Summary
We study the law of a stochastic differential equation
where the drift anticipates the future behavior of the Brownian path ω, for example the endpoint. We first investigate anticipation of the endpoint, using a conditional Girsanov transformation and methods of Malliavin calculus. A combination with results of Buckdahn [2] leads to new versions of the anticipating Girsanov transformation of Ramer and Kusuoka, and in particular to explicit formulas for the Carleman-Fredholm determinant.
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Buckdahn, R., Föllmer, H. A conditional approach to the anticipating Girsanov transformation. Probab. Th. Rel. Fields 95, 311–330 (1993). https://doi.org/10.1007/BF01192167
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DOI: https://doi.org/10.1007/BF01192167