Summary
Let X be a forward diffusion and Y a backward diffusion, both defined on [0,1], X t and Y tbeing 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.
which generalizes the backward and forward Itô 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].
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Supported in part by NSF grant #DMS-8500997; part of the research for this work was performed while this author was visiting the Université de Provence at Marseille
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Pardoux, E., Protter, P. A two-sided stochastic integral and its calculus. Probab. Th. Rel. Fields 76, 15–49 (1987). https://doi.org/10.1007/BF00390274
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DOI: https://doi.org/10.1007/BF00390274