On the Supremum of Certain Families of Stochastic Processes

2008/12/21 by Wenbo V. Li, Li, Wenbo V., Natesh S. Pillai +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0812.4062

openalex publication_date 2008/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a family of stochastic processes \X_t^ε, t ∈ T\ on a metric space T, with a parameter ε\downarrow 0. We study the conditions under which lim_{\e → 0} ¶(sup_{t ∈ T} |X_t^\e| < δ) =1 when one has the a priori estimate on the modulus of continuity and the value at one point. We compare our problem to the celebrated Kolmogorov continuity criteria for stochastic processes, and finally give an application of our main result for stochastic intergrals with respect to compound Poisson random measures with infinite intensity measures.

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