2020/06/25 by Michael B. Marcus, Marcus, Michael B., Jay Rosen +1
Economics, Econometrics and Finance · Mathematics · #60E07 #60G15 #60G17 #60G99 #60J25 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2006.14457
openalex publication_date 2020/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let B=\ Bt,t∈ R1 \ be Brownian motion killed after an independent exponential time with mean 2/λ2. The process B has potential densities, u(x,y) =e-λ|y-x|\over λ, x,y∈ R 1, which is also the covariance of an Ornstein--Uhlenbeck process. Let f be an excessive function for B. Then, e-λ|y-x|\over λ+f(y), x,y∈ R 1, is the kernel of an α-permanental process Xα=\ Xα(t), t∈ R 1\ for all α>0. It is shown that for all k≥ 1 and intervals Δ⊆ [0,1] , \limsuph→ 0sup_\stackrel|u-v|≤ h u,v∈Δ \frac|Xk/2 (u)-Xk/2 (v)| 2 ( |u-v| log 1/|u-v|)1/2= √ 2 supt∈ΔXk/21/2(t) a.s. The local modulus of continuity of Xk/2 for all k≥ 1 is also obtained. Local and uniform moduli of continuity are also obtained for chi--square processes which are defined by, Yk/2(t)=∑i=1k\fracη2i(t)2, t∈ [0,1], where η=\η(t);t∈ [0,1]\ is a mean zero Gaussian process and \ηi;i=1,…, k\ are independent copies of η.