Integrating a Bessel Bridge - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:01:36Z http://mathoverflow.net/feeds/question/112010 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112010/integrating-a-bessel-bridge Integrating a Bessel Bridge GB 2012-11-10T18:40:37Z 2012-11-10T19:06:48Z <p><strong>Preliminaries</strong></p> <p>An order-3 <em>Bessel Process</em> 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.</p> <p>A <em>Bessel Bridge</em> is a Bessel Process on time interval $[0, 1]$, conditioned to have start point $(0, x_0)$ and end point $(1, x_f)$.</p> <p><strong>My Question</strong></p> <p>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.</p> <p><strong>A Possibly Useful Fact</strong></p> <p>When $x_0 = x_f = 0$, the Bessel Bridge is called a <em>Brownian Excursion Process</em>, and the density function for its integral is known.</p> <p>Thanks!</p>