Introduction to stochastic calculus***

Usually, in most textbooks and research papers, the evolution of the stock price in the Black-Scholes-Merton (BSM) model is given by a so-called stochastic differential equation. To better understand these concepts, this chapter provides a heuristic introduction to stochastic calculus. Stochastic calculus arises naturally in continuous-time actuarial finance. The overall objective of this chapter is to provide a heuristic introduction to stochastic calculus based on Brownian motion by defining Ito's stochastic integral and stochastic differential equations. This is a rather complex topic so the presentation focuses on providing a working knowledge of the material. The chapter helps to understand the definition of a stochastic integral and its basic properties, compute the mean and variance of a given stochastic integral, apply Ito's lemma to simple situations and understand how a stochastic process can be the solution to a stochastic differential equation.