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Given stochastic differential equation $$dx = \mu(x,t)dt+\sigma_1(x,t)dB_1+\sigma_2(x,t)dB_2 \tag1$$ where $dB_1$ and $B_2$ are orthogonal 1-dimentional Brownian motions. It is equivalent to $$dx = \mu(x,t)dt+\sigma(x,t)dB_3 \tag2$$ where $\sigma = \sqrt{\sigma_1^2+\sigma_2^2}.$

We are to solve these equations by Monte Carlo simulation. Equation (1) needs simulation of two Brownian motion whereas Equation (2) needs one.

Questions:

1) What is the comparison in error measure between the two versions?

2) Is there a substantial saving in running time and memory using Equation (2) over Equation (1)?

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I doubt you will ever get a satisfying answer for this question. The reason is because nobody knows. In general virtually 99% of scientific software is heavily memory bound. That means, the CPU is underutilized in a sense that it could handle more but the memory bus is too slow to feed it. Very very often software is so insanely memory bound that memory bus is utilized by nearly 100% but you have maybe 0.001% peak performance as FLOP.

Long story short, expert software developer would try out both. Novices and inexperienced programmer should go for the one that requires less memory transaction (you want to trade off memory transactions with floating point operations).

I would give option 2 a shot.

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