A Two-Sided Stochastic Integral and its Calculus

Let X be a forward diffusion and Y a backward diffusion, both defined on [-0, 1], X t and yt being respectively adapted to the past of a Wiener process W(.), and to its future increments. We construct a "two-sided" stochastic integral of the form. t q~(u, X,, Y") dW(u) 0 which generalizes the backward and forward It6 integrals simultaneously. Our construction is quite intuitive, and leads to a generalized stochastic calculus. It is also shown that for each fixed t, our integral coincides with that defined by Skorohod in [18].