An introduction to stochastic integration

Abstract

[en] The main purpose of this master’s thesis is to continue and extend the study of the stochastic integral seen in the subject of Stochastic Calculus, providing (hopefully) an introductory text that will allow the average student of the subject (and to anyone who is already familiar with the stochastic integral with respect to the Brownian motion) to expand his knowledge. To do so, in the second chapter we briefly review the construction of the Itô integral to then see how we exploit these ideas to generalize the construction to other processes, this is done following the construction provided in the third chapter [7]. Many of the presented results and notions of this part (and the following ones) have been completed with additional explanations and comparisons with the already seen objects in order to make the understanding of these clearer. In the third chapter, we discuss the topic of stochastic integration with respect to random fields. We first treat the integral with respect to the space-time Gaussian white noise, following the construction presented in the first two chapters of [6], since it deals with objects which might be a bit more familiar to the intended audience as its construction uses the already studied Itô integral with respect to the Brownian motion. Before doing so, we introduce two crucial Gaussian processes (the isonormal process and the white noise), which generalize the Brownian motion and are crucial when it comes to define the stochastic integral with respect to the space-time white noise. Next, and following the second chapter of [11], we introduce a wider class of random fields (which contains the ones already seen) that can be used as integrators and show how one constructs integrals with respect to such objects. During this process, we use the already studied Gaussian white noise as a canonical example that will serve us as a model to compare the new construction. Finally, and to keep the reader entertained, in the Introduction and in Section 2.1 we pose a problem regarding the validity of the models that use the Brownian motion and the space-time Gaussian white noise as driving noises which is treated in Sections 2.3.4 and 3.2.4, providing results regarding the approximation in law of the stochastic integral with respect to the Brownian motion (Theorem 2.3.11, of which we could not find a statement nor a proof elsewhere) and the one with respect to the Brownian sheet (Theorem 3.2.2, which is a simplification of one of the results in [2]), respectively.