Initialization-Dependent Solution Selection by the Euler-Maruyama Scheme for a Degenerate SDE

Abstract

We study the Euler-Maruyama approximation of the degenerate stochastic differential equation $dX_t = |X_t|^\alpha\,dW_t$, $0<\alpha<1/2$, $X_0 = 0$, for which uniqueness in law fails. The Euler scheme started exactly from zero is trapped by the absorbing solution, whereas a nonzero initialization $x_h \to 0$ can select a different weak solution. We identify the initialization-dependent limiting regimes. The critical spatial scale is $r_h = h^{1/[2(1-\alpha)]}$, and the relevant depth parameter is $\Theta_h = h\log(1+\log^+(r_h/|x_h|))$. If $\Theta_h \to 0$, the scheme converges to the non-sticky weak solution. If $\Theta_h \to \theta \in (0,\infty)$, it converges to the solution that stays at zero for the deterministic time $\theta/|\log\alpha|$ and then evolves as the non-sticky solution. If $\Theta_h \to \infty$, the scheme converges to the absorbing solution. The proof is based on an exact scale-invariant Euler chain, a logarithmic exit-time law of large numbers, a discrete Tanaka occupation estimate, and a compact-uniform restart argument. Every finite deterministic delay can be obtained by a suitable vanishing initialization, while oscillating depth parameters yield process-level nonconvergence. Thus a vanishing initialization can survive in the limit as a macroscopic deterministic delay and thereby act as a solution selector for the Euler-Maruyama approximation.