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Suppose we are given $\mu$ and $\sigma$ for a lognormal distribution with random variable $X$. $\mu$ is the mean of the variable's logarithm and $\sigma$ is the standard deviation of the variable's logarithm. Then if $Z$ is sampled from a standard normal distribution ($Z$ has mean 0 and std deviation 1),

$$X = e^{\mu + \sigma Z}$$

Now, I am given access to a uniform random generator, and use Box-Muller to generate a sample $Z$, can I generate a sample from $X$ by applying the above formula, or will there be issues with doing this?

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Yes, that should work fine – Glen_b Aug 6 '14 at 23:52
Yes. You're generating GBM? Correct. – Student T Aug 7 '14 at 3:20

X is log-normal means log X is normal. Since the exponential is a monotone-transformation if you correctly sample Z, you will correctly sample X so yes, the method you suggested works

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