This master thesis is to fulfill the curiosity in how special characteristics of Brownian motion motivate the development of Ito calculus from Stieltjes integral. We will see the general Stieltjes integral theories and how the concept of p-variation plays a determining role and the case where the fractional Brownian motion can also be an integrator under the Stieltjes scope. Then we will give a tour of the distinctive traits of Brownian motion and how they mark the necessity to evolve a new Ito integral. Subsequently, we expand the integrator in Ito integral to jump-diffusion processes.
Thereafter, we see how these new stochastic processes play a role in modelling financial assets and how sample paths of these stochastic processes can be generated using simulation and examplify the fact that Jump-Diffusion models are improvements from classical Black-Scholes-Merton model to incorporate the fat tail effects exhibited in empricial financial data.