The area of a self-similar fragmentation

2011/01/20 by Jean Bertoin, Bertoin, Jean · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.1101.3965

arxiv created 2011/01/20 · openalex publication_date 2011/01/20 · arxiv updated 2011/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the area A=∫₀^∞(∑_i=1^∞ X_i(t)) \d t of a self-similar fragmentation process \X=(\X(t), t≥ 0) with negative index. We characterize the law of A by an integro-differential equation. The latter may be viewed as the infinitesimal version of a recursive distribution equation that arises naturally in this setting. In the case of binary splitting, this yields a recursive formula for the entire moments of A which generalizes known results for the area of the Brownian excursion.

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