The theory of integration is well known to most mathematics students. We are interested here in the theory of 'good integrators'. The Riemann-Stieltjes construction of the integral allows only integrators that have finite variation to be integrators of continuous integrands. This is a problem for even the most fundamental stochastic process-the Brownian Motion-and it seems impossible to build a theory of stochastic integration with this approach. In this talk, I first show how the Riemann-Stieltjes approach fails, and then introduce the semimartingale as the general 'good integrator' for stochastic integration, and finally recover the integral for the Brownian Motion.

For this talk, it will be useful for the audience to know basic measure theory and probability theory (in particular, the meaning of the term 'convergence in probability'). Due to the nature of the topic, some ideas are very technical. I will sketch these ideas very lightly. The intention of the talk is to give audience a grasp of the particularities of the theory of stochastic integrals.