Are there any explicit path-wise solutions for a 3 dimensional Bessel process?

E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.


1 Answer 1


As mentioned in the surveys https://almostsuremath.com/2010/07/28/bessel-processes/ and https://www.math.uchicago.edu/~lawler/bessel18new.pdf, one starts in reverse i.e.

the absolute value of a d-dimensional Brownian motion is a Bessel process with parameter a = (d − 1)/2 for $dX_{t}=\frac{a}{X_{t}}dt+dW_{t}$.

And they use properties of this Bessel-sde to understand the norm $|B_{t}|$.


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