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And this can be proved using Ito's lemma, but the following gives a different solution.
divide throughout by S_t:
dS_t/S_t = mu dt + sigma dW_t
divide throughout by dt:
dS_t/dt * 1/ S_t = mu + sigma dW_t/dt
integrate both sides with respect to dt and limits 0 and t:
ln(S_t) - ln(S_0) = mu*t + sigma dW_t
add ln(S_0) to both sides:
ln(S_t) = mu*t + sigma dW_t + ln(S_0)
take the exponential of both sides:
S_t = exp(mu*t + sigma dW_t + ln(S_0))
S_t = exp(mu*t + sigma dW_t) * exp(ln(S_0)))
S_t = S_0 * exp(mu*t + sigma dW_t)
which is not the same solution that Ito's Lemma gives, what am I doing incorrectly?
Any help much appreciated.