5
$\begingroup$

I want to simulate a nonlinear stochastic differential equation $$ {\rm d}X_t = f(X_t) {\rm d}t + g(X_t){\rm d}B_t $$ where $f,g \in C^{\infty}({\mathbb R}^n ,{\mathbb R})$ and $B_t$ is one-dimensional standard brownian motion. How can I do it by MATLAB?

A good simulation file for a similar equation might be helpful.

$\endgroup$
12
$\begingroup$

Check out this paper by Des Higham and the SDETools MATLAB toolbox.

$\endgroup$
  • 6
    $\begingroup$ @Mohammad Khosravi: I was just in the process of typing out my own answer - I'm the author of SDETools so I can't say much more. From your notation, I assume that you have an Itô SDE. First, learn about the very simple Euler-Maruyama method. The Higham paper is a gentle intro, and then if you'e at all familiar with Matlab's ODE suite (e.g., ode45) you should find SDETools easy to use. $\endgroup$ – horchler Sep 2 '13 at 15:27
  • $\begingroup$ @Mohammad : the Des Higham paper cited by Juan is excellent. You can probably find it for free if you do a little Googling (I did). Warning: there may be a link to some Matlab files that is out-of-date. The files are now at personal.strath.ac.uk/d.j.higham/algfiles.html . Second warning: I am having a lot of trouble getting the Octave versions of those files (there is a link to the Octave files at the link I just gave) to work. I am an Octave/Matlab novice. I have not tried the Matlab versions yet. $\endgroup$ – Stefan Smith Sep 4 '13 at 22:47
  • $\begingroup$ @Mohammad : I just downloaded and executed the Matlab files at the link I gave above, in Matlab 2013, and they all ran just fine, despite my trouble with the Octave files I mentioned. $\endgroup$ – Stefan Smith Sep 5 '13 at 19:10
1
$\begingroup$

$$dx_{t}=2x_{t} dt +x_{t} dw_{t}\\x_{0}=1$$

% EM Euler-Maruyama method on linear SDE  
%  
% SDE is dX = lambda*X dt + mu*X dW, X(0) = Xzero,  
% where lambda = 2, mu = 1 and Xzero = 1.  
%  
% Discretized Brownian path over [0,1] has dt = 2^(-8).  
% Euler-Maruyama uses timestep R*dt.  

state=randn(100)  
lambda = 2;  
mu = 1;  
Xzero = 1;   % problem parameters  
T = 1;  
N = 2^8;  
dt = 1/N;  
dW = sqrt(dt)*randn(1,N); % Brownian increments  
W = cumsum(dW); % discretized Brownian path  
Xtrue = Xzero*exp((lambda-0.5*mu^2)*([dt:dt:T])+mu*W);  
plot([0:dt:T],[Xzero,Xtrue]), hold on  
R = 4;  
Dt = R*dt;  
L = N/R; % L EM steps of size Dt = R*dt  
Xem = zeros(1,L); % preallocate for efficiency  
Xtemp = Xzero;  
for j = 1:L  
Winc = sum(dW(R*(j-1)+1:R*j));  
Xtemp = Xtemp + Dt*lambda*Xtemp + mu*Xtemp*Winc;  
Xem(j) = Xtemp;  
end  
plot([0:Dt:T],[Xzero,Xem]), hold off 
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.