The point in these stochastic calculus notes has been reached where the theory of stochastic integration is sufficiently well developed to apply in a wide range of situations.
Results such as Ito’s lemma, properties of quadratic variations and existence and uniqueness of solutions to stochastic differential equations followed quite directly from the definition of stochastic integration. Then, once it was shown that integration with respect to martingales is well-defined, results such as preservation of the local martingale property and Ito’s isometry also followed without too much effort.
Over the next few posts, I will take a break from further development of the general theory. Instead, I look at certain special processes, applying the calculus developed so far and gaining a few examples to motivate further development of the theory.
This will include properties of Brownian motion, such as Lévy’s characterization, Girsanov transforms, stochastic time changes and martingale representation. Other important processes which I take a brief look at include Bessel processes, the Poisson process and the Cauchy process. We will also derive a general description of processes with independent increments, including the Lévy-Khintchine formula characterizing Lévy processes.