Stochastic models for finance and insurance : relative entropy
This thesis is devoted to the analysis of the notion of relative entropy in the frame of models based on (geometric) Lévy processes. It is well-known that financial markets where the stock prices follow a (geometric) Lévy process are incomplete except for the cases that the Lévy process is a Brownian motion or a Poisson process. In arbitrage-free and complete financial markets, there exists a unique martingale measure. We are interested in such probability measures Q that the price process S = S_0 exp(X) is Q-martingale and Q is equivalent (or, at least, absolutely continuous) to the original measure P. In financial mathematics the notion of a martingale measure is very important because of the no-arbitrage property and hedging. The famous BlackScholes model, which was already studied for decades is an example of the case where exists a unique absolutely continuous martingale measure, but it is not the case for the general (geometric) Lévy process. The set of the absolutely continuous martingale measures M may have one of three different forms: (1) the set M could be empty; (2) the set M consists of just one measure (case without jumps, the classic BlackScholes model); (3) the set M consists of an infinite number of martingale measures. We focus our attention on the case (3). It is used the notion of relative entropy I(P,Q) as an analogue of the distance between measures P and Q. The measure Q* from the class of absolutely continuous martingale measures M which minimizes the relative entropy, is called the minimal entropy martingale measure (MEMM). There is also introduced the notion of Esscher transformation and the Esscher martingale measure (EMM), which are already well-known from actuarial mathematics. One of the main results of the thesis states the identity of the notions MEMM and EMM for our basic model. The thesis is divided in four chapters which are followed by two appendices. In Chapter 1 we collect main results of measure theory and stochastic analysis. In Chapter 2 we give an introduction to the problem of the MEMM. In Chapter 3 we investigate the model of the price processes based on the exponential (geometric) compound Poisson process and show its connection with the one-step model. In Chapter 4 the main objects of the investigations are models driven by the linear and exponential (geometric) Lévy processes. There are stated the main results of the thesis, including a sufficient condition of the existence of the EMM, coincidence of the EMM and the MEMM for the models driven by the linear and exponential Lévy processes and a series of important explicit equalities for the value of the relative entropy of the MEMM with respect to the original probability measure P. In Appendix A are collected the most important properties of the technical functions which are widely used in the main body of the thesis. In Appendix B we apply the general theory developed in Chapter 2, to a particular model, the one-step model.
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