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 dXt = |Xt|α dWt, 0 < α< 1/2, X0 = 0, for which uniqueness in law fails. The Euler scheme started exactly from zero is trapped by the absorbing solution, whereas a nonzero initialization xh → 0 can select a different weak solution. We identify the initialization-dependent limiting regimes. The critical spatial scale is rh = h1/[2(1−α)], and the relevant depth parameter is Θh = hlog(1+log+(rh/|xh|)). If Θh → 0, the scheme converges to the non-sticky weak solution. If Θh → θ∈ (0,∞), it converges to the solution that stays at zero for the deterministic time θ/|logα| and then evolves as the non-sticky solution. If Θh → ∞, 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.
Related
- Degenerate parabolic equations in divergence form: fundamental solution and Gaussian bounds
- Initialization-dependent BFGS trial rates for tilted absolute values
- Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
- Strong convergence of Euler-Maruyama schemes for additive fractional SDEs with Lipschitz drift
- Convergence of Densities for the Euler--Maruyama Approximation of Brownian-Driven McKean--Vlasov Equations
BibTeX
@misc{tanaka2026initializationdependent,
title = {{Initialization-Dependent Solution Selection by the Euler-Maruyama Scheme for a Degenerate SDE}},
author = {Akihiro Tanaka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13800},
eprint = {2610.13800},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13800},
}
RIS
TY - UNPB
TI - Initialization-Dependent Solution Selection by the Euler-Maruyama Scheme for a Degenerate SDE
AU - Akihiro Tanaka
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13800
UR - https://arxiv.org/abs/2610.13800
AB - 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 → 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−α)], and the relevant depth parameter is Θ_h = hlog(1+log^+(r_h/|x_h|)). If Θ_h → 0, the scheme converges to the non-sticky weak solution. If Θ_h → θ∈ (0,∞), it converges to the solution that stays at zero for the deterministic time θ/|logα| and then evolves as the non-sticky solution. If Θ_h → ∞, 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.
ER -