Risk Modelling and Survival Analysis · Stochastic processes

Stochastic Processes: Basics and Classification by State Space and Time

Updated 11 October 2026 · Fact-checked

A stochastic process is a collection of random variables {X_t : t ∈ T} indexed by time. The set of values X_t can take is the state space S. The index set T is the time set. You classify a process as discrete or continuous in time and in state, giving four types.

Understand Stochastic Processes: Basics and Classification

A stochastic process is a family of random variables {X_t : t ∈ T}, all defined on the same probability space. Think of it as a random quantity that changes with time. Examples are the number of claims received so far, the state of a policyholder (healthy, sick, dead), or the daily value of an index.

The time set (or index set) T is the set of times at which you observe the process. If T is countable, such as {0, 1, 2, ...}, the process is discrete time. If T is an interval such as [0, ∞), the process is continuous time.

The state space S is the set of all values the random variables can take. If S is finite or countable, the process has a discrete state space. If S is an interval of real numbers, the state space is continuous. Be careful: the word discrete or continuous refers to the set of values, not to how often the process changes.

This gives four combinations:

Discrete time, discrete state: a no claims discount level at each policy renewal; a simple random walk.
Discrete time, continuous state: the closing price of a share each day; an AR(1) time series.
Continuous time, discrete state: the number of claims up to time t (Poisson process); a multi-state health model.
Continuous time, continuous state: Brownian motion; an interest rate model in continuous time.

A single run of the process, X_t for each t, is called a sample path or realisation. In exams, the first thing you do with any process is name its time set and state space. That choice decides which tools you can use later, such as Markov chains or jump processes.

Key rules to remember

Definition of a stochastic process: {X_t : t ∈ T}, with X_t taking values in S; T is the time set and S is the state space. State both whenever you describe a process.
Discrete time set: T = {0, 1, 2, ...} (or any countable set); Process is observed at separate points, so you write X_n.
Continuous time set: T = [0, ∞) (or any interval); Process is defined at every instant, so you write X_t.
Simple random walk: X_n = X_0 + Y_1 + Y_2 + ... + Y_n, with Y_i independent and identically distributed; Discrete time. The state space is discrete if the Y_i take integer values.
Counting process: N(t) = number of events in (0, t], N(0) = 0; Continuous time, discrete state space {0, 1, 2, ...}. Sample paths never decrease.

How to solve Stochastic Processes: Basics and Classification questions

Use this method for any question that asks you to define, describe or classify a stochastic process.

1Identify what is being observed: a count, a price, a status, a balance.
2Write down the time set T. Ask whether the process is defined only at separate points or at every instant.
3Write down the state space S. List the possible values. Ask whether they are countable or fill an interval.
4Combine the answers into one of the four types: discrete or continuous in time, discrete or continuous in state.
5Check the detail the question gives: independence, dependence on the past, or whether the process can go down as well as up.
6Justify your classification in one sentence for each of time and state. Marks are given for reasons, not only labels.

Quickest way: Two-question test

When to use it: Use this in multiple-choice questions and when you have under a minute to classify a process.

Ask: can I list the times at which the process is observed? If yes, discrete time. If it runs for every instant, continuous time.
Ask: can I list the values the process takes? If yes (including infinite lists like 0, 1, 2, ...), discrete state. If the values fill an interval, continuous state.
Match to a known example: counts of events are continuous time and discrete state; prices and returns are continuous state; yearly status levels are discrete in both.

Common mistakes in Stochastic Processes: Basics and Classification

Calling a process continuous in state because the number of values is large or infinite.
Students confuse infinite with uncountable.
Fix: Values like 0, 1, 2, ... are countable, so the state space is discrete. Only intervals of real numbers give continuous state.
Mixing up time set and state space.
Both are described using the words discrete and continuous.
Fix: Always write T and S as two separate lines before naming the type.
Classifying a Poisson process as discrete time.
Students see that it counts whole events and assume time is also in steps.
Fix: Events can occur at any instant, so time is continuous. Only the state space {0, 1, 2, ...} is discrete.
Deciding the type from the frequency of data collected.
Daily or monthly data makes a process look discrete in time.
Fix: Classify the model, not the data. If the model is defined for all t ≥ 0, it is continuous time even if you observe it only monthly.
Giving a label with no justification in a written answer.
Students treat classification as a one-word answer.
Fix: State T and S explicitly and give a reason, for example: S = {0, 1, 2, ...} so state is discrete.

Worked examples

Example 1

Classify each process by time set and state space: (a) the number of claims reported to an insurer up to time t, t ≥ 0; (b) the daily closing value of a stock index; (c) the no claims discount level of a motor policyholder at each annual renewal, with levels 0%, 20%, 40%.

Show the solution

(a) Time: claims can arrive at any instant, so T = [0, ∞), continuous time.
(a) State: the count takes values 0, 1, 2, ..., a countable set, so discrete state.
(b) Time: observed once per day, T = {1, 2, 3, ...}, discrete time.
(b) State: an index value can be any positive real number, so continuous state.
(c) Time: levels change only at annual renewals, T = {0, 1, 2, ...}, discrete time.
(c) State: S = {0%, 20%, 40%}, a finite set, so discrete state.

Answer: (a) continuous time, discrete state; (b) discrete time, continuous state; (c) discrete time, discrete state.

Example 2

A simple random walk starts at X_0 = 0. Each step Y_i equals +1 with probability 0.6 and −1 with probability 0.4, independently. State the time set and state space, and find E[X_10].

Show the solution

Time set: steps happen at n = 0, 1, 2, ..., so T = {0, 1, 2, ...}, discrete time.
State space: X_n is an integer, so S = the set of all integers, which is countable. The state is discrete.
Write X_10 = Y_1 + Y_2 + ... + Y_10 since X_0 = 0.
Find E[Y_i] = (+1)(0.6) + (−1)(0.4) = 0.2.
By linearity, E[X_10] = 10 × 0.2 = 2.

Answer: Discrete time, discrete state (the integers); E[X_10] = 2.

Exam tips

Begin every answer on this topic by writing T and S. It earns marks and keeps you organised.
In multiple-choice questions, check the state space first. The options often differ only in this.
Learn one standard example for each of the four types. Examiners reuse them: random walk, AR(1), Poisson process, Brownian motion.
When asked to give an example of a process, add a short reason so the classification is clear.
Link the classification to later topics. Discrete time and state suggests a Markov chain; continuous time and discrete state suggests a Markov jump process.

Practice questions from Stochastic processes

Stochastic Processes: Basics and Classification: frequently asked questions

What is the difference between a discrete time and a continuous time stochastic process?

In a discrete time process the time set is countable, such as {0, 1, 2, ...}, so you observe it at separate points. In a continuous time process the time set is an interval such as [0, ∞), so the process is defined at every instant.

What is the state space of a stochastic process?

It is the set of all values the random variables X_t can take. It may be finite, like {healthy, sick, dead}, countably infinite, like {0, 1, 2, ...}, or an interval of real numbers.

Is a Poisson process discrete or continuous?

It is continuous in time and discrete in state. Events can happen at any moment, but the count N(t) takes only the values 0, 1, 2, and so on.

Why does classification matter for the exam?

It tells you which model and tools apply. Discrete time and state points to Markov chains, while continuous time and discrete state points to jump processes and Poisson processes. Classification questions are also quick marks.