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