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

TL;DR

This paper generalizes the Kunita-Watanabe decomposition for $L^2$ spaces involving nonlinear stochastic integrals with continuous martingales, linking regularity properties and solving an associated optimization problem.

Contribution

It introduces a novel decomposition for $L^2$ spaces with nonlinear stochastic integrals, connecting regularity conditions and providing explicit solutions to related optimization problems.

Findings

Established a generalized decomposition for nonlinear stochastic integrals.

Demonstrated the relation between martingale regularity and integrand regularity.

Provided an explicit example solving the optimization problem.

Abstract

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.