  We investigate distributions of hyperbolic Bessel processes. We find links
between the hyperbolic cosine of hyperbolic Bessel processes and functionals of
geometric Brownian motion. We present an explicit formula for the Laplace
transform of the hyperbolic cosine of a hyperbolic Bessel process and some
other interesting probabilistic representations of this Laplace transform. We
express the one-dimensional distribution of a hyperbolic Bessel process in
terms of other, known and independent processes. We present some applications
including a new proof of Bougerol's identity and its generalization. We
characterize the distribution of the process which is the hyperbolic sine of
hyperbolic Bessel process.
