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Strict SDE Comparison for Cusp Coefficients and Counterexamples
Abstract
We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation d Xtx=σ(Xtx) d Bt, X0x=x∈ I=(ℓ,r), for σ>0 and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in W1,ploc(I), 1≤ p<2. We illustrate using σ(x):=1+|x|β for x∈ℝ, β∈ (0,1), and show that strict comparison holds if and only if β∈[1/2,1). We give examples showing that neither σ∈ W1,ploc(ℝ), p∈(1,2), nor σ∈ Cβ(ℝ), β∈[1/2,1), is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.
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BibTeX
@misc{larsen2026strict,
title = {{Strict SDE Comparison for Cusp Coefficients and Counterexamples}},
author = {Kasper Larsen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19389},
eprint = {2609.19389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19389},
}
RIS
TY - UNPB
TI - Strict SDE Comparison for Cusp Coefficients and Counterexamples
AU - Kasper Larsen
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.19389
UR - https://arxiv.org/abs/2609.19389
AB - We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation d X_t^x=σ(X_t^x) d B_t, X_0^x=x∈ I=(ℓ,r), for σ>0 and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in W^1,p_loc(I), 1≤ p<2. We illustrate using σ(x):=1+|x|^β for x∈ℝ, β∈ (0,1), and show that strict comparison holds if and only if β∈[1/2,1). We give examples showing that neither σ∈ W^1,p_loc(ℝ), p∈(1,2), nor σ∈ C^β(ℝ), β∈[1/2,1), is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.
ER -