A two-sided stochastic integral and its calculus

Abstract

Let X be a forward diffusion and Y a backward diffusion, both defined on [0,1], X t and Y t 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. $$\mathop \smallint \limits_0^t \Phi (u,X_u ,Y^u )dW(u)$$ which generalizes the backward and forward Ito 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].