# State Space

## Contents

## Definition

A **State Space** is a fundamental concept in probability theory^{[1]} representing the possible configurations for a modelled system.

Any given state in this state space are understood to summarize and represent more complex internal system states, i.e. it is an abstraction. The usefulness of state space based models is determined by the degree to which they can represent and help analyse (e.g. forecast) what happens in the actual system.

The state space is normally assumed shared by more than one entities. In complex situations it may be that entities share parts of the state space. For example some states may be meaningless for part of the population. Such systems can be emulated at the level of transition probabilities by introducing sets of states that do not communicate with each other.

## Discrete versus Continuous State Spaces

The state space of a stochastic system can be discrete or continuous (or a mix). Furthermore, a discrete state space may have an infinite or finite number of states

Computer representations of state spaces are always finite and discrete state space. Continuous space may be approximated.

In credit risk management applications it is common to consider a *finite state space*, where a limited number of distinct states represent all possible states the system (or process) can be in.

## Hidden versus Observed States

In general the state space concerns observable (visible, measurable) states. When modelling stochastic systems for risk management purposes we may also introduce states that are deemed *hidden* (non-observable). This may due to complexity, information barriers or other reasons. A general class of models applicable in this case are Hidden Markov Chain Models

## Examples

- The Rating Scale of a Credit Rating System is the finite state space of a Credit Risk process.
- The set of financial ratios and other Balance Sheet figures may be considered a continuous state space representing the internal state of a firm

## Path Space

The space of all possible journeys of a system through its state space is called the path space.

## References

- ↑ Encyclopedic Dictionary of Mathematics