Some people here like @jsoloff are big fans of stochastic calculus so I here are some fun facts on Wiener processes, Ito's lemma, Feynman-Kac, backward Kolmogorov, and the Black-Scholes PDE
Going back to basics, the Wiener process is a continuous-everywhere, differentiable-nowhere stochastic process, that is the limit of an infinitely fast random walk. You flip coins infinitely fast and it becomes continuous. (Note the notation and math I do here is handwavey) 
The thing that makes a Wiener process so special is that it has quadratic variation of T. This means if you slice up the process into finite chunks, take the amount it goes up or down in each chunk, and square it, you get T as the number of slices goes to infinity
I really don't like the mathematical notation for this because it makes it confusing but here you go P is a partition, the limit is as the partition widths become infinitely small
So what does this mean for a typical stochastic process? This is where Ito's Lemma comes in
Let's say we have a function of this stochastic process, f(t, Xt), and we want to find df(t,Xt) based on what we know about dXt. In normal calculus, you just use the chain rule and Taylor polynomial expansion
If you substitute in the stochastic process, you get this Taylor polynomial expansion
dt^2, dt dBt both go to zero - it is something infinitesimally(spellcheck?) small times something else that is infinitesimally small
However, because of the quadratic variation of dWt (annoyingly Wikipedia uses dBt which is lame), dBt dBt = dt. This is handwavey and not rigorous but just bear with me, nerds who actually know this stuff
When we do that, the small terms go away, dBt dBt goes to dt, and we get this fun equation. Using Ito's lemma is how we turned geometric Brownian motion into the formula for St = S0 * exp((mu - sigma^2/2)*t + sigma*Wt) in my last long thread.
Ito's lemma also helps us move from stochastic differential equations with a final terminal value (like an option) to a closed-form PDE via the Kolmogorov backwards equation
You'll notice this looks a lot like the Black-Scholes PDE!
This is why I hate the 'perfectly delta hedged portfolio' proof of the Black-Scholes PDE. You can use Ito's Lemma and the Kolmogorov backwards equation and it's a two line proof. Or you go through all this nonsense
The last bit is the Feynman-Kac equation, which basically says that the discounted value at expiry of an option is equal to the solution to a PDE. Looks a lot like taking the discounted expected value at expiry of an option payoff!
