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 $\alpha = \beta = (2p - 1)/p$.