Strong rate for one-dimensional driftless continuous strict local martingales

Paper: arXiv 2610.39896

Authors: 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 $γ=\frac{1}{2}$).

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on mathematical proofs for strong convergence rates of a numerical scheme. While it includes a ‘Numerical validation’ section, the empirical rigor is limited to demonstrating the theoretical results rather than extensive backtesting or real-world data application. The novelty lies in establishing the first strong rate for discretisation schemes of strict local martingales.

Research Flowchart

  flowchart TD
    A[Research Goal: Strong Rate for 1D Driftless SLMs] --> B{Key Methodology: Truncated Discrete Scheme};
    B --> C[Model/Data: dX_t = σ(X_t) dW_t, X_0=x_0>0];
    C --> D{Computational Process: Analyze E[|X^h_N - X_T|] as h -> 0};
    D --> E[Outcome 1: Strong rate E[|X^h_N - X_T|] ~ O(h^min{θα, 1/2})];
    E --> F[Outcome 2: For CEV model σ(x)=x^(1+γ), rate is 1/2 for γ in (0,1)];