SDE
Stochastic Differential Equations

SDE is a MATLAB library which illustrates the properties of stochastic differential equations and some algorithms for handling them, by Desmond Higham.

The original version of these routines is available at "http://www.maths.strath.ac.uk/~aas96106/algfiles.html".

Languages:

SDE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BLACK_SCHOLES, a MATLAB library which implements some simple approaches to the Black-Scholes option valuation theory, by Desmond Higham.

BROWNIAN_MOTION_SIMULATION, a MATLAB program which simulates Brownian motion in an M-dimensional region.

CNOISE, a MATLAB library which generates samples of noise obeying a 1/f^alpha power law, by Miroslav Stoyanov.

COLORED_NOISE, a MATLAB library which generates samples of noise obeying a 1/f^alpha power law.

CORRELATION, a MATLAB library which contains examples of statistical correlation functions.

PCE_BURGERS, a MATLAB program which defines and solves a version of the time-dependent viscous Burgers equation, with uncertain viscosity, using a polynomial chaos expansion in terms of Hermite polynomials, by Gianluca Iaccarino.

PCE_LEGENDRE, a MATLAB program which assembles the system matrix associated with a polynomal chaos expansion of a 2D stochastic PDE, using Legendre polynomials;

PCE_ODE_HERMITE, a MATLAB program which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.

PINK_NOISE, a MATLAB library which computes a "pink noise" signal obeying a 1/f power law.

STOCHASTIC_DIFFUSION, MATLAB functions which implement several versions of a stochastic diffusivity coefficient.

STOCHASTIC_GRADIENT_ND_NOISE, a MATLAB program which solves an optimization problem involving a functional over a system with stochastic noise.

STOCHASTIC_RK, a MATLAB library which applies a Runge Kutta (RK) scheme to a stochastic differential equation.

Author:

Desmond Higham

Reference:

1. Desmond Higham,
   An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations,
   SIAM Review,
   Volume 43, Number 3, September 2001, pages 525-546.

Source Code:

• bpath.m computes one simulation of discretized Brownian motion over the time interval [0,1] using 500 time steps and a user specified random number seed.
• bpath_vectorized.m, a second version of BPATH using faster, vectorized commands.
• bpath_average.m, displays the average of 1000 Brownian paths.
• chain.m tests the stochastic Chain Rule.
• em.m, applies the Euler-Maruyama method to integrate a linear SDE.
• emstrong.m tests the strong convergence of the Euler-Maruyama method.
• emweak.m tests the weak convergence of the Euler-Maruyama method.
• milstrong.m tests the strong convergence of the Milstein method.
• stab_asymptotic.m examines the asymptotic stability of the Euler-Maruyama method applied to a stochastic differential equation.
• stab_meansquare.m examines the mean-square stability of the Euler-Maruyama method applied to a stochastic differential equation.
• stochastic_integral_ito.m, approximates the stochastic integral W(t) dW using the Ito integral.
• stochastic_integral_strat.m, approximates the stochastic integral W(t) dW using the Stratonovich integral.
• timestamp.m, prints the YMDHMS date as a timestamp.

Examples and Tests:

• sde_test.m, calls all the tests.
• sde_test_output.txt, the output file.
• stochastic_integral_ito_test.m, tests stochastic_integral_ito.
• stochastic_integral_strat_test.m, tests stochastic_integral_strat.

A number of graphics images are created by the example programs:

• bpath.png, an image of the computed path for BPATH.
• bpath_vectorized.png, an image of the computed path for BPATH_VECTORIZED.
• bpath_average.png, an image of the averaged paths for BPATH_AVERAGE.
• chain.png, an image comparing solutions done with and without the chain rule.
• em.png, an image of a true solution versus the Euler-Maruyama estimate.
• emstrong.png, an image of the strong convergence of the Euler-Maruyama error with stepsize.
• emweak0.png, an image of the weak convergence of the standard Euler-Maruyama method.
• emweak1.png, an image of the weak convergence of the weak Euler-Maruyama method.
• milstrong.png, an image of the strong convergence of the Milstein method.
• stab_asymptotic.png, an image of an asymptotic stability check.
• stab_meansquare.png, an image of a meansquare stability check.

You can go up one level to the MATLAB source codes.