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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.

MSC:
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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