= Solution
Put $\mathcal F_n=\sigma(X_0,\ldots,X_n)$ and $\tau=T_0\wedge T_y$. Before $\tau$, the integer-valued increment belongs to $\{-1,0,1\}$. The <martingale> property gives
$$
\mathbb P(X_{n+1}-X_n=1\mid\mathcal F_n)
=\mathbb P(X_{n+1}-X_n=-1\mid\mathcal F_n).
$$
Their sum is at least $1/2$, so each conditional probability is at least $1/4$. From any state in $\{1,\ldots,y-1\}$, a run of at most $y$ upward moves reaches $y$ and has conditional probability at least $4^{-y}$. Applied in successive blocks of $y$ steps, this gives
$$
\mathbb P(\tau>ky)\leq(1-4^{-y})^k,
$$
so $\tau<\infty$ almost surely.
The stopped process $X_{n\wedge\tau}$ takes values in $[0,y]$, hence is a bounded martingale and has <uniform integrability>. The <optional sampling theorem for a supermartingale> gives
$$
x=\mathbb E[X_\tau]
=y\mathbb P(T_y<T_0),
$$
because $X_\tau$ is zero or $y$. Therefore
$$
\mathbb P(T_y<T_0)=\frac{x}{y}.
$$
Back to article page
