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

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