1
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
3
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.