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It can be verified that Q; and Q2 both qualify as generators, and exp(Q1) = exp(Q2) = P. Thus, Q, and Proof. Let b = e~47, and let Proposition 2. There exist transition matrices P which have more than one valid generators Remark. This last result raises this question: when a transition matrix P has multiple valid generators Q,

Figure 1 It can be verified that Q; and Q2 both qualify as generators, and exp(Q1) = exp(Q2) = P. Thus, Q, and Proof. Let b = e~47, and let Proposition 2. There exist transition matrices P which have more than one valid generators Remark. This last result raises this question: when a transition matrix P has multiple valid generators Q,

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exactly, to at least six decimal places (i.e., to within 10~°). Furthermore, our Q has row-sums 0, as it must Let us now instead use the series method presented in Theorem 1. By Theorem 2, we automatically
exactly, to at least six decimal places (i.e., to within 10~°). Furthermore, our Q has row-sums 0, as it must Let us now instead use the series method presented in Theorem 1. By Theorem 2, we automatically
This Q now has non-negative off-diagonal entries and still has row-sums 0. However, it no longer exactly satisfies that exp(Q) = P. In fact, equation (2), to obtain
This Q now has non-negative off-diagonal entries and still has row-sums 0. However, it no longer exactly satisfies that exp(Q) = P. In fact, equation (2), to obtain
which is very close to, but not exactly equal to, the original transition matrix P. It is useful to know which approximation is more accurate. One method of comparison is to compute the
which is very close to, but not exactly equal to, the original transition matrix P. It is useful to know which approximation is more accurate. One method of comparison is to compute the
We again see from Theorem 3(c) that there does not exist an exact generator for this transition matrix. following transition matrix: Using the method of Jarrow et al. in equation (3) to compute the approximate generator Q si7T, we cal We first consider the average one-year transition matrix from Moody’s which covers the period of 1980-
We again see from Theorem 3(c) that there does not exist an exact generator for this transition matrix. following transition matrix: Using the method of Jarrow et al. in equation (3) to compute the approximate generator Q si7T, we cal We first consider the average one-year transition matrix from Moody’s which covers the period of 1980-
Once again, we see by Theorem 3(c) that an exact generator for this P does not exist. Rated” weights, we obtain the following annual transition matrix:
Once again, we see by Theorem 3(c) that an exact generator for this P does not exist. Rated” weights, we obtain the following annual transition matrix:
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Applied MathematicsMarkov ProcessesMathematical FinanceCredit RatingMarkov ProcessFinance and Investment Banking
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