# MATH40780 Stochastic Calculus (online)

This module is a introduction to stochastic calculus. It begins with fundamental concepts in probability theory, stochastic processes, Brownian motion and martingales, leads on to stochastic integrals, chiefly Itô integrals and their properties, and Itô’s formulae, and finishes with a treatment of stochastic differential equations and a few applications in finance. The existence and properties of Brownian motion, Itô integrals and formulae will be addressed heuristically.

The following topics will be covered.

1. Preliminaries - review of fundamental concepts, stochastic processes, Brownian motion, conditional expectation and martingales.
2. Stochastic Integrals - Riemann integrals and variants, Itô integrals and Itô formulae.
3. Multi-dimensional Itô Calculus - Definition of N-dimensional Itô process and properties, multi-dimensional Itô formulae; correlated Weiner processes.
4. Stochastic Differential Equations - review of deterministic ODEs, Itô SDEs and general linear SDEs.

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this module, the student is expected to be able to understand the definitions, theorems and examples covered in all of the topics listed above, and carry out associated computations.

Student Effort Hours:
Student Effort Type Hours
Autonomous Student Learning

150

Online Learning

50

Total

200

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations
Learning Requirements:

Students are required to have taken MATH40790, MATH30360 or MATH40430.

Module Requisites and Incompatibles
Incompatibles:
MATH40690 - Stochastic Calculus

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: Written examination. 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

80

Continuous Assessment: Continuous assessment. Throughout the Trimester n/a Standard conversion grade scale 40% No

20

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour