Stochastic Differential Equations - A Few General Questions I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I wanted to try something different. I have really enjoyed it and am actually seriously considering going to graduate school to study this stuff. The main issue is that we didn't do a measure theory approach to it. The professor used a lot of ideas directly from Wiener and we always took a random walk approach.
Anyway, I just wanted to ask:
a) What is the point of simulating SDEs if the solution is always different due to the randomness of the Wiener process? I have been simulating Geometric Brownian Motion and read that it is used in the Black-Scholes model in finance, so how do they actually price stocks based on the SDE?
b) What are the methods used in determining the coefficients in an SDE to calibrate it to data?
c) What is a good textbook for a very applied and computational approach to stochastic calculus with a crash course on measure theory?
 A: The single best introduction to SDE from numerical angle is this Higham's paper. It will probably give you an approximation to answers to your three questions.
a) In finance the assets are not priced on the basis of knowing exactly their cash flows. Moreover, they're not priced on the basis of knowing their expected cash flows either. Ideally, you need to know the whole distribution of future prices. It's often expressed as a pair "risk-return". The risk part is a quantified uncertainty about the return. That's why stochastic calculus seems to fit finance applications so well, it's precisely because it appears to capture our understanding of uncertainty about the future cash flows from the assets. 
The sampling would represent possible cash flow paths. Each path is a possible realization of the future. In Monte Carlo methods you explicitly sample paths, and obtain the distribution of cash flows, which allows you to price the assets.
However, under certain conditions, you can formulate and solve the SDE as partial differential equations (PDE) - non-stochastic. That's what Merton did with Black-Scholes (BS) PDE approach: he linked them to SDE. Original BS paper formulated option pricing problem as a heat transfer equation from physics.
In BS equation for an option price, you can see that there are 5 inputs: asset price, volatility, strike price, risk free return and time to maturity. Even before BS equation, these were all known to be determinants of the option prices. That's why when the paper came out it immediately made a sense to practitioners. Note, now that there's nothing about the future price of the asset. The only information about the future price is volatility, which represents the uncertainty about the future. 
So, intuitively, what BS equation does is it expresses the option price as a function of the distribution of future prices, namely its standard deviation. That's how SDEs are used: you express your price through the distributions of future outcomes, and if you're lucky you solution will have something simple like the standard deviation in it.
b) Monte Carlo is used a lot, but as I wrote above if you can convert the problem into PDE, then all kinds of methods such as finite elements can be used. 
c) I'm not sure there's such a book, i.e. computational with measure theory. If you're mathematician I can recommend the one I used: Shreve's text "Stochastic Calculus for Finance II: Continuous-Time Models". There's no software coming with it though, it's quite theoretical, may not work for you, if you're not strong in math.
UPDATE
I want to add a physics example to a). Look at diffusion process. You can think of a single atom's path as a single path in SDE, maybe in its Monte Carlo sampling. It's totally unpredictable. However, when you look at the diffusion of large quantities of atoms, the diffusion process is very predictable in terms of the speed with wich one material goes into another in a macro level. 
A: 
a) What is the point of simulating SDEs if the solution is always
  different due to the randomness of the Wiener process? I have been
  simulating Geometric Brownian Motion and read that it is used in the
  Black-Scholes model in finance, so how do they actually price stocks
  based on the SDE?

The Black-Scholes model is for evaluating the price of stock options (where the underlying stock is assumed to follow Geometric Brownian Motion). In terms of probability theory the idea is that the option price is the (discounted) expectation (as of today) of a function of the stock price  at the maturity date of the option. So you are averaging over all the paths. 
The expectation is under what is called the risk-neutral measure- the idea being that to avoid arbitrage[basically having two prices for the same financial product] - all bets(options and other [financial] derivatives)  on the future value of the stock price must be representable as an expectation  with respect to a single pricing measure which is absolutely continuous [agrees on what states are impossible]  with the real measure.If you think of the pdf of the stock price at maturity,$S_T$, then the pdf in the pricing measure is the price today of receiving a dollar at time $T$ for each value of the stock at time $T$.
b) What are the methods used in determining the coefficients in an SDE to calibrate it to data?
AFAIK you cannot even estimate the drift of a "gaussian" SDE $dX_t=\mu dt+\sigma dW_t$ , whereas the diffusion term is easy. However, the drift term is irrelevant to pricing options where the drift term (for the pricing measure) is set at the risk free rate of interest.the diffusion term ($\sigma$), on the other hand, determines the possible pricing measures so it should tie in with historical estimates...[but things get complicated]
c) What is a good textbook for a very applied and computational approach to stochastic calculus with a crash course on measure theory? 
Although I think it would be a nice idea to combine the theoretical issue of dealing with infinite dimensional spaces [namely in the time dimension] with computations, I don't think you will find a book that takes that approach. 
you might want to read the parable of the bookmaker from baxter and rennie's book, Financial Calculus to read about arbitrage pricing and how probability is used in derivative pricing.
A: *

*"What is the point of sampling random variables at all if the solution is always different due to the randomness..."? When you're simulating an SDE (or any stochastic process in general), you're sampling from a certain distribution of sample paths. (More precisely, a probability measure defined on an infinite dimensional space---e.g. the Wiener measure is defined on $C[0, \infty)$). 

*In the GBM case you cite, estimation of parameters reduces to classical parametric models. In general, see Statistics of Random Processes I by Liptser and Shiryaev for starters.

*Stochastic Differential Equations: An Introduction with Applications by Øksendal has 6 editions. You can't go wrong with that.
