Some remarkable pure martingales (Q1868122)

A continuous martingale \((M_t)\) is said to be pure if the increasing process \((\langle M\rangle_t)\) is measurable w.r.t. its DDS-Brownian motion \(\gamma\) (defined by the relation \(M_t= \gamma_{\langle M,M\rangle_t}\), \(t\geq 0\)). For some Brownian motion \(B\) and \(n\in\mathbb{N}\), set \(M_t= \int^t_0 B^n_s dB_s\). \textit{D. W. Stroock} and \textit{M. Yor} [in: Séminaire de probabilités XV. Lect. Notes Math. 850, 590-603 (1981; Zbl 0456.60048)] have shown that, if \(n\) is odd, the martingale \((M_t)\) is pure. The question about the pureness of \((M_t)\) when \(n\) is even remained open till now and is solved finally in the present paper. This is done by using the relation between the martingale \((M_t)\) and a stochastic Bessel equation \[ dX_t= dB_t+ a{dt\over X_t}\tag{1} \] of dimension \(d= 2a+ 1\in ]1,2[\). Some other properties of the solutions of (1) are investigated.