engleski

Enlargement of a filtration with a conditional Markov chain – the $(\mathcal{;H};)$ hypothesis and rating-based modeling

The problem of enlargements of filtrations was studied already in the late 1970's by the French probability school. If $(\mathcal{;F};_{;t};)_{;t\geq 0};$ and $(\mathcal{;G};_{;t};)_{;t\geq 0};$ are two filtrations such that $\mathcal{;F};_{;t}; \subset \mathcal{;G};_{;t};$ for every $t \geq 0$, i.e. $(\mathcal{;G};_{;t};)$ is an enlargement of $(\mathcal{;F};_{;t};)$, it does not hold in general that every $(\mathcal{;F};_{;t};)$-local martingale is a $(\mathcal{;G};_{;t};)$-local martingale. In case this property holds, we say that the two filtrations satisfy the $(\mathcal{;H};)$ hypothesis. The theory of enlargements of filtrations has been recently applied to mathematical finance, especially in models of insider trading and credit risk models. We study the progressive enlargement of a filtration with a conditional Markov chain (Bielecki and Rutkowski, 2000) and show that the $(\mathcal{;H};)$ hypothesis holds in this case. Also we obtain formulas for conditional expectations with respect to $\mathcal{;G};_{;t};$. The $(\mathcal{;H};)$ hypothesis remains valid under certain equivalent changes of measure and we show that the conditional Markov chain retains its properties as well under such a change of measure. These results will be used to construct the rating based L\'{;e};vy Libor model, which extends the defaultable L\'{;e};vy Libor model (Eberlein, Kluge, Sch\"{;o};nbucher, 2006) to the multiple credit rating setting.

conditional Markov chain; enlargement of filtration; credit rating; Lévy Libor model

nije evidentirano