On symmetric and skew Bessel processes

We consider the one-dimensional stochastic differential equation Xt=x0+Bt+∫0tδ−12Xsds, where δ∈(1,2)δ∈(1,2), x0∈Rx0∈R and BB is a Brownian motion. For x0≥0x0≥0, this equation is known to be solved by the δδ-dimensional Bessel process and to have many other solutions. The purpose of this paper is to identify the source of non-uniqueness and, from this insight, to transform the equation into a well-posed problem. In fact, we introduce an additional parameter and for each admissible value of this parameter we construct a unique (in law) strong Markov solution of this equation. These solutions are the skew and symmetric Bessel processes, respectively. Moreover, we uncover an alternative way to introduce the δδ-dimensional Bessel process.