Levy modelling of defaultable bonds

A market with defaultable bonds modelled by equations with L\'evy noise will be considered during the talk. The main aim is to derive conditions under which the market with defaultable bonds, issued by firms with time dependent and random rating classes, is free of arbitrage. Obtained theorems provide HJM conditions for the arbitrage-free property. It is assumed that the Levy process might be infinite dimensional. Importance of treating models with infinite number of factors was stressed in recent papers of Carmona and Tehranchi (A characterization of hedging portfolios for interest rate contingent claims, Annals of Applied Probability, 14(3), 1267-1294, 2004) and Ekeland and Taflin (A theory of bond portfolios, Annals of applied probability 15, no. 2, 1260-1305, 2005). In my talk it will be considered fractional recovery of market value, fractional recovery of treasury value and fractional recovery of par value. Several default times will be discussed as well. The rating classes change according to a conditional, continuous time Markov chains and the default time is equal to the moment of entering by the firm the worst rating class. My talk is based on joint work with Mariusz Nieweglowski and Jerzy Zabczyk.