Let $(X_{t})_{t\geq 0}$ be a Bessel Process starting at $x>0$ of dimension $\delta>0$. Namely \begin{align*} X_{t}=x+W_{t}+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{X_{s}}\, ds. \end{align*} where $(W_{t})_{t\geq 0}$ is a Brownian Motion. I am interested in how to find the distribution of the Hitting Time $\tau:=\inf\{t>0\,|\, X_{t}=0\}$.

Given that the Bessel Process can be expressed as a time changed Brownian Motion, up to the first hitting of the boundary, is it possible to obtain the distribution of $\tau$ by utilising the Reflection Principle for Brownian Motion?

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    $\begingroup$ Did you take a look at the paper Hitting times of Bessel processes by my colleagues Byczkowski, Małecki and Ryznar? Google also suggests several other papers, including this and this. $\endgroup$ – Mateusz Kwaśnicki Jun 19 '18 at 12:31
  • $\begingroup$ Oh, only now I noticed that the question asks about the hitting time of zero, not of an arbitrary point. The references that I gave above deal with the general case. $\endgroup$ – Mateusz Kwaśnicki Jun 20 '18 at 18:04

Since the Bessel process has Brownian scaling, we have $$ \mathbb{P}^x(\tau<t)=F(xt^{-\frac12}) $$ for some unknown function $F$, where $\mathbb{P}^x$ denotes the probability for the Bessel process started from $x$. Now, either from Fokker-Plank equation, or by Ito calculus, or conditioning on the exit time and position from an $\epsilon$-neighborhood of the starting point and sending $\epsilon$ to zero, you can see that $F$ satisfies the following differential equation: $$ F'(y)\left(y+\frac{\delta-1}{y}\right)+F''(y)=0, $$ which can be easily solved: $$ F(y)=c_1\int^ys^{1-\delta}e^{-s^2/2} ds +c_2. $$ It remains to choose the constants so that $F(0)=1$ and $F(+\infty)=0$. This is of course not possible for $\delta\geq 2$, due to the fact that in this range the process never hits zero.

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  • $\begingroup$ This is a great answer! Can you recommend a reference on where this derivation is done? $\endgroup$ – fast_and_fourier Jun 24 '18 at 6:48

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