Dynamics of semimartingales with jumps: decomposition and delay

Author(s)

Leandro Batista Morgado

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2015-05-25

Examining board members

Paulo Regis Caron Ruffino; Eduardo Garibaldi; Pedro Jose Catuogno; Marcelo Sobottka; Fabiano Borges da Silva

• Advisor: Paulo Regis Caron Ruffino

Abstract

The main subject of this thesis is the theory of stochastic differential equations driven by semimartingales with jumps. We consider applications in the decomposition of stochastic flows in differentiable manifolds, and geometrical aspects about these equations. Initially, in a differentiable manifold endowed with a pair of complementary distributions, we discuss the decomposition of continuous stochastic flows, that is, flows generated by SDEs driven by Brownian motion. Previous results guarantee that, under some assumptions, there exists a decomposition in diffeomorphisms that preserves the distributions up to a stopping time. Using the so called Marcus equation, and a technique that we call 'stop and go' equation, we construct a stochastic flow close to the original one, with the property that the constructed flow can be decomposed further on the stopping time. After, we deal with the decomposition of stochastic flows in the discontinuous case, that is, processes generated by SDEs driven by semimartingales with jumps. We discuss the existence of this decomposition, and obtain the SDEs for the respective components, using an extension of the Itô-Ventzel-Kunita formula. Finally, we propose a model of stochastic differential equations including delay and jumps. The idea is to describe some phenomena such that the information comes to the receptor by different channels: continuously, with some delay, and in discrete times, instantaneously. We deal with geometrical aspects related with this subject: parallel transport in càdlàg curves, and lifting of solutions of these equations to the linear frame bundle of a differentiable manifold (AU)

• FAPESP's process: 11/14797-2 - Stochastic dynamics in foliated spaces
• Grantee: Leandro Batista Morgado
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)