2014/07/17 by Francesco Russo, Russo, Francesco, Frédéri Viens +1
Economics, Econometrics and Finance · Physics and Astronomy · #Stochastic processes and financial applications #Financial Risk and Volatility Modeling #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1407.4568
We consider a class of stochastic processes X defined by X( t) =∫0TG( t,s) dM( s) for t∈\lbrack0,T], where M is a square-integrable continuous martingale and G is a deterministic kernel. Let m be an odd integer. Under the assumption that the quadratic variation [ M] of M is differentiable with E[ \vert d[ M] (t)/dt\vert m] finite, it is shown that the mth power variation limε→0ε-1∫0Tds( X( s+ε) -X( s) ) m 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)) . When M is the Wiener process, X is Gaussian; the class then includes fractional Brownian motion and other Gaussian processes with or without stationary increments. When X is Gaussian and has stationary increments, δ is X's univariate canonical metric, and the condition on δ is proved to be necessary. In the non-stationary Gaussian case, when m=3, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô formula is established for all functions of class C6.