Examples of numerical solution of stochastic differential equation(SDE)?

Question

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.

Answer 1

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

Answer 2

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

% 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