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, 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. 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.
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 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 to iterate over the exposures.
I welcome comments and criticism of this approach.
