Chapter 2 Flashcards
Stochastic process
A model for a time-dependent random phenomenon. So, just as a single random variable describes a static random phenomenon, a stochastic process is a collection of random variables Xt, one for each time t in some set J.
State Space of a stochastic process, S
The set of values that the random variables Xt are capable of taking on.
Example of a process with discrete state space and discrete time changes
Motor insurance company reviewing the status of its customers annually. 3 levels of discount are possible depending on the accident record of the driver.
In this case the appropriate state space is S = {0, 25, 40}, and the time set is J = {0,1,2,…} where each interval represents a year.
Example of a process with discrete state space and continuous time changes
A health insurance company classifies its policyholders as Healthy, Sick or Dead. Hence the state space S = {H, S, D}. As for the time set, it is natural to take J = [0, ∞) as illness or death can occur at any time.
Example of a process with continuous state space
Claims of unpredictable amounts reach an insurance company at unpredictable times; the company needs to forecast the cumulative claims over [0,n] in order to assess the risk that it might be able to meet its liabilities. It is standard practice to use [0, ∞) for both S and J in this problem.
Processes of a mixed type
Just because a stochastic process operates in continuous time does not mean that it cannot also change value at predetermined discrete instants; such processes are said to be of mixed type.
Example of a mixed type process
Consider a pension scheme in which members have the option to retire on any birthday between the ages 60 and 65. The number of people electing to take retirement at each year of age between 60 and 65 cannot be predicted exactly, nor can the time and number of deaths among active members. Hence the number of contributors to the pension scheme can be modelled as a stochastic process of mixed type with state space S = {1, 2, 3, …} and time set J = [0, ∞).
Meaning: Strict Stationarity
The statistical properties of the process remain unchanged as time elapses.
A stochastic process is said to be strictly stationary if the joint distributions of
Xt1 , Xt2 ,…, Xtn and Xt+t1 , Xt+t2 ,…, Xt+tn are identical for all t,t1,t2,…,tn in the time set J and
for all integers n.
Conditions of weak stationarity
- E[Xt] is constant for all t
- cov[ Xt, X(t+k) ] depends only on the lag, k.
Increment
An increment of a process is the amount by which its value changes over a period of time. eg Xu - Xt (u>t)
Independent increments
If the increment Xu - Xt is independent of all the past increments of the process (with u>t)
A process with independent increments has the Markov property.
Markov property
If the future development of a process can be predicted from its present state alone, without any reference to its past.
Sample Path (of a process)
A joint realisation of the random variables Xt, for all t in J.
Result w.r.t. independent increments & Markov property
A process with independent increments has the Markov property.
Give the conditions under which a Markov chain has at least one stationary distribution.
A Markov chain with a finite state space has at least one stationary probability distribution.
