Martingale decomposition of a $L^2$ space with nonlinear stochastic integrals

This paper presents a generalization of the Kunita-Watanabe decomposition of a $L^2$ space with nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$. To get the result, a useful relation between the regularity of the martingale family respect to its parameter and the regularity of the integrand in its martingale decomposition is shown.The decomposition presented in the main result is also the solution of an optimization problem in $L^2$. Finally, an example is given where the optimization problem is solved explicitely.