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Is there a way to go from this

$$\ln S_t=\ln S_0+(\mu-\sigma^2/2)t+\sigma W_t $$

$$\ln S_t\sim N[\ln S_0+(\mu-\sigma^2/2)t, (\sigma^2)t]$$

To the Black-Scholes partial differential equation?

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  • $\begingroup$ Hi and welcome to Quant.SE! I tried to reformat your question but it is still unclear. I can only guess what you ask and the answer is: Try to take your process and see if its an Ito process and what the ito formula looks like. I wouldnt be surprised if you end up with the black scholes SPDE. (please take a look at your question as well. something is still not right there!!) $\endgroup$ – vanguard2k Feb 21 '17 at 10:31
  • $\begingroup$ Your equation looks quite off. For $t = 0$ it implies that $S_0 \sim \mathcal{N} \left( \ln S_0, 0 \right)$. Also - your question is not clear to me. Is $S_t$ supposed to follow a geometric Brownian motion? If yes, what do you want to know that goes beyond the standard derivation of the PDE that you find in most textbooks? $\endgroup$ – LocalVolatility Feb 21 '17 at 10:33
  • $\begingroup$ Im writing on my ipad and its hard to write the equations correctly $\endgroup$ – krkrkrkr Feb 21 '17 at 10:34
  • $\begingroup$ Your equations are looking OK now (I rolled back one of your edits that didn't make sense). My previous question still stands: what do you want to know that goes beyond the standard derivation of the PDE that you find in most textbooks? $\endgroup$ – LocalVolatility Feb 21 '17 at 10:52
  • $\begingroup$ I want to go from this to the BS partial differential equation but no book ive found goes this route. Should i apply itos lemma on ln St? $\endgroup$ – krkrkrkr Feb 21 '17 at 10:54
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What you wrote in your question is simply one of the assumptions of the Black-Scholes framework.

The whole interesting part about getting to the Black-Scholes PDE (i.e. constructing a replicating portfolio) comes next and is the most common derivation of the Black-Scholes formula, which you can find the derivation on the Wikipedia page.

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