  In this paper, we model the rating process of an entity by using a
geometrical approach. We model rating transitions as an SDE on a Lie group.
Specifically, we focus on calibrating the model to both historical data (rating
transition matrices) and market data (CDS quotes) and compare the most popular
choices of changes of measure to switch from the historical probability to the
risk-neutral one. For this, we show how the classical Girsanov theorem can be
applied in the Lie group setting. Moreover, we overcome some of the
imperfections of rating matrices published by rating agencies, which are
computed with the cohort method, by using a novel Deep Learning approach. This
leads to an improvement of the entire scheme and makes the model more robust
for applications. We apply our model to compute bilateral credit and debit
valuation adjustments of a netting set under a CSA with thresholds depending on
ratings of the two parties.
