Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum

In this article, we consider an m-dimensional stochastic differential equation with coefficients which depend on the maximum of the solution. First, we prove the absolute continuity of the law of the solution. Then we prove that the joint law of the maximum of the ith component of the solution and the i′th component of the solution is absolutely continuous with respect to the Lebesgue measure in a particular case. The main tool to prove the absolute continuity of the laws is Malliavin calculus.