Symmetric stochastic integrals and their approximations

The paper deals with a definition and an approximation of symmetric (Stratonovich) stochastic integral with respect to Brownian motion B using the representation of a square‐integrable process X in the form . We show that a natural definition is possible provided a certain Hilbert‐Schmidt operator related to the kernel L is nuclear (trace class). Denote BΔ a polygonal approximation of B corresponding to a partition Δ of [0,1] and . We prove that and converge in mean to as the mesh (the later case needs additional assumption or non-randomness of L).