Stochastic Process Lecture 09

 Stochastic Process 

Lecture 09

Introduction

The word "stochastic" originates from the Greek word "stokhastikos", which means random or by chance.

A stochastic process or stochastic model is a statistical phenomenon in which the collection of a random variable changes over time according to some probability law. The values of the variable at each time point are determined by a probability distribution. 

Mathematically, a stochastic process can be represented by Xt(s); t ϵ T, s ϵ S or X(s, t) on some probability space.

Where:

X(t) denotes the random variable at time t.

t is the time index, which can either be discrete (t = 1, 2, ...) or continuous (0 < t < ∝). 

T and S are the index sets that define the range of time and sample space of a random variable Xt(s). The T and S are called the parameters of the stochastic model.

Examples:

i. To describe the GDP and the yearly income of a company as a stochastic model.

ii. The average temperature and months can be described as a stochastic model, etc. 

The two approaches can be employed to generate stochastic processes. They are given below:

Let Xt(s) be a stochastic process of s, t. t ϵ T and s ϵ S.

Case I: A family of random variables {X(s, •), t ϵ T} made a function on s, and t is fixed.

Examples:

i. The number of customers (Xt) in the cash counter of a bank at 10.00 AM on the nth day of the week.

ii. The maximum temperature (Xt) of a city recorded at 2.00 pm on n^th day

This is the easier approach to creating a stochastic process.

Case II: A family of random variables {X(•, s), s ϵ S} made a function on t, and s is fixed.

Examples:

i. The exchange rate (Xt) of US $ in Pakistani rupees during a day in the nth week.

ii. The price of a stock in a stock (Xt) market at any time “t”.

Notations and their Meanings

1. If s and t are fixed, then Xt(s) is a number.

2. If t ϵ T is fixed, then Xt(s) is a single random variable defined on s.

3. If s ϵ S is fixed, then Xt(s) is a single time function.

4. If s & t are variable, then Xt(s) is a collection of random variables that are time functions and constitute a stochastic process.

Explanation of the concept of stochastic process by taking a practical example:

In Covid-19, a professor uploaded a video lecture on social media and counted the number of viewers at various points throughout the day on different days of the week. The professor uploaded the video at 9:00 am and counted the viewers at 9:00 am on day 1 through day 6, at 11:00 am on day 1 through day 6, at 1:00 pm on day 1 through day 6, and at 4:00 pm on day 1 through day 6.

Notations

X1(t): Number of viewers at time “t” (09.00 am), called sample point 1 on Day – 1

X2(t): Number of viewers at time “ ” (09.00 am), called sample point 1 on Day – 2

 X3(t): Number of viewers at time “ ” (09.00 am), called sample point 1 on Day – 3

 X4(t): Number of viewers at time “ ” (09.00 am) called sample point 1 on Day – 4

 X5(t): Number of viewers at time “ ” (09.00 am), called sample point 1 on Day – 5

 X6(t): Number of viewers at time “ ” (09.00 am), called sample point 1 on Day – 6

Note: time is constant here

Similarly, Xi(tj) i = 1, 2, 3, 4, 5, 6 and j = 1, 2, 3, 4 for 09.00 am, 11.00 am, 01.00 pm, and 04.00 pm, respectively.

The professor observed the numbers of viewers at various points in time (called sample points):

The diagrammatic representation of the above time series is given below:

Explanation of some Basic Terms

Sample Space

The collection of all sample points (realisation or sample path) at a particular point of time “t”. In the above example, the collection of sample points at 09.00 am is the sample space for 09.00 am.

Similarly, the sample space for 11.00 am, 01.00 pm and 04.00 for each fixed t ϵ T, Xt(s) is a single random variable on S.

Ensemble

The collection of sample functions or realisations at various points of time on various sample spaces. In the above example, the total number of viewers at 09.00 am, 11.00 am, 01.00 pm and 04.00 on a particular day.

In the above example, the collection samples function on a particular day.

• Read More: Basic Concepts in Stochastic Process