The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.

This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.