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?