Stochastic Calculus and Introduction to Stochastic Differential Equations

In this chapter we introduce the basic notions of stochastic calculus, starting from the Brownian motion. Stochastic processes describe time-dependent random phenomena, generalizing the usual deterministic evolution. The description of the latter requires the notions of differential and integral, which need to be properly extended to stochastic properties. Stochastic calculus is the branch of mathematics dealing with this important topic. The reason why traditional calculus is not suitable for stochastic processes is revealed by the Brownian motion. Since \(Var(B_t) = t\), implying that \(B_t\) “scales” as \(\sqrt{t}\), its trajectories are not differentiable in the usual sense. The stochastic calculus allows us to introduce generalized notions of differential and integral, notwithstanding this difficulty. Moreover, it allows to write differential equations involving stochastic processes, providing thus a powerful generalization of ordinary differential equations to study phenomena evolving in time in a non deterministic way.