1) = p_i, \quad
P(i, i - 1) = q_i, \quad p_i + q_i = 1 \]
\[ h_i = \PP_i(T_0 < \infty), \quad h_0 = 1 \]
\[ h_i = p_i h_{i + 1} + q_i h_{i - 1} \]
\[ \implies p_i (h_{i + 1} - h_i) = q_i (h_i -
h_{i - 1}) \]
Set $u_i = h_i - h_{i - 1}$.
\[ u_{i + 1} = \frac{q_i}{p_i} u_i = \cdots =
\prod_{k = 1}^i \frac{q_k}{p_k} u_1 \quad u_1 =
h_1 - 1 \]
\begin{align*}
    h_i
    &= \sum_{j = 1}^i (h_j - h_{j - 1}) + 1 \\
    &= 1 + \sum_{j = 1}^i u_j \\
    &= 1 + u_1 + \sum_{j = 2}^i u_1 \prod_{k =
    1}^{j - 1} \frac{q_k}{p_j} \\
    \implies h_i
    &= 1 + (h_1 - 1) + (h_1 - 1) \sum_{j = 2}^i
    \prod_{k = 1}^{j - 1} \frac{q_k}{p_k}
\end{align*}
Set $\gamma_j = \prod_{k = 0}^j \frac{q_k}{p_k}$,
$\gamma_0 = 1$. Then
\[ h_i = 1 - (1 - h_1) \sum_{j = 0}^{i - 1}
\gamma_j \]
We want $(h_i)$ to be the minimal non-negative
solution, implies:
\[ (1 - h_1) \le \frac{1}{\sum_{j = 0}^\infty
\gamma_j} \]
Minimality implies
\[ h_1 = 1 - \frac{1}{\sum_{j = 0}^\infty \gamma_j} \]
\begin{itemize}
    \item $\sum_{j = 0}^\infty \gamma_j < \infty$,
        then
        \[ h_i = \frac{\sum_{j = 1}^\infty
        \gamma_j}{\sum_{j = 0}^\infty \gamma_j} \]
    \item $\sum_{j = 0}^\infty \gamma_j = \infty$,
        then $h_i = 1$ for all $i \ge 1$.
\end{itemize}

\subsubsection*{Mean hitting times}

$A \subseteq I$, $\tau_A = \inf\{n \ge 0 : X_n \in
A\}$. $k_i^A = \EE_i[\tau_A]$.

\begin{flashcard}
    \begin{theorem*}
        The vector $(k_i^A : i \in I)$ is \cloze{the
        minimal non-negative solution to the
        system
        \[ k_i^A = \begin{cases}
            0 & \text{if $i \in A$} \\
            1 + \sum_{j \not\in A} P(i, j) k_j^A &
            i \not\in A
        \end{cases} \]}
    \end{theorem*}
\end{flashcard}

\begin{hiddenflashcard}
    What does $k_i^A$ represent?
    \[ k_i^A = \cloze{\EE_i[\tau_A]} \]
\end{hiddenflashcard}

\begin{hiddenflashcard}
    What does $h_i^A$ represent?
    \[ h_i^A = \cloze{\PP_i(\tau_A < \infty)} \]
\end{hiddenflashcard}

\begin{hiddenflashcard}
    What does $\tau_A$ represent?
    \[ \tau_A = \cloze{\inf\{n \ge 0 : X_n \in A\}} \]
\end{hiddenflashcard}

\begin{proof}
    If $i \in A$, then $k_i^A = 0$. Assume $i
    \not\in A$. Then
    \begin{align*}
        k_i^A
        &= \EE_i[\tau_A] \\
        &= \sum_{n = 0}^\infty \PP_i(\tau_A > n)
        \\
        &= \sum_{n = 0}^\infty \PP_i(X_0 \not\in
        A, \dots, X_n \not\in A) \\
        &= 1 + \sum_{n = 1}^\infty \PP_i(X_1
        \not\in A, \dots, X_n \not\in A) \\
        &= 1 + \sum_{n = 1}^\infty \sum_j
        \PP_i(X_1 = j, X_2 \not\in A, \dots, X_n
        \not\in A) \\
        &= 1 + \sum_{n = 1}^\infty \sum_j P(i, j)
        \PP(X_1 \not\in A,\dots, X_n \not\in A
        \mid \cancel{X_0 = i}, X_1 = j) \\
        &= 1 + \sum_{n = 1}^\infty \sum_j P(i, j)
        \PP_j(X_0 \not\in A, \dots, X_{n -  1}
        \not\in A) \\
        &= 1 + \sum_j P(i, j) \sum_{n = 0}^\infty
        \PP_j(X_0 \not\in A, \dots, X_n \not\in A)
        \\
        &= 1 + \sum_j P(i, j) \EE_j[\tau_A] \\
        &= 1 + \sum_j P(i, j) k_j^A \\
        &= 1 + \sum_{j \not\in A} P(i, j) k_j^A
    \end{align*}
    Minimality: Let $(x_i)$ be another
    non-negative solution. Then $x_i = 0$, $i \in
    A$. If $i \not\in A$, then
    \begin{align*}
        x_i
        &= 1 + \sum_{j \not\in A} P(i, j) x_j \\
        &= 1 + \sum_{j \not\in A} P(i, j) +
        \sum_{j \not\in A} \sum_{k \not\in A} P(i,
        j) P(j, k) x_k \\
        x_i
        &= 1 + \sum_{j_1 \not\in A} P(i, j_1) +
        \cdots + \sum_{j_1, \dots, j_{n + 1}
        \not\in } P(i, j) \cdots P(j_{n - 2}, j_{n
        - 1}) + \sum \text{non-negative terms} \\
        x_i
        &\ge 1 + \PP_i(\tau_A > 1) + \PP_i(\tau_A
        > 2) + \cdots + \PP_i(\tau_n)
    \end{align*}
    So $x_i \ge \sum_{k = 0}^n \PP_i(\tau_A > k)$
    for all $n$. So
    \[ x_i \ge \sum_{k = 0}^\infty \PP_i(\tau_k) =
    \EE_i[\tau_A] = k_i^A \qedhere \]
\end{proof}

\subsubsection*{Simple Markov Property}

Recall that the Simple Markov property states that
if $m \in \NN$, $i \in I$, $X \sim
\Markov(\lambda, P)$ then conditional on $X_m =
i$, $(X_{n + m})_{n \ge 0}$ is $\Markov(\delta_i,
P)$ and is independent of $X_0, \dots, X_m$. \\
We would like to generalise this to a value of $m$
that is randomly picked.

\begin{flashcard}[stopping-time]
\begin{definition*}
    A random variable $T : \Omega \to \{0, 1,
    \dots\} \cup \{\infty\}$ is called a
    \emph{stopping time} if \cloze{the event $\{T = n\}$
    depends on $X_0, \dots, X_n$ for all $n \in
    \NN$.}
\end{definition*}
\end{flashcard}

\begin{example*}
    $A \subseteq I$, $\tau_A = \inf\{n \ge 0 : X_n
    \in A\}$. Then $\{\tau_A = n\} = \{X_0 \not\in
    A, \dots, X_{n - 1} \not\in A, X_n \in A\}$ so
    first hitting times are always stopping times.
    What about last hitting time:
    \[ L_A = \sup\{n \le 10 : X_n \in A\} \]
    Then $L_A$ is \emph{not} a stopping time,
    because for example $\{L_A = 5\}$ does not
    depend on $X_0, \dots, X_5$ only.
\end{example*}
