How to derive the Black Scholes partial differential equation from a stock log-normal distribution?

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?

• 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!!) – vanguard2k Feb 21 '17 at 10:31
• 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? – LocalVolatility Feb 21 '17 at 10:33
• Im writing on my ipad and its hard to write the equations correctly – krkrkrkr Feb 21 '17 at 10:34
• 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? – LocalVolatility Feb 21 '17 at 10:52
• 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? – krkrkrkr Feb 21 '17 at 10:54