Log-dynamics of a geometric Brownian motion

MediumStochastic Processes~5m

A price follows $dS_t = 3\,S_t\,dt + 4\,S_t\,dW_t$ for a standard Brownian motion $W_t$ . Writing the dynamics of $X_t = \ln S_t$ as $dX_t = a\,dt + b\,dW_t$ , find $a^2 + b^2$ .

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Apply Itô's lemma to $\ln S$ : the drift becomes $\mu - \tfrac12\sigma^2$ , the diffusion stays $\sigma$ .
With $\mu=3,\sigma=4$ : $a=3-8=-5$ , $b=4$ .

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Log-dynamics of a geometric Brownian motion

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