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I have proven the distribution of a discrete time model, evolving over a uniform mesh with $\delta t = T/L$ is given by

$$S(t_{i+1}) = S(t_i) + \mu \delta t S(t_i) + \sigma\sqrt{\delta t}S(t_i)Y_i,$$

for $i = 0, . . . , M − 1$, where $Y_i$ is an i.i.d. N(0, 1) sequence, converges to that of a log-normal random variable as $\delta\to0$(and hence as $L\to \infty$). This required an application of the Central Limit Theorem.

So now if I let $X_1, X_2, . . . , X_n$ be a set of independent Normal random variables, with means $\mu_i$ and variances $θ^2_i$ for $i = 1, . . . n$ respectively is it possible to make changes to use the affine property of Normal random variables, rather than the Central Limit Theorem?

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