Strong rate for one-dimensional driftless continuous strict local martingales

Antoine Jacquier

Papers from arXiv.org

Abstract: For one-dimensional strict local martingales of the form $\mathrm{d} X_t=\sigma(X_t) \mathrm{d} W_t$ starting from $X_{0}=x_{0}>0$, we prove the strong rate $\mathbb{E}\left[|X^h_N-X_T|\right] \lesssim h^{\min\{\theta\alpha,\frac{1}{2}\}}$ for a discrete scheme~$X^h$ truncated at a level $h^{-\alpha}$, with time step~$h$, as $h\downarrow 0$, where~$\theta$ depends on the truncation property. For the CEV model $\sigma(x)\equiv x^{1+\gamma}$ in particular, the rate is~$\frac{1}{2}$ for any $\gamma \in (0,1)$ (up to a logarithmic pre-factor at $\gamma=\frac{1}{2}$).

Date: 2026-10
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