I'm looking for information on how to compute the distribution of the random vector


$$Z = \int\_0^t f(B\_s) ds$$


where $t>0$ is fixed, $B\_s$ is a 2D Brownian bridge with $B\_0 = 0$, $B\_t=b \in \mathbb{R}^2$, and $f : \mathbb{R}^2 \rightarrow \mathbb{R}^K$ has components $f\_k : \mathbb{R}^2 \rightarrow \mathbb{R}$ that are positive, bounded, and in $C^\infty$.


Here's a reference to the [local time of a 1D Brownian bridge](http://www.stat.berkeley.edu/tech-reports/539.pdf), that gives an explicit density for the local-time $L\_t(x)$ so a time-integral can be computed as $\int\_\mathbb{R} f(x)L\_t(x) dx$. But it doesn't appear to work for processes in the plane. 


Any pointers to relevant papers or textbook chapters would be appreciated. 


The motivation is a problem in experimental biophysics, which I'm happy to outline if others are interested. This is my first venture into stochastic integration.


  

**Edit: By request, a summary of the motivating problem**:


The overall problem is optical tracking of a microscopic biological process consisting of many component parts, some diffusing and others not. It requires estimating the size and rate of diffusion of an ensemble of particles in a membrane, under unfavorable conditions: particle sizes smaller than the optical resolution, other objects in the field, unknown per-particle brightness, and most significantly, a hidden process that adds new particles to the ensemble at random times. 


The random vector $Z$ is one piece of our probability model of this (rather complicated) biological process. It represents the light due to a single particle, collected for a brief exposure $[0,t)$ by a digital camera attached to our [microscope](http://en.wikipedia.org/wiki/Total_internal_reflection_fluorescence_microscope). Each of the scalar components $f\_k$ is the parameter of a Poisson random variable modeling the number of photons detected by pixel $k$ during the exposure. A simple model of the optics has 
$$f\_k(x) = \int\_{A\_k} g(x - y) dy$$
where $A\_k \subset \mathbb{R}^2$ is the rectangular area "seen" by pixel $k$ (so the $A\_k$ are disjoint) and $g$ is a Gaussian approximating the microscopes' [point spread function](http://en.wikipedia.org/wiki/Point_spread_function).


The fixed end-point $B\_t=b$ arises from the way we move from this continuous-time model to our discrete time experimental data. We use a [Hidden Markov Model](http://en.wikipedia.org/wiki/Hidden_Markov_model) in which each discrete time $i \in \mathbb{N}$ corresponds to an instant $t\_i$ between camera exposures; and the HMM emissions $Z$ are conditioned upon a transition $b\_i \mapsto b\_j$. Then we can evaluate the likelihood of an experimental observation, a complete digital movie containing $n$ exposures, in only $O(n)$ operations by using the [forward algorithm](http://en.wikipedia.org/wiki/Forward_algorithm) to iterate over the exposures.


I welcome comments and criticism of this approach.



Time-integral of a smooth, vector-valued function of a planar Brownian bridge
To compute the *distribution* of $Z$ for a general function $f$ might be difficult. To evaluate it, one could rewrite the process $(B\_s)\_{0\le s\le t}$ as $B\_s=W\_s+(b-W\_t)(s/t)$ for every $0\le s\le t$, where $(W\_s)\_{s\ge0}$ is a standard 2D Brownian motion starting from $W\_0=0$.


