A simple PDE and Wiener‐Hopf Riccati equations

We investigate the PDE (1.1), concentrating on the case when σ < in which the boundary condition (1.1b) is not of Feller's type and we lose the minimum principle. Investigation of nonnegative solutions leads us to Wiener‐Hopf theory and to a Riccati equation. A much‐studied Markov chain analogue is developed further in the hope that it will illuminate all aspects of the PDE case.