# OrnsteinUhlenbeckBridge package

Version **1.0.0** by Lawrence Murray

Ornstein--Uhlenbeck model for diffusion bridge sampling.

## Synopsis

This simulates a number of data sets for testing. GNU Octave is
required. Running it is optional, as a number of simulated data sets are
already included.

This runs a particle filter as well as samples from the posterior
distribution for a data set of a single observation at time 1.

This runs tests on the bootstrap and bridge particle filters on the simulated
data sets. Alternatively, these tests may be run as an array job on a cluster:

```
qsub -t 0-15 qsub_test_bridge.sh
qsub -t 0-15 qsub_test_bootstrap.sh
```

Finally, results may be plot with:

```
octave --path oct/ --eval "plot_and_print"
```

GNU Octave and OctBi are required.

Note that, as of version 1.1.0 of LibBi, running any of these gives the
warnings:

```
Warning (line 29): 'obs' variables should not appear on the right side of actions in the 'transition' block.
Warning (line 42): 'obs' variables should not appear on the right side of actions in the 'lookahead_transition' block.
```

This is normal.

## Description

This package includes an Ornstein–Uhlenbeck model that is observed
directly. The task is to simulate diffusion bridges between the observed
values. The form of the model is as studied in Aït-Sahalia (1999):

\[dx=(\theta_{1}-\theta_{2}x)\, dt+\theta_{3}\, dW,\]
with fixed parameters $\theta_1 = 0.0187$, $\theta_2 = 0.2610$ and $\theta_3 =
0.0224$.

It was used as a test case in Del Moral & Murray (2014). The package may be
used to reproduce the results in that paper.

## References

Aït-Sahalia, Y. Transition Densities for Interest Rate and Other Nonlinear
Diffusions. *The Journal of Finance*, 1999, 54, 1361–1395.

Del Moral, P. & Murray, L. M. Sequential Monte Carlo with Highly Informative
Observations, 2014. [arXiv]