Unit outline_

# MATH4512: Stochastic Analysis

## Overview

Capturing random phenomena is a challenging problem in many disciplines from biology, chemistry and physics through engineering to economics and finance. There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. Hence it is of crucial importance that applied mathematicians are equipped with tools used to analyse and quantify random phenomena. This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics.

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 None None None Students should have a sound knowledge of probability theory and stochastic processes from, for example, STAT2X11 and STAT3021 or equivalent Yes

### Teaching staff

Coordinator Ben Goldys, beniamin.goldys@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final exam
Written calculations
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
Written assignment
20% Week 06
Due date: 11 Sep 2022 at 23:59
15 standard maths assignment
Outcomes assessed:
Assignment Assignment 2
Written assignment
20% Week 12
Due date: 30 Oct 2022 at 23:59
15 standard maths assignment
Outcomes assessed:
= 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.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## 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 Introduction to continuous time processes. Filtrations and stopping times. Lecture and tutorial (4 hr)
Week 02 Martingales, continuous local martingales, and Doob’s optional sampling theorem Lecture and tutorial (4 hr)
Week 03 Supermartingales and the Doob-Meyer decomposition Lecture and tutorial (4 hr)
Week 04 Introduction to Brownian motion and its fundamental properties Lecture and tutorial (4 hr)
Week 05 The Girsanov theorem and stochastic exponential. Lecture and tutorial (4 hr)
Week 06 Quadratic variation and applications (Dambis, Dubins and Schwarz theorem). Lecture and tutorial (4 hr)
Week 07 The Itˆo integral for continuous local martingales. Lecture and tutorial (4 hr)
Week 08 Itˆo’s lemma. The Brownian local time and the Itˆo-Tanaka formula. Lecture and tutorial (4 hr)
Week 09 Predictable representation property Lecture and tutorial (4 hr)
Week 10 Notions of strong and weak solutions to SDEs. Lecture and tutorial (4 hr)
Week 11 Existence and Uniqueness of solutions to SDEs. Lecture and tutorial (4 hr)
Week 12 Feynman-Kac formula and PDEs. 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. Demonstrate an understanding of the concepts of continuous time processes, stopping times and martingales.​
• LO2. ​Apply the properties of Brownian motion (such as martingale property, Markov property, behaviour under the equivalent change of measure and predictable representation property). ​
• LO3. Describe the construction of Ito integral and apply Ito integration correctly in relevant examples.​
• LO4. Identify the similarities and differences between strong and weak solutions to SDEs and apply existence and uniqueness results to some specific SDEs.​
• LO5. Check the martingale property of stochastic processes using Ito's lemma.
• LO6. Use the Feymann Kac formula to analyse real world problems including arbitrage pricing.
• LO7. Identify, formulate and solve original practical problems that can be addressed using mathematical techniques learnt in this unit and examine the implementations and provide an interpretation of the results.​
• LO8. Independently research sources provided by lecturers such as journal articles and working papers and evaluate the real-world plausibility and effectiveness of various approaches and tools.​

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.