Smoothness of the distribution of the supremum of a multi-dimensional diffusion process (Q1935422)

Consider a multi-dimensional diffusion process \((X_t^i;\;0\leq t\leq T,\,1\leq i\leq d)\) and assume that the associated vector fields are commutative. Under some assumption, one can prove that almost surely, the first component \(X^1\) attains its maximum \(F^1=\sup_{0\leq t\leq T}X_t^1=X_{\theta^1}^1\) at a unique random time \(\theta^1\). The authors apply Malliavin's calculus and prove that the random vector \(F=(X^i_{\theta^1};\;1\leq i\leq d)\) has a smooth density on \((X_0^1,\infty)\times\mathbb{R}^{d-1}\).