Unit outline_

# STAT4021: Stochastic Processes and Applications

## Overview

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. In this unit you will rigorously establish the basic properties and limit theory of discrete-time Markov chains and branching processes and then, building on this foundation, derive key results for the Poisson process and continuous-time Markov chains, stopping times and martingales. You will learn about various illustrative examples throughout the unit to demonstrate how stochastic processes can be applied in modeling and analysing problems of practical interest, such as queuing, inventory, population, financial asset price dynamics and image processing. By completing this unit, you will develop a solid mathematical foundation in stochastic processes which will become the platform for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control and financial mathematics.

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 None None STAT3011 or STAT3911 or STAT3021 or STAT3003 or STAT3903 or STAT3005 or STAT3905 or STAT3921. STAT2011 or STAT2911, and MATH1003 or MATH1903 or MATH1907 or MATH1023 or MATH1923 or MATH1933 or equivalent. That is, students are expected to have a thorough knowledge of basic probability and integral calculus and to have achieved at credit level or above in their studies in these topics. Yes

### Teaching staff

Coordinator Qiying Wang, qiying.wang@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final exam
Written exam
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Homework assignment 1
written assignments
5% Week 04 days
Outcomes assessed:
In-semester test (Record+) Quiz 1
In class quiz
25% Week 07
Due date: 19 Apr 2021 at 11:00
50 minutes
Outcomes assessed:
Assignment Homework assignment 2
written assignments
10% Week 12 days
Outcomes assessed:
= Type B in-semester exam
= Type D final exam

### Assessment summary

Detailed information for each assessment can be found on Canvas.

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Preliminaries: moment generating function, joint distribution, conditional probability and conditional expectation, random sum and basic concepts of stochastic processes. Lecture and tutorial (4 hr)
Week 02 Markov chain: transition probabilities, Chapman-Kolmogorov equations, and classification of states. Lecture and tutorial (4 hr)
Week 03 Markov Chain: Periodicity, recurrence and transience, positive and null recurrences. Lecture and tutorial (4 hr)
Week 04 Markov Chain: limit distribution, stationary distribution, absorption probability and mean return time. Lecture and tutorial (4 hr)
Week 05 Random walk and Branching processes: Gambler's ruin problem, expected duration, extinction probability Lecture and tutorial (4 hr)
Week 06 Random walk and Branching processes: expected duration, extinction probability, Weeks 1-6 revision Lecture and tutorial (4 hr)
Week 07 Quiz 1, basic properties of Poisson distribution and exponential distribution Lecture and tutorial (4 hr)
Week 08 Poisson processes: definition, interarrival and arrival (waiting) times, conditional distribution of arrival time. Lecture and tutorial (4 hr)
Week 09 Poisson processes: splitting and merging a Poisson process, nonhomogeneous Poisson process and compound Poisson process Lecture and tutorial (4 hr)
Week 10 Continuous-time Markov chains: definition and basic properties, the Embedded Markov Chain and the generatorContinuous-time Markov chains: definition and basic properties, the Embedded Markov Chain and the generator matrix, forward and backward Equations, stationary and limit distributions. Lecture and tutorial (4 hr)
Week 11 Simple queuing theory: the \$M/M/1\$ queue, the \$M/M/1\$ queue with finite capacity and the \$M/M/k, k\ge 1,\$ queue Lecture and tutorial (4 hr)
Week 12 Introduction to BM and martingale and Weeks 7-12 revision Lecture and tutorial (4 hr)
Week 13 Quiz 2, revision, Assignments revisited and pass exams. Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Explain and apply the theoretical concepts of probability theory and stochastic processes.
• LO2. Construct a discrete-time Markov chain and identify its transition probability matrix from practical problem settings.
• LO3. Explain the limit theorems for discrete-time Markov chains to identify and interpret predictions of Markov chain models.
• LO4. Explain Gambler's ruin problem and calculate extinction probability
• LO5. Explain the properties of the Poisson and exponential distributions.
• LO6. Construct a Poisson process and identify its single parameter from practical problem settings in a diverse range of applications.
• LO7. Explain the definition of the continuous-time Markov chain based on discrete-time Markov chains and the Poisson process.
• LO8. Explain the length in the queue and solve simple waiting time problems
• LO9. Explain definitions of Brownian and martingales
• LO10. Write clear, complete and logical proofs for mathematical hypotheses that are new to the student.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory