2007/07/26 by Michael B. Marcus, Michael Marcus, Marcus, Michael +2
Economics, Econometrics and Finance · Environmental Science · Mathematics · #60 F25 #Analysis of environmental and stochastic processes #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60 #msc:F25
paper · pdf · doi:10.48550/arxiv.0707.3928
openalex publication_date 2007/07/26 · arxiv created 2009/10/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G=\G(x),x≥ 0\ be a mean zero Gaussian process with stationary increments and set σ2(|x-y|)= E(G(x)-G(y))2. Let f be a function with Ef2(η)<\ff, where η=N(0,1). When σ2 is regularly varying at zero and limh→ 0h2\over σ2(h)= 0 and limh→ 0σ2(h)\over h= 0 but (d2\over ds2σ2(s))j0 is locally integrable for some integer j0≥ 1, and satisfies some additional regularity conditions, \bea && ∫abf((G(x+h)-G(x))/(σ(h))) dx \nn && = ∑j=0j0 (h/σ(h))j E(Hj(η) f(η))\over√ j! :(G')j:(I[a,b]) +o(h\overσ(h))j0\nn \eea in L2. Here Hj is the j-th Hermite polynomial. Also :(G')j:(I[a,b]) is a j -th order Wick power Gaussian chaos constructed from the Gaussian field G'(g) , with covariance E(G'(g)G'(\wt g)) = ∫ ∫ ρ(x-y)g(x)\wt g(y) dx dy, where ρ(s)=1/2d2\over ds2σ2(s).