Category Archives: Stochastic Calculus lectures

We now turn to the theory of stochastic differential equations. Stochastic differential equations are the differential equations corresponding to the theory of the stochastic integration. As usual, we consider a filtered probability space which satisfies the usual conditions and on … Continue reading →

In this section, we describe a theorem which has far reaching consequences in mathematical finance: The Girsanov theorem. It describes the impact of a probability change on stochastic calculus. Let be a filtered probability space. We assume that is the … Continue reading →

In this section, we study some of the most important martingale inequalities: The Burkholder–Davis–Gundy inequalities. Interestingly, the range of application of these inequalities is very large and they play an important role in harmonic analysis and the study of singular … Continue reading →

In the previous Lecture, we proved that any martingale which is adapted to a Brownian filtration can be written as a stochastic integral. In this section, we prove that any martingale can also be represented as a time changed Brownian … Continue reading →

In this Lecture we show that, remarkably, any square integrable integrable random variable which is measurable with respect to a Brownian motion, can be expressed as a stochastic integral with respect to this Brownian motion. A striking consequence of this … Continue reading →

In the next few Lectures we will illustrate through several examples of application the power of the stochastic integration theory. We start with a study of the multidimensional Brownian motion. As already pointed out, a multidimensional stochastic process , is … Continue reading →

Itō’s formula is certainly the most important and useful formula of stochastic calculus. It is the change of variable formula for stochastic integrals. It is a very simple formula whose specificity is the appearance of a quadratic variation term, that … Continue reading →

The goal of this Lecture is to extend the domain of definition of the Itō integral with respect to Brownian motion. The idea is to use the fruitful concept of localization. We will then be interested in the wider class … Continue reading →

In the same way that a stochastic integral with respect to Brownian motion was constructed, a stochastic integral with respect to square integrable martingales may be defined. We shall not repeat this construction, since it was done in the Brownian … Continue reading →

It turns out that stochastic integrals may be defined for other stochastic processes than Brownian motions. The key properties that were used in the above approach were the martingale property and the square integrability of the Brownian motion. As above, … Continue reading →