2009/12/04 by Francesco Russo, Russo, Francesco, Frédéri Viens +1
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.0912.0782
openalex publication_date 2009/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
This paper considers the class of stochastic processes X which are Volterra convolutions of a martingale M. When M is Brownian motion, X is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let m be an odd integer. Under some technical conditions on the quadratic variation of M, it is shown that the m-power variation exists and is zero when a quantity δ2(r) related to the variance of an increment of M over a small interval of length r satisfies δ(r) = o(r1/(2m)) . In the case of a Gaussian process with homogeneous increments, δ is X's canonical metric and the condition on δ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when m=3, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô's formula is proved to hold for all functions of class C6.