Ornstein Uhlenbeck Simulations and Descretisation error

Ornstein Uhlenbeck simulations based on simple discretisation and compared to Gillespie solution

1. Introduction
The attached matlab code simulates the Ornstein Uhlenbeck process and the example shows how the paths mean revert.

The aim of the script is to highlight how the discretization error in the finite difference approximation affects the results by examining the first and second moments of the expected results.

The simulations from the simple discretisation approach are compared with the Gillespie solution that works for any time step.

2. Running the code

The script:
OrnsteinUhlenbeckControl.m

can be run to show how to the model is initialised that will output several figures highlighting simulations and validation.

The main engines determining the simulations:

Simple discretisation:
OrnsteinUhlenbeck.m

Gillespie solution:
OrnsteinUhlenbeckGillespie.m

Simple discretisation with option to reduce time step:
OrnsteinUhlenbeckReducedTimeStep.m

3. References

[1] “The Stochastic Behaviour of Commodity Prices: Implications for Valuation and Hedging,” E.S. Schwartz, “J. of Finance, V. 52, Issue 3, 1997.

[2] “Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral,” D. T. Gillespie, Physical Review E, August 1996.