Let $\alpha$ and $\beta$ be real numbers and $f \in C_0 [0,
\infty)$. We study the existence and uniqueness of solutions g of the
equation $g(t) = f(t) + \alpha \sup_{0 \leq s \leq t} g(s) + \beta \inf_{0 \leq
s \leq t} g(s)$. Carmona, Petit, Le Gall, and Yor have shown existence or
nonexistence and uniqueness for some $\alpha, \beta$. We settle the remaining
cases. We study the nearest neighbor walk on the integers, which behaves just
like fair random walk unless one neighbor has been visited and the other has
not, when it jumps to the unvisited neighbor with probability p. If $p
< 2/3$, we show these processes, scaled, converge to the solution of the
equation above for Brownian paths, with $\alpha = \beta = (2p - 1)/p$.