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Initialization-Dependent Solution Selection by the Euler-Maruyama Scheme for a Degenerate SDE
We study the Euler-Maruyama approximation of the degenerate stochastic differential equation $dX_t = |X_t|^α\,dW_t$, $0 < α< 1/2$, $X_0 = 0$, for which uniqueness in law fails. The Euler scheme started exactly from zero is trapped by the absorbing solution, whereas a nonzero initialization $x_h \to
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