Stochastic Processes

2.1 Definition

A stochastic process involves a random element and a time element.

Any collection {X_t : t Î T} of random variables may be considered as a stochastic process.

The set of all values X can ever take is the state space, denoted S.
The collection T of possible times is usually either the non-negative integers (discrete time) or the non-negative reals (continuous time).

An unpredictable (random) observable process may be modelled to predict its future behaviour:

• short-term movements (exchange rates)
• medium-term extrema (storm damage claims, derivatives)
• long-term asymptotics (steady-state costs, eg health insurance)

Usefulness for modelling is determined by the nature of the dependence of X_n (or X(t)) on preceding X values.

2.2 Classification

• Based on time parameter: discrete-time or continuous-time.
  Use notation {X₁, X₂, . . .} in discrete time, { X(t): t ³ 0 } in continuous
• Based on state space: discrete or continuous.
  A counting process is discrete, taking only integer values; the size of an insurance claim is treated as continuous.

Nothing useful can be said about all stochastic processes. But a stochastic process may possess certain properties which make it easier to study.

2.3 The History

Note: this subsection uses continuous-time notation but is equally true in discrete time.

H_t -- the history of X up until time t -- is the collection of answers to all questions about the behaviour of X in that time range.
We can write H_t^X to avoid confusion.
In formal notation,

The history is most often used for conditional expectations: E(X_s  |  H_t) is the expected value of X_s given all the information up until time t. Therefore

{H_t^X : t ³ 0} is the filtration generated by X.

In general a filtration need not be generated by a single process: it may contain the histories of many processes at once. In that case we say that all the processes are adapted to the filtration.

2.4 Stationary Processes

A process X is stationary if

• the distribution of X_t is the same for all t.
• for each k the distribution of the vector (X_t, X_t+1, . . . X_t+k) is the same for all t.

We shall be meeting many stationary models later on.

2.5 Processes with Stationary, Independent Increments

A process X has stationary increments if the distribution of (X_t+s - X_t) is the same as that of (X_s - X₀) for all t.

X has independent increments if (X_t+s - X_t) and (X_u+s - X_u) are independent r.v.s whenever the intervals (t,t+s) and (u,u+s) do not overlap.

The random walk model has stationary independent increments, as does the Poisson process and some other continuous-time processes.

2.6 The Markov property

This is about memorylessness.
'The future is independent of the past, given the present'
Formally, X has the Markov property if the distribution of X_t+s given H_t is the same as the distribution of X_t+s given X_t.

Example: if we want to predict tomorrow's weather and know today's weather, then yesterday's weather is irrelevant.