SDE
Stochastic Differential Equations

SDE is a FORTRAN77 library which illustrates the properties of stochastic differential equations and some algorithms for handling them, 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 described and 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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BLACK_SCHOLES, a FORTRAN77 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 FORTRAN77 library which generates samples of noise obeying a 1/f^alpha power law.

CORRELATION, a FORTRAN77 library which contains examples of statistical correlation functions.

GNUPLOT, examples which illustrate the use of the gnuplot graphics program.

PCE_BURGERS, a FORTRAN77 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 FORTRAN77 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 FORTRAN77 library which computes a "pink noise" signal obeying a 1/f power law.

QR_SOLVE, a FORTRAN77 library which computes the least squares solution of a linear system A*x=b.

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 FORTRAN77 library which applies a Runge Kutta (RK) scheme to a stochastic differential equation.

Author:

Original MATLAB version by Desmond Higham. FORTRAN77 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.f, the source code.
• sde.sh, BASH commands to compile the source code.

Examples and Tests:

• sde_prb.f, a sample calling program.
• sde_prb.sh, BASH commands to compile and run the sample program.
• sde_prb_output.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 FORTRAN77 source codes.