Applications of Stochastic calculus in economy and statistics: Extensions of the Kyle-Back model. Ambit processes and power variation.

Abstract

This thesis deals with three possible applications of stochastic calculus: modelling prices by supply and demand in a financial market where there is an informed trader, turbulence and financial models using ambit processes and the asymptotic analysis of certain power variation processes.

The thesis is organized as follows. Part I contains the basic facts and techniques of mathematics used in the latter parts. Part II deals with the markets with asymmetric information, Chapter 2 presents the basic models by Kyle and Back, and Chapter 3 presents the new results of Kyle’s model with L´evy noise: [Cor14b] and a General Model: [Cor14a], and also a short summary of other related models. Part III is dedicated to ambit processes. Chapter 4 introduces ambit fields and processes and bond markets, summarizes the new results of some applications of ambit processes on energy markets and turbulence: [CFV14], and on a short rate model: [CFSV13]. In Part IV, power variation processes are introduced and new results of [CF10] are summarized in Section 5.3. Finally, the above mentioned articles are included in the appendices.