2023/04/12 by Hubert Lacoin, Lacoin, Hubert · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2304.05781
openalex publication_date 2023/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given d≥ 1, we provide a construction of the random measure - the critical Gaussian Multiplicative Chaos - formally defined e√(2d)Xd μ where X is a log-correlated Gaussian field and μ is a locally finite measure on \mathbb Rd. Our construction generalizes the one performed in the case where μ is the Lebesgue measure. It requires that the measure μ is sufficiently spread out, namely that for μ almost every x we have ∫B(0,1)\fracμ(d y)|x-y|deρ(log (1)/(|x-y|) )lt;∞, for any compact set where ρ:\mathbb R+→ \mathbb R+ can be chosen to be any lower envelope function for the 3-Bessel process (this includes ρ(x)=xα with α∈ (0,1/2)). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure μ is in a sense optimal.