medium · Quantitative Finance
An analyst models the movement of a stock between three regimes: 1 (Growth), 2 (Stagnation), and 3 (Crisis). Regime 3 is absorbing. The transition matrix for the transient regimes is Q = beginpmatrix 0.5 & 0.4 0.1 & 0.8 endpmatrix.
If the fundamental matrix is N = (I - Q)^-1, what is the interpretation of the sum of the entries in the first row of N?
- The expected total time spent in all transient states before absorption, given the process starts in regime 1.
- The long-run share of time the process spends within regime 1 before its first transition into crisis.
- The expected number of direct transitions observed between the Growth and Stagnation states before absorption occurs.
- The probability that the process eventually reaches the absorbing crisis regime 3, given that it starts out in regime 1.
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