Category Archives: Stochastic Calculus lectures

In this lecture, we study properties of the Itō integral that was defined in the previous lecture. The following proposition is easy to prove and its proof is left to the reader as an exercise. Proposition: Let , show that … Continue reading →

Since a Brownian motion does not have absolutely continuous paths, we can not directly use the theory of Riemann-Stieltjes integrals to give a sense to integrals like for every continuous stochastic process . However, if is regular enough in the … Continue reading →

If , is a subdivision of the time interval , we denote by , the mesh of this subdivision. Proposition. Let be a standard Brownian motion. Let . For every sequence of subdivisions such that , the following convergence takes … Continue reading →

We already observed that as a consequence of Kolmogorov’s continuity theorem, the Brownian paths are -Hölder continuous for every . The next proposition, which is known as the law of iterated logarithm shows in particular that Brownian paths are not … Continue reading →

In this Lecture we present some basic properties of the Brownian motion paths. Proposition. Let be a standard Brownian motion. Proof Since the process is also a Brownian motion, in order to prove that we just have to check that … Continue reading →

Definition: Let be a probability space. A continuous real-valued process is called a standard Brownian motion if it is a Gaussian process with mean function and covariance function It is seen that is a covariance function, because it is symmetric … Continue reading →

In this post, we prove some fundamental martingale inequalities that, once again, are due to Joe Doob Theorem (Doob’s maximal inequalities) Let be a filtration on probability space and let be a continuous martingale with respect to the filtration . … Continue reading →

When dealing with stochastic processes, it is often important to work with versions of these processes whose paths are as regular as possible. In that direction, the Kolmogorov’s continuity theorem (see Lecture 6) provided a sufficient condition allowing to work … Continue reading →

Let us first remind some basic facts about the notion of uniform integrability which is a crucial tool in the study of continuous time martingales. Definition. Let be a family of random variables. We say that the family is uniformly … Continue reading →

The next four lectures will be devoted to the foundational theorems of the theory of continuous time martingales. All of these theorems are due to Joseph Doob. The following first theorem shows that martingales behave in a very nice way … Continue reading →