Fun With Stochastic Process
Previously we have seen the Brownian motion; How all of it started, it's mathematical form, various properties and how all these are connected to the field of Finance. Now we will move another step forward in that direction with Stochastic process.
Itô Calculus, named after famous mathematician Kiyosi Itô extends the method of calculus to stochastic process like Brownian motion. Itô Calculus has huge application in mathematical finance, for example in- Black-Scholes-Merton model, Vasicek's interest rate model etc. which we will discuss.
Introduction to Ito integral, Ito integral for simple integrands, construction of Ito integral. Properties of Ito integral: Ito integral is a martingale...
Properties of Ito integral: Ito Isometry, Quadratic variation of Ito integral up to time t...
Ito integral for general integrands
and it's properties...
Using various properties calculate Ito integral of W(t) with respect to W(t). How Stochastic calculus is different from Ordinary calculus. Stratonovich integral: how mid-point sum is not equal to the Left-hand sum, unlike Riemann sum in ordinary calculus. Why left-hand sum definition is more preferable in finance rather than the Mid-point sum definition?
Ito- Doeblin formula; Ito- Doeblin formula for Brownian motion and it's proof. A simple example of Ito- Doeblin formula...
Ito process: Definition. Quadratic Variation of Ito Process and it's proof...