Projections of scaled bessel processes

Let X and Y denote two independent squared Bessel processes of dimension m and n-m, respectively, with n ≥ 2 and m ∈ [0, n), making X+Y a squared Bessel process of dimension n. For appropriately chosen function s, the process s(X + Y) is a local martingale. We study the representation and the dynamics of s(X + Y), projected on the filtration generated by X. This projection is a strict supermartingale if, and only if, m < 2. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of X at zero.