在 MATLAB 中模拟 1,000 个几何布朗运动

Question

我目前拥有模拟几何布朗运动的代码，由 http://www-math.bgsu.edu/~zirbel/sde/matlab/index.html 提供。

但是，我想生成 1,000 个模拟并将它们显示在图表中。

我目前生成单个模拟的代码如下：

% geometric_brownian(N,r,alpha,T) simulates a geometric Brownian motion 
% on [0,T] using N normally distributed steps and parameters r and alpha

function [X] = geometric_brownian(N,r,alpha,T)

t = (0:1:N)'/N;                   % t is the column vector [0 1/N 2/N ... 1]

W = [0; cumsum(randn(N,1))]/sqrt(N); % S is running sum of N(0,1/N) variables

t = t*T;
W = W*sqrt(T);

Y = (r-(alpha^2)/2)*t + alpha * W;

X = exp(Y);

plot(t,X);          % plot the path
hold on
plot(t,exp(r*t),':');
axis([0 T 0 max(1,exp((r-(alpha^2)/2)*T+2*alpha))])
title([int2str(N) '-step geometric Brownian motion and its mean'])
xlabel(['r = ' num2str(r) ' and alpha = ' num2str(alpha)])
hold off

Answer 1

该代码不能直接用于模拟 1,000 条路径/模拟。不幸的是，它还没有被矢量化。做你想做的最简单的方法是使用for 循环：

N = 1e3;
r = 1;
alpha = 0.1;
T = 1;
npaths = 1e3;          % Number of simulations

rng(0);                % Always set a seed
X = zeros(N+1,npaths); % Preallocate memory
for i = 1:n
    X(:,i) = geometric_brownian(N,r,alpha,T);
    hold on
end
t = T*(0:1:N).'/N;
plot(t,exp(r*t),'r--');

这是相当缓慢和低效的。您将需要大量修改该函数以对其进行矢量化。可以提高性能的一件事是，如果您至少从函数内部删除了绘图代码并在循环之后单独运行它。

另一种选择可能是在我的SDETools toolbox 中使用sde_gbm 函数，它是完全矢量化的并且速度更快：

N = 1e3;
r = 1;
alpha = 0.1;
T = 1;
npaths = 1e3;        % Number of simulations

t = T*(0:1:N)/N;     % Time vector
y0 = ones(npaths,1); % Vector of initial conditions, must match number of paths
opts = sdeset('RandSeed',0,'SDEType','Ito'); % Set seed
y = sde_gbm(r,alpha,t,y0,opts);

figure;
plot(t,y,'b',t,y0*exp(r*t),'r--');
xlabel('t');
ylabel('y(t)');
title(['Geometric Brownian motion and it's mean: ' int2str(npaths) ...
       ' paths, r = ' num2str(r) ', \alpha = ' num2str(alpha)]);

在任何一种情况下，都会得到一个看起来像这样的图

              

Answer 2

要执行 1000 次模拟，直接的方法是：

Nsims = 1000;
N=10^15;         % set to length of individual sim                    
r = 1;
alpha = 0.1;
T = 1;

t = (0:1:N)'/N;                   
t = (T*(r-(alpha^2)/2))*t;
W = cat(1,zeros(1,Nsims),cumsum(randn(N,Nsims))); 
W = W*(sqrt(T)*alpha/sqrt(N));
Y = repmat(t,1,Nsims) + W;
X = exp(Y);

绘图和以前一样

plot(t,X);             % plots ALL 1000 paths
%   plot(t,X(:,paths));   % use instead to show only selected paths (e.g. paths =[1 2 3])
hold on
plot(t,exp(r*t),':');
axis([0 T 0 max(1,exp((r-(alpha^2)/2)*T+2*alpha))])
title([int2str(N) '-step geometric Brownian motion and its mean'])
xlabel(['r = ' num2str(r) ' and alpha = ' num2str(alpha)])
hold off

对于相对较短（小）的模拟集，循环您的代码或执行上述操作应该可以。对于重型模拟，您可能会受益于 Horchler 承诺的速度优势。