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MA30371: Stochastic processes and martingales

[Page last updated: 22 May 2025]

Academic Year: 2025/26
Owning Department/School: Department of Mathematical Sciences
Credits: 12 [equivalent to 24 CATS credits]
Notional Study Hours: 240
Level: Honours (FHEQ level 6)
Period:
Semester 1
Assessment Summary: EXCB 100%
Assessment Detail:
  • MA32071 Closed-book written examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA22039 OR take MA20225 OR take MA20301
In taking this module you cannot take MA30125 OR take MA40058
Learning Outcomes: On completing the unit, you will be able to:
  • Formulate appropriate Markovian models for a variety of real world problems;
  • Classify a birth-death process as explosive or non-explosive;
  • Make effective use of time reversal to describe departures from queues;
  • Perform standard computations for simple Markov processes in continuous time;
  • Verify the martingale property for various examples;
  • Apply the optional stopping theorem to effectively compute first passage probabilities.



Synopsis: In this unit you will explore two fundamental classes of modern stochastic processes. You will learn about Markov process in continuous time and discover some of its many applications in areas such as genetics, queuing theory and ruin theory. Moreover, you will find out why martingales, which are the mathematical equivalent of a fair game, are used in many areas of modern probability.

Content: Markov processes in continuous time: Q-matrices and waiting time distributions; Examples, including Poisson processes, birth and death processes, compound Poisson processes; Equilibrium distributions and ergodicity; Stopping times and the strong Markov property; Explosions; Reversibility. Theory of Martingales: Filtrations, definitions and simple examples, optional stopping theorem, statement of convergence theorem, discrete-time version of stochastic integral. Examples will be chosen from: Queuing networks: M/M/s queue, departure process, series of M/M/s queues, open and closed migration networks; Ruin problems in insurance; Blocking probabilities in telecommunication networks; Simple genetics models: Wright-Fisher and Moran models. Kingman's coalescent process; Population models and branching processes; First passage problems; Fair games and betting strategies.

Course availability:

MA30371 is Optional on the following courses:

Department of Mathematical Sciences
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)

Notes:

  • This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
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