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Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0>0$ is the solution of the SDE $$d X_t = 2\,\sqrt{X_t}\,d W_t+\delta_0\,d t.$$ A naive application of the Girsanov Theorem seems to imply that, in some equivalent new measure (i.e., absolutely continuous w.r.t. the old one), the same process may have a different value $\delta_1$ of the dimension parameter.

I believe that this is not really possible to do rigorously, at least not for an arbitrary pair of values $\delta_0,\delta_1\geq 0$. What is the exact statement of result here --- is it ever possible to change the dimension of a Bessel process by an absolutely continuous change of measure, and if so, under what conditions on $\delta_0$ and $\delta_1$?

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  • $\begingroup$ I think this is Proposition 2.2 of Lawler's notes. $\endgroup$
    – Nate Eldredge
    Jun 1, 2020 at 22:49
  • $\begingroup$ Thank you, but up to the hitting time of 0 it seems much easier: the process can be bounded from below, so the drift correction remains bounded from above and Girsanov Theorem applies. So it probably implies that if both old and new dimensions are at least 3, then the answer to my question is "yes" because the hitting time is then infinite a.e. And in the other cases? $\endgroup$
    – Serge Resnick
    Jun 2, 2020 at 1:53

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