Abstract
Leta, b beC2(R1)-functions with bounded derivatives of first and second order. We study stochastic differential equations $$dX_t = a(X_t )dW_t + b(X_t )dt,0 \leqq t \leqq 1,$$ whose initial valueX0 is a Frechet differentiable random variable which may depend on the whole path of the driving Brownian motion (Wt). This anticipation requires to pass from the Ito-integral to the Skorohod-integral. We show that the equation has a unique local solution {Xt(ω), 0≦t≦t0(ω)}, for sufficiently smallt0(ω)>0, and we provide conditions for the existence of a global solution {Xt(ω), 0≦t≦1}.
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