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Projections of scaled Bessel processs. (English) Zbl 07088984
Summary: Let \(X\) and \(Y\) denote two independent squared Bessel processes of dimension \(m\) and \(n-m\), respectively, with \(n\geq 2\) and \(m \in [0, n)\), 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.

60G44 Martingales with continuous parameter
60G48 Generalizations of martingales
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J55 Local time and additive functionals
60J60 Diffusion processes
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