Multiple stochastic fractional integrals: A transfer principle for multiple stochastic fractional integrals
The single Wiener integral representation of fractional Brownian motion \(B^H_t = \int K(t,s) dW_s\) using a deterministic two-parameter fractional kernel \(K\) is extended to a multiple Wiener integral representation of the fractional multiple stochastic integrals of \textit{A. Dasgupta} and \textit{G. Kallianpur} [Probab. Theory Relat. Fields 115, 505-525 (1999; Zbl 0948.60022)] and \textit{T. E. Duncan, Y. Hu} and \textit{B. Pasik-Duncan} [SIAM J. Control Optimization 38, 582-612 (2000; Zbl 0947.60061)], using a continuous fractional mapping on \(L^2([0,T]^n)\). These integrals are then related to multiple fractional Stratonovich integrals via a Hu-Meyer type formula. A fractional anticipating integral is defined through chaos decompositions, and a related anticipating affine equation is solved. Finally, a strong law of large numbers is formulated for the fractional Brownian motion.
scientific article; zbMATH DE number 1933102
scientific article