| CODE | SOR3121 | ||||||||
| TITLE | Stochastic Processes 2 | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 6 | ||||||||
| DEPARTMENT | Statistics and Operations Research | ||||||||
| DESCRIPTION | - Brownian Motion - Distributional results; - Reflection principle; - Stopping times and Hitting times; - Special types of BM like geometric BM and the Brownian bridge; - Diffusion processes. - Conditional Expectation - Precise definition; - Properties of the conditional expectation operator; - Deriving results using the conditional expectation. - Martingales - Discrete parameter martingales: definitions; - Stopping times; - Standard results for submartingales like: - Upcrossing lemma; - Doob's Convergence Theorem; - Doob-Meyer decomposition; - Uniformly integrable martingales. - Branching Processes - Generating functions, definitions; - Extinction probabilities; - Martingales recuperated from branching processes and intro to more complicated examples. Study-Unit Aims: The main aim of this study-unit is that of familiarizing the students with the theoretical and practical framework underlying a number of stochastic processes namely: random walks, Poisson processes, Markov chains, renewal processes and continuous-time Markov Chains. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Acquire the theoretical foundations required for all types of stochastic processes covered in this unit; - Be able to visualise the different stochastic processes within the probabilistic context, i.e., in terms of its probability space and its probability measure; - Gain appreciation of the use of conditional expectation, and its relevance to various aspects of martingale and semi-martingale theory; - Obtain knowledge of the different settings and applications where the stochastic processes covered can be used; - Be able to connect the theoretical commonalities between different types of stochastic processes, covered in this unit and prior, and specific types of stochastic processes and their generalisations. 2. Skills: By the end of the study-unit the student will be able to: - Use the theoretical knowledge gained in the study unit to identify which type of stochastic process should be used in specific contexts; - Apply stochastic process to real-life applications; - Use various statistical packages to perform computations for estimation, simulation and problem-solving related to stochastic processes; - Use the material learnt as foundations to other important topics in stochastic processes, statistical modelling and computational statistics. Main Text/s and any supplementary readings: Suggested Texts: Shiryaev A.N. (1996) Probability, Springer Karlin, Samuel and Taylor, Howard, M. (1975) A First Course in Stochastic Processes, Academic Doob, J.L. (1953) Stochastic Processes, Wiley Williams, D. (2001) Probability with Martingales, Cambridge Ross, S. (1996) Stochastic Processes, Wiley Billingsley, P. (1995) Probability and Measure, Wiley Karlin, Samuel and Taylor, Howard M. (1998) An Introduction to Stochastic Modeling, Academic Resnick and Sidney I. (2002) Adventures in Stochastic Processes, Birkhäuser |
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| ADDITIONAL NOTES | Pre-Requisite Study-Units: SOR1110, SOR2211 and SOR3110 | ||||||||
| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Mark A. Caruana David Paul Suda |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2023/4. It may be subject to change in subsequent years. |
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