Summary
Leta, b beC 2(R 1)-functions with bounded derivatives of first and second order. We study stochastic differential equations
whose initial valueX 0 is a Fréchet differentiable random variable which may depend on the whole path of the driving Brownian motion (W t ). This anticipation requires to pass from the Itô-integral to the Skorohod-integral. We show that the equation has a unique local solution {X t (ω), 0≦t≦t 0(ω)}, for sufficiently smallt 0(ω)>0, and we provide conditions for the existence of a global solution {X t (ω), 0≦t≦1}.
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Buckdahn, R. Skorohod stochastic differential equations of diffusion type. Probab. Th. Rel. Fields 93, 297–323 (1992). https://doi.org/10.1007/BF01193054
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DOI: https://doi.org/10.1007/BF01193054