Credit risk models under partial information

This Ph.D. thesis consists of five independent parts (Introduction included) devoted to the modeling and to studying problems related to default risk, under partial information. The first part constitutes the Introduction. The second part is devoted to the computation of survival probabilities of a firm, conditionally to the information available to the investor, in a structural model, under partial information. We exploit a numerical hybrid technique based on the application of the Monte Carlo method and of optimal quantization. As an application, we trace the credit spreads curve for zero coupon bonds for different maturities, showing that (as in practice on the market) the spreads in the neighborhood of the maturity are not null, i.e., under partial information there is some residual risk on the market, even if we are close to maturity. Calibration to real data completes this second part. In the third part we deal, by means of the Dynamic Programming, with a discrete time maximization of the expected utility from terminal wealth problem, in a market where defaultable assets are traded. Contagion risk between the default times is modeled, as well as model uncertainty, by working under partial information. In the part devoted to numerics we study the robustness of the solution found under partial information. In the fourth part we are interested in studying the problem linked to the uncertainty of the investment horizon. In particular, in a complete market model subject to default risk, we solve, both with a direct martingale approach and with the Dynamic Programming, three different consumption maximization problems. More specifically, denoting by the default time, where is an exogenous positive random variable, we consider three problems of maximization of expected utility from consumption: when the investment horizon is fixed and equal to T, when it is finite, but possibly uncertain, equal to T ^, and when it is infinite. First we consider the general stochastic coefficients case, then, in order to obtain explicit results in the logarithmic and power utility cases, we pass to the constant coefficients case. Finally, in the fifth part we deal with a totally different problem, given that it is purely theoretical. In the context of enlargement of filtrations our aim is to retrieve, in a specific setting, the already known results on martingales’ characterization, on the decomposition of martingales with respect to the reference filtration as semi-martingales in the progressively and in the initially enlarged filtrations and the Predictable Representation Theorem. Some of these results were used in the fourth part of this thesis. The interest in this study is pedagogical: in our specific context most of the results are found more easily, by exploiting "basic" tools, such as Girsanov’s Theorem and by computing conditional expectations.