×

White noise based stochastic calculus associated with a class of Gaussian processes. (English) Zbl 1255.60117

The purpose is to define a stochastic integral and integrating processes taking values in some space of stochastic distributions, namely the so-called Kondratiev space \(S_{-1}\); as integrator some Gaussian process \(X_n(t)\) having stationary increments is taken (\(X_n\) can be a fractional Brownian motion, for example).
The so-called Vage inequality allows to extend continuously the Wick product (defined for Hermite polynomials \(H_\alpha\) by \(H_\alpha\diamondsuit H_\beta:= H_{\alpha+\beta}\)) to Kondratiev stochastic distributions, to which the distributional derivative \(X_m'(t)\) pertains.
Therefore, the authors can define an Itô-Skorokhod-like stochastic integral as \(\int^t_0 Y(t)\diamondsuit dX_m(t)\). Then, they show that such Wick integral is also the \(S_{-1}\)-limit of the Riemannian sums \(\sum_i Y(t_i)(X_m(t_{i+1})- X_m(t_i))\), and, finally, they prove that the following Itô formula holds almost surely \[ df(X_m(t))= f'(X_m(t))\diamondsuit dX_m(t)+ {1\over 2} f''(X_m(t))\,dr(t), \] where \(r(t)\) denotes the Lévy-Khintchine function associated to the covariance function of \(X_m(t)\).

MSC:

60H40 White noise theory
60H05 Stochastic integrals
60G15 Gaussian processes
60G22 Fractional processes, including fractional Brownian motion
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv