Transition Rate
From Open Risk Manual
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Definition
A Transition Rate is a key property of a multi-state stochastic system (e.g. a Markov Chain). It measures the probability (per unit of time) that an event (state transition) occurs within an infinitesimally small time interval.
Mathematically, if is a stochastic process, its transition rates are defined as follows:
- Lets assume a state space with D+1 distinct states: .
- The rate of moving from state m to state n in an infinitesimal dime is , represented as:
Properties
- Since the transition rates are refer to the probabilities of transitions, they must be positive (but need not be less than unity)
- The collection of all transition rates forms a Transition Rate Matrix that satisfies further properties
See Also
Issues and Challenges
The terminology around transition matrix quantities can be confusing as they are used in slightly different contexts:
- When modelling stochastic processes in continuous time, the transition rate is distinct from the Transition Probability which measures transition frequencies over a finite time period
- When estimating transition phenomena the accumulation of statistics is always over a finite period, yet frequently one still uses the term "transition rate"
References