PreprintCC0arXiv
Strong rate for one-dimensional driftless continuous strict local martingales
Abstract
For one-dimensional strict local martingales of the form d Xt=σ(Xt) d Wt starting from X0=x0>0, we prove the strong rate 𝔼[|XhN−XT|] ≲ hmin{θα,1/2} for a discrete scheme~Xh truncated at a level h−α, with time step~h, as h↓ 0, where~θ depends on the truncation property. For the CEV model σ(x)≡ x1+γ in particular, the rate is~1/2 for any γ∈ (0,1) (up to a logarithmic pre-factor at γ=1/2).
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BibTeX
@misc{jacquier2026strong,
title = {{Strong rate for one-dimensional driftless continuous strict local martingales}},
author = {Antoine Jacquier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39896},
eprint = {2610.39896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39896},
note = {CC0},
}
RIS
TY - UNPB
TI - Strong rate for one-dimensional driftless continuous strict local martingales
AU - Antoine Jacquier
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39896
UR - https://arxiv.org/abs/2610.39896
AB - For one-dimensional strict local martingales of the form d X_t=σ(X_t) d W_t starting from X_0=x_0>0, we prove the strong rate 𝔼[|X^h_N−X_T|] ≲ h^min{θα,1/2} for a discrete scheme~X^h truncated at a level h^−α, with time step~h, as h↓ 0, where~θ depends on the truncation property. For the CEV model σ(x)≡ x^1+γ in particular, the rate is~1/2 for any γ∈ (0,1) (up to a logarithmic pre-factor at γ=1/2).
ER -