Multi-Period Transition Matrix

Definition

A Multi-Period Transition Matrix is a collection of square (n x n) matrixes representing the transition probabilities of a stochastic system (e.g. a Markov Chain) over several successive periods.

Representations

Cumulative Form

For each period k, encode the transition rates from initial time 0: $T^{mn}_{0k}$.

Incremental Form

For each period k, encode the transition rates from previous time k-1: $T^{mn}_{k-1,k}$.

Special Cases

Markov Chains

In the special case where the system state at a given time is the sole determinant of the probabilities of future transitions we talk about Markov Chains. The Markov assumption is quite strong in general and should be used with care (although its limitations can be sometimes relaxed by extending the state-space)

Time Homogeneous Transitions

As a subset of the Markov Chain case, if the matrix happens to be the same over all periods we talk of time-homogeneous transitions. The assumption of time homogeneity (T is independent of k) would normally not be satisfied.

Open Source Implementations

• https://github.com/open-risk/transitionMatrix

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