如何用矩阵乘法替代依赖前序列的递推for循环？

Hey there! I get why you're stuck—this recursive step feels like it has to be a loop at first glance, but we can absolutely refactor it using matrix operations by leaning into cumulative products. Let's walk through this.

First, let's recap what your original code does #

Your loop generates paths for a geometric Brownian motion (GBM) process, where each step depends on the previous value:

S= zeros(Sim,n+1); 
S(:,1)=S_0; 
for i=1:round(n) 
    S(:,i+1) = S(:,i) .* cte_exp .* exp(sigma.*sqrt(dt).*normrnd(0,1,Sim,1)); 
end

Every iteration multiplies the current value by cte_exp and a random exponential term. This is equivalent to multiplying the initial value by the product of all those step-wise factors.

The Matrix-Based Solution #

Here's how to replace the loop with vectorized operations:

• Generate all random perturbations at once
  Instead of generating one column of random numbers per loop iteration, create an entire Sim×n matrix of normal random values upfront:

  Z = normrnd(0, 1, Sim, n);
  

• Compute the multiplicative factor for each step
  Calculate the full matrix of step-wise growth factors (combining the deterministic cte_exp term and the random exponential):

  step_factors = cte_exp * exp(sigma * sqrt(dt) * Z);
  

• Compute cumulative products across steps
  To replicate the recursive multiplication, we need the product of factors up to each step. Add a column of 1s (for the initial state, where no multiplication has happened yet) and use cumprod to compute row-wise cumulative products:

  cumulative_factors = cumprod([ones(Sim, 1), step_factors], 2);
  

  The 2 in cumprod tells Matlab to compute the product along columns (i.e., step-by-step for each simulation path).

• Calculate the full S matrix
  Multiply the initial value S_0 by the cumulative factors to get all paths in one go:

  S = S_0 .* cumulative_factors;
  

Why This Works #

Each row of cumulative_factors represents the product of all step factors up to that point. For example:

• cumulative_factors(:,1) is 1 → matches S(:,1) = S_0
• cumulative_factors(:,2) is step_factors(:,1) → matches S(:,2) = S_0 * step_factors(:,1)
• cumulative_factors(:,3) is step_factors(:,1)*step_factors(:,2) → matches S(:,3) = S(:,2)*step_factors(:,2) = S_0*step_factors(:,1)*step_factors(:,2)

This exactly replicates your original loop's behavior, but with vectorized operations that are often faster in Matlab, especially for large Sim or n.

内容的提问来源于stack exchange，提问作者user9609639