We investigate a stochastic integral equation of the form $x'(s) = y'(s) + \int_0^\alpha {K(s,t)dx(t)} $, where $y( s )$ is a process with orthogonal increments on the interval $T_\alpha = [0,\alpha ]$ and $K(s,t)$ is a continuous Fredholm or Volterra kernel on $T_\alpha \times T_\alpha $. Since $y'(s)$ need not exist, we must first decide upon a rigorous interpretation for this schematic equation. We say that a process $x(s)$ on $T_\alpha $ satisfies this integral equation if, for a suitable class $H_\alpha $ of functions on $T_\alpha $, the equality\[ \int_0^\alpha {g(s)dx(s)} = \int_0^\alpha {g(s)dy(s) } + \int_0^\alpha {g(s)ds } \int_0^\alpha {K(s,t)dx(t) } \] holds for $g \in H_\alpha $. With this interpretation, a formal integration of the schematic equation yields a meaningful equality. A process $x(s)$ is exhibited which satisfies the integral equation by a consideration of the (formal) reciprocated form of the integral equation. A similar interpretation is also obtained for the reciprocated form....