Preliminaries

An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. It is known that this process is equivalent to a Brownian Motion conditioned to always be positive.

A Bessel Bridge is a Bessel Process on time interval $[0, 1]$, conditioned to have start point $(0, x_0)$ and end point $(1, x_f)$.

My Question

I am trying to find a density function for the random variable $\int_0^1 \beta_3(t) dt$, where $\beta_3(t)$ is a random realization of a three-dimensional Bessel Bridge.

A Possibly Useful Fact

When $x_0 = x_f = 0$, the Bessel Bridge is called a Brownian Excursion Process, and the density function for its integral is known.

Thanks!

Preliminaries

An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. It is known that this process is equivalent to a Brownian Motion conditioned to always be positive.

A Bessel Bridge is a Bessel Process on time interval $[0, 1]$, conditioned to have start point $(0, x_0)$ and end point $(1, x_f)$.

My Question

I am trying to find a density function for the random variable $\int_0^1 \beta_3(t) dt$, where $\beta_3(t)$ is a random realization of an order-3 Bessel Bridge.

A Possibly Useful Fact

When $x_0 = x_f = 0$, the Bessel Bridge is called a Brownian Excursion Process, and the density function for its integral is known.

Thanks!