Stochastic Differential Equation: Approximations Method

This project implements the Euler-Maruyama scheme to approximate the true solution of an Ito stochastic differential equation (SDE).

Building

First, clone the repository:

git clone https://github.com/bdosse-jovian/approximations-eds.git
cd approximations-eds

Build the project:

make

Start the calculation:

cd bin/
./compute_approximation.exe

Without any modification of the source file, the program will compute the pathwise approximation of an Ito process over the $[0, 1]$ time interval. The process is solution of the Ito SDE:

$$ dX_t = -X_tdt + dW_t $$

where $W_t$ is a standard Wiener process. The data are stored in the ./data/data_1.csv file. The file may be imported in any data manipulation program like R or LibreOffice Calc. The precision is, by default, set to $2^{-7}$.

The CSV file has no header line, and two (or three, see the COMPARE directive in config.h for more details) columns. The first column contains time information, whereas the second contains position information. The third and optional one contains the position according to a reference process.

A file named calculation.R computes empirical means and deviation of the absolute error at time $T$ ($T$ being the time boundary) for given datasets.

Configuration

To configure this program, you need to modify the config.h file in the src folder, and recompile the program.

The configuration file describes each variable in a meaningful way, so read the comments carefully before any modification. In particular, it is in the config.h file that you define the functions of your Ito SDE.

Debug, cleaning, etc.

Enabling debug symbols is made by using the following recipe:

make debug

In order to get rid of object files:

make clean

It is possible to get rid of every generated files by issuing:

make mrproper

How to help?

There are things I'm unable to do at the moment. Here is a list of some of them:

• Implement the Milstein scheme (with numerical differentiation utility)
• Implement a Runge-Kutta scheme (is there something better than second strong order of convergence?)
• Implement something to play with Ito-Taylor expansions
• Connection with a plotting software like gnuplot
• Compute pathwise approximations and reference process given an array of seed for the pseudo-random number generator

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References

KLOEDEN, Peter E. and Eckhard PLATEN. Numerical Solution of Stochastic Differential Equations. 1st ed. Vol. 23. Berlin : Springer, 1992. XXXVI, 636 p. (Applications of Mathematics). ISBN 978-3-662-12616-5