2016 Applied Probability

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Academic unit or major
Graduate major in Mathematical and Computing Science
Miyoshi Naoto  Nakano Yumiharu 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(W832)  Fri3-4(W832)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
Access Index

Course description and aims

This course focuses on stochastic processes and its applications. In particular, topics include one dimensional point processes and dynamic optimization of discrete time stochastic processes, as well as its applications.

Student learning outcomes

At the end of this course, students will be able to:
1) Understand point processes, a fundamental class of stochastic processes. In particular, understand Poisson processes and renewal processes, and apply them to analyses of stochastic models.
2) Understand the theory and numerical methods for dynamic optimization of discrete time stochastic processes, and apply them to optimization problems in finance.


Point processes, stationary point processes, Poisson processes, renewal processes, renewal equations, Palm probability, dynamic optimization, optimal stopping problems, risk hedging problems.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Classes 1--7 is devoted to point processes. After an achievement confirmation, the last 7 classes deal with dynamic optimization of discrete time stochastic processes.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Definition of point processes Understand the definition of point process
Class 2 Stationarity Understand the concept of stationarity for point processes.
Class 3 Poisson process and M/G/1 queues Understand the definition of Poisson processes and apply one to the analysis of M/G/1 queues.
Class 4 Renewal processes and renewal theorem Understand renewal processes and renewal theorem.
Class 5 Renewal equations and their applications Understand renewal equations and apply them to a simple example.
Class 6 Regenerative processes Understand regenerative processes.
Class 7 General stationary point processes on the line Understand the role of shift operators for stationary point processes.
Class 8 Palm theory and its applications Understand the Palm probability and apply it to a simple example.
Class 9 Preliminaries of probability theory and stochastic processes Review the fundamentals of conditional expectations and discrete time Markov processes.
Class 10 Dynamic programming principle Explain the dynamic programming principle.
Class 11 Estimation of the conditional expectation Explain the estimation methods of conditional expectations.
Class 12 Optimal stopping problems Explain the derivation of general solutions of optimal stopping problems.
Class 13 Numerical solutions of optimal stoppong problems Explain numerical methods of optimal stopping problems.
Class 14 Risk hedging problems Explain risk hedging problems and derive its solution.
Class 15 Numerical solutions of the risk hedging problems Explain numerical methods of risk hedging problems.



Reference books, course materials, etc.

F. Baccelli and P. Bremaud, Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, Springer
D.P. Bertsekas, Dynamic Programming and Optimal Control I, II, Athena Scientific

Assessment criteria and methods

Several reports.

Related courses

  • MCS.T212 : Fundamentals of Probability
  • MCS.T312 : Markov Analysis

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None required.

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