The paper deals with a definition and an approximation of symmetric (Stratonovich) stochastic integralwith 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 BΔa polygonal approximation of B corresponding to a partition Δ of [0,1] and. We prove thatandconverge in mean toas the mesh(the later case needs additional assumptionor non-randomness of L).