Summary
Consider a stochastic differential equation on ℝd with smooth and bounded coefficients. We apply the techniques of the quasi-sure analysis to show that this equation can be solved pathwise out of a slim set. Furthermore, we can restrict the equation to the level sets of a nondegenerate and smooth random variable, and this provides a method to construct the solution to an anticipating stochastic differential equation with smooth and nondegenerate initial condition.
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Malliavin, P., Nualart, D. Quasi sure analysis and Stratonovich anticipative stochastic differential equations. Probab. Th. Rel. Fields 96, 45–55 (1993). https://doi.org/10.1007/BF01195882
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DOI: https://doi.org/10.1007/BF01195882