SDE
Stochastic Differential Equations

SDE is a FORTRAN90 library which illustrates the properties of stochastic differential equations and some algorithms for handling them, making graphics files for processing and display by gnuplot, by Desmond Higham.

The library requires access to the QR_SOLVE library as well.

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

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

BLACK_SCHOLES, a FORTRAN90 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 FORTRAN90 library which generates samples of noise obeying a 1/f^alpha power law.

CORRELATION, a FORTRAN90 library which contains examples of statistical correlation functions.

GNUPLOT, FORTRAN90 programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

ORNSTEIN_UHLENBECK, a FORTRAN90 library which approximates solutions of the Ornstein-Uhlenbeck stochastic differential equation (SDE) using the Euler method and the Euler-Maruyama method.

PCE_BURGERS, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which computes a "pink noise" signal obeying a 1/f power law.

QR_SOLVE, a FORTRAN90 library which computes the least squares solution of a linear system A*x=b.

STOCHASTIC_DIFFUSION, a FORTRAN90 library which implements 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 FORTRAN90 library which applies a Runge Kutta (RK) scheme to a stochastic differential equation.

Author:

Original MATLAB version by Desmond Higham. FORTRAN90 version by John Burkardt.

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:

• sde.f90, the source code.

Examples and Tests:

• sde_test.f90, a sample calling program.
• sde_test.txt, the output file.

The calling program creates various data and command files which can be used with GNUPLOT to create images.

• bpath_commands.txt, commands to plot the data from BPATH.
• bpath_data.txt, data from BPATH.
• bpath.png, a plot of the BPATH data.
• bpath_average_commands.txt, commands to plot the data from BPATH_AVERAGE.
• bpath_average_data.txt, data from BPATH_AVERAGE.
• bpath_average.png, a plot of the BPATH_AVERAGE data.
• chain_commands.txt, commands to plot the data from CHAIN.
• chain_data.txt, data from CHAIN.
• chain.png, a plot of the CHAIN data.
• em_commands.txt, commands to plot the data from EM.
• em1_data.txt, data from EM.
• em2_data.txt, data from EM.
• em.png, a plot of the EM data.
• emstrong_commands.txt, commands to plot the data from EMSTRONG.
• emstrong_data.txt, data from EMSTRONG.
• emstrong.png, a plot of the EMSTRONG data.
• emweak0_commands.txt, commands to plot the data from EMWEAK with the standard EM method.
• emweak0_data.txt, data from EMWEAK with the standard EM method.
• emweak0.png, a plot of the data from EMWEAK with the standard EM method.
• emweak1_commands.txt, commands to plot the data from EMWEAK with the weak EM method.
• emweak1_data.txt, data from EMWEAK with the weak EM method.
• emweak1.png, a plot of the data from EMWEAK with the weak EM method.
• milstrong_commands.txt, commands to plot the MILSTRONG data.
• milstrong_data.txt, data from MILSTRONG.
• milstrong.png, a plot of the strong convergence data from MILSTRONG.
• stab_asymptotic_commands.txt, commands to plot the STAB_ASYMPTOTIC data.
• stab_asymptotic_data1.txt, data from STAB_ASYMPTOTIC.
• stab_asymptotic_data2.txt, data from STAB_ASYMPTOTIC.
• stab_asymptotic_data3.txt, data from STAB_ASYMPTOTIC.
• stab_asymptotic.png, a plot of the asymptotic stability results from STAB_ASYMPTOTIC.
• stab_meansquare_commands.txt, commands to plot the STAB_MEANSQUARE data.
• stab_meansquare_data1.txt, data from STAB_MEANSQUARE.
• stab_meansquare_data2.txt, data from STAB_MEANSQUARE.
• stab_meansquare_data3.txt, data from STAB_MEANSQUARE.
• stab_meansquare.png, a plot of the meansquare stability results from STAB_MEANSQUARE.

List of Routines:

• BPATH performs a Brownian path simulation.
• BPATH_AVERAGE: displays the average of 1000 Brownian paths.
• BPATH_AVERAGE_GNUPLOT writes a GNUPLOT input file to plot BPATH_AVERAGE data.
• BPATH_GNUPLOT writes a GNUPLOT input file to plot BPATH data.
• CHAIN tests the stochastic Chain Rule.
• CHAIN_GNUPLOT writes a GNUPLOT input file to plot CHAIN data.
• EM applies the Euler-Maruyama method to a linear SDE.
• EM_GNUPLOT writes a GNUPLOT input file to plot EM data.
• EMSTRONG tests the strong convergence of the EM method.
• EMSTRONG_GNUPLOT writes a GNUPLOT input file to plot EMSTRONG data.
• EMWEAK tests the weak convergence of the Euler-Maruyama method.
• EMWEAK_GNUPLOT writes a GNUPLOT input file to plot EMWEAK data.
• FILENAME_INC increments a partially numeric filename.
• GET_UNIT returns a free FORTRAN unit number.
• MILSTRONG tests the strong convergence of the Milstein method.
• MILSTRONG_GNUPLOT writes a GNUPLOT input file to plot MILSTRONG data.
• R8_NORMAL_01 returns a unit pseudonormal R8.
• R8_SIGN returns the sign of an R8.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_NORMAL_01 returns a unit pseudonormal R8MAT.
• R8VEC_MEAN returns the mean of an R8VEC.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• STAB_ASYMPTOTIC examines asymptotic stability.
• STAB_MEANSQUARE examines mean-square stability.
• STOCHASTIC_INTEGRAL_ITO approximates the Ito integral of W(t) dW.
• STOCHASTIC_INTEGRAL_STRAT approximates the Stratonovich integral of W(t) dW.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.