On symmetric and skew Bessel processes

We consider the one-dimensional stochastic differential equation X t = x 0 + B t + ∫ 0 t δ − 1 2 X s d s , where δ ∈ ( 1 , 2 ) , x 0 ∈ R and B is a Brownian motion. For x 0 ≥ 0 , this equation is known to be solved by the δ -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 δ -dimensional Bessel process.