SDE Toolbox

by Umberto Picchini

News - Screenshots - Requirements - Download - Intentions - Documentation - Restrictions - Credits - License - References

Warning! SDE Toolbox is no more developed but it's still downloadable. Its inferential capabilities can be considered surpassed (at best). Actually the parameter estimation methods were already far from the state-of-art when the project began in 2007 (!). The considered implemented parametric and non-parametric Monte Carlo likelihood methods were chosen for their ability to treat both one-dimensional and multivariate SDE systems, although the quality of the inferential results can't match those obtained using more advanced techniques.
Nevertheless the toolbox capabilities to simulate numerical solutions of SDE systems are still valid and can serve as a useful starting point to those willing to simulate stochastic dynamical models easily.

SDE Toolbox is a free MATLAB® package to simulate the solution of a user defined Itô or Stratonovich stochastic differential equation (SDE), estimate parameters from data and visualize statistics; users can also simulate an SDE model chosen from a model library.

More in detail, the user can specify:
- the Itô or the Stratonovich SDE to be simulated. This can be a user defined SDE or an SDE chosen from the SDE Toolbox models library;
- the SDE structural parameter values;
- the number of the SDE's solution trajectories to be simulated;
- the numerical integration method (Euler-Maruyama or Milstein);
- the time interval [t0,T] to be considered;
- the integration stepsize;
- the parameter estimation method;

to obtain (see also the screenshots):
- parameter estimates and confidence intervals;
- plot(s) of the solution trajectories;
- plot(s) of the trajectories empirical mean, together with their 95% confidence bands and the 1st and 3rd quartiles;
- histogram(s) of the trajectories distribution at the endtime T;
- Monte-Carlo statistics of the solution process at the endtime T, i.e. mean, moments, skewness, kurtosis, 95% confidence bands etc.;


5th December 2007: Version 1.4.1 is available here. (i) This version includes faster implementations of the parameter estimation procedures: the parametric estimation procedure (SDE_PSML.m) speed has been boosted for the case of multi-dimensional SDEs, now it is 14x-27x times faster, depending on the machine; negligible improvement for the non-parametric estimation procedure (SDE_NPSML.m); no improvement for the estimation of one-dimensional SDEs. (ii) The parametric estimation with Milstein integration scheme (SDE_PSML_milstein.m) has been removed: in fact the parametric SML procedure, as presented in Pedersen (1995) and Brandt-Santa-Clara (2002), is defined only with respect to the Euler-Maruyama (EM) approximation. Scroll this page for the complete list of the changes.
26th November 2007: Version 1.4.0 is available here. With this new version the use of global variables is avoided (except for the demo files): this makes the present version much more capable of interfacing with other Matlab programs, but the structure of the 'sdefiles' is different. Thus sdefiles created under previous versions will not work with v. 1.4.0 (however, it is straightforward to adapt them, just look at the new sdefiles in the "models_library" folder); as a consequence the Toolbox can be used without necessarily running SDE_library_run.m, and can be integrated into user defined Matlab programs; several examples are provided in the User's Guide. Approximated parameters 95% intervals can now be calculated. Scroll this page for the complete list of the changes.
10th September 2007: Version 1.3.0 is available here. With this new version it is possible to estimate the parameters of the SDE model from data. Two methods are implemented: a parametric and a non-parametric one. See the User's Guide for details.

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Only MATLAB base is required to run the toolbox. SDE Toolbox has been developed and tested with MATLAB 6.5 (R13) and 7.3.0 (R2006b) for Windows; however it should work under different platforms/versions as well.


SDE Toolbox is free software (with some restrictions) hosted on SourceForge and can be downloaded here.


SDE Toolbox is not intended to provide a complete package for the numerical treatment of SDEs: this is a free toolbox for simulating sample paths of an SDE solution, computing statistics and estimating the parameters from data. Other important issues (e.g. stability of the solutions) are not treated. This has to be intended as a customizable piece of code which, in the author's intentions, should furnish ideas to stimulate users in developing their own SDEs package, and give some programming hints to SDEs newbies.
From a numerical point of view, users are highly encouraged in giving a look into the References section, in particular the excellent monographies [1,2] and the article [3], the latter giving a MATLAB-based introduction to SDE simulation. Other useful references for numerical methods are [4,5,6,7]. Highly specialistic references for SDE theory and stochastic calculus are [8,9,10,11]; important references for parameter estimation of diffusion processes are [12,13]. See also the Toolbox User's Guide and references therein.


Take a look at the pdf User's Guide (~3.4 Mb). I suggest to download it (click the right mouse button and select "save target as") instead of open it with a browser.


This program is free software (read the License); however if you have used it in your researches and if you have published any results, please give me credit and cite my work as:

Umberto Picchini. SDE Toolbox: Simulation and Estimation of Stochastic Differential Equations with MATLAB,

Furthermore you are encouraged to send me a corresponding reprint.


Copyright (C) 2007-2013, Umberto Picchini.
Early versions of this toolbox have been created with the support of the Biomathematics Laboratory at the Institute for Systems Analysis and Informatics "A. Ruberti", organ of the Italian National Research Council (CNR); though, possible bugs, errors and misprints are on my own responsibility.


This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see


[1] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations. Springer, 1992.
[2] Peter E. Kloeden, Eckhard Platen and Henri Schurz. Numerical solution of SDE through computer experiments. Springer, 1994.
[3] Desmond J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM J. Numer. Anal., 43(3):525–546, 2001.
[4] K. Burrage and P.M. Burrage. High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations. Applied Numer. Mathematics, 22:81–101, 1996.
[5] P.M. Burrage and K. Burrage. A variable stepsize implementation for stochastic differential equations. SIAM J. Sci. Comput., 24(3):848–864, 2002.
[6] Pamela Marion Burrage. Runge–Kutta methods for stochastic differential equations. PhD thesis, Department of Mathematics, University of Queensland (Australia), 1999.
[7] Andreas Rößler. Runge–Kutta methods for the numerical solution of stochastic differential equations. PhD thesis, Department of Mathematics, University of Darmstadt, 2003.
[8] Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus. Springer- Verlag, 1991.
[9] Bernt Øksendal. Stochastic differential equations: an introduction with applications. Springer, second edition, 2000.
[10] L.C.G. Rogers and David Williams. Diffusions, Markov processes and martingales. Volume 2: Itô calculus. John Wiley & Sons, 1987.
[11] D.W. Stroock and S.R.S. Varadhan. Multidimensional Diffusion Processes. Springer-Verlag, 1979.
[12] B.L.S. Prakasa Rao. Statistical Inference for Diffusion Type Processes. Arnold, London and Oxford University press, New York, 1999.
[13] Y.A. Kutoyants. Statistical Inference for Ergodic Diffusion Processes. Springer, London, 2004.