Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 65

An interesting application of matrices and matrix ODEs is to the modelling of credit rating applications (Wilmott (2006), vol. 2, pp. 665–73). To this end, we define a so-called transition matrix P, which is a table whose elements are probabilities representing migrations from one credit rating to another credit rating. For example, a company having a B rating has a probability 0.07 of getting a BB rating in a small period of time. More generally, we are interested in continuous-time transitions between states using Markov chains, and we have the following Kolmogorov forward equation:

(3.30)

where and unit matrix and is the transition rate matrix having the following properties:

The Kolmogorov backward equation is:

(3.31)

The objective is to compute the transition rate matrix Q (that is, states that are in one-to-one correspondence with the integers).

In the case of countable space, the Kolmogorov forward equation is:

(3.32)

where Q(t) is the transition rate matrix (also known as generator matrix), while the Kolmogorov backward equation is:

(3.33)