Bessel process conditioned to stay positive

Question

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Suppose the stochastic process $(X_t)_{t\ge 0}$ with start in $X_0:=x>0$ is the solution of the SDE $$ dX_t = dB_t + \frac{\rho-1}{2X_t} \, dt $$ with $B_t$ denoting Brownian motion known as the generalized Bessel process with dimension parameter $\rho\in\mathbb R$. Let us assume we have $\rho\in (0,2)$ such that it is possible to construct the reflected Bessel process.

Can anything be said about the law of the Bessel process conditioned to never hit 0? More specifically, is there a Law of the Iterated Logarithm? (If $\rho=1$ it holds $X_t=B_t$, so the desired law is that of a 3-dimensional Bessel process. But what about general $\rho\in(0,2)$?)

Answer 1

This conditioning of a Bessel process is a Doob transform. For $\rho$ in $(0,2)$ it leads to a Bessel process of dimension $4-\rho$. See Goeing-Jaeschke, A., Yor, M. (2003) A Survey on some generalizations of Bessel processes. Bernoulli 9, 313–349 and the book by Revuz-Yor, Continuous Martingales and BM. the LIL for Bessel processes is the same as for Brownian motion and is thus not affected by the conditioning. See (Revuz and Yor, 1994, Exercise X1.1.20).