### 2017　Applied Probability

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Graduate major in Mathematical and Computing Science
Instructor(s)
Miyoshi Naoto  Nakano Yumiharu
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(W832)  Fri3-4(W832)
Group
-
Course number
MCS.T410
Credits
2
2017
Offered quarter
4Q
Syllabus updated
2017/4/19
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

This course focuses on stochastic processes and its applications. In particular, topics include compound Poisson processes and the optimal stopping of stochastic processes, as well as its applications.

### Student learning outcomes

At the end of this course, students will be able to:
1) Understand compound Poisson processes, a fundamental class of stochastic processes, and apply them to evaluation of ruin probability in risk theory.
2) Understand the theory and numerical methods for the optimal stopping of stochastic processes, and apply them to the pricing problems in finance.

### Keywords

Poisson processes, compound Poisson processes, risk processes, ruin probability, optimal stopping problems, pricing, American options.

### 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 compound Poisson processes and risk analysis. After an achievement confirmation, the last 7 classes deal with the optimal stopping of stochastic processes.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Ruin problem and homogeneous Poisson processes Understand the definition of ruin problem and Poisson processes
Class 2 Compound Poisson processes Understand the definition of compound Poisson processes
Class 3 Ruin probability in compound Poisson risk model Understand the ruin probability in compound Poisson model
Class 4 Ruin probability and renewal theory Deepen the understanding of ruin probability in compound Poisson model
Class 5 Asymptotic property of ruin probability: Light-tailed claim sizes Understand the asymptotic property of ruin probability for light-tailed claim sizes
Class 6 Asymptotic property of ruin probability: Subexponential claim sizes Understand the asymptotic property of ruin probability for subexponential claim sizes
Class 7 Duality in risk and queueing processes Understand a dual property in risk and queueing processes
Class 8 Achievement confirmation Deepen the understanding of the first part
Class 9 Preliminaries of probability theory and stochastic processes Review the fundamentals of conditional expectations and discrete time Markov processes.
Class 10 Estimation of the conditional expectation Explain the estimation methods of conditional expectations.
Class 11 Optimal stopping problems Explain the derivation of general solutions of optimal stopping problems.
Class 12 Numerical solutions of optimal stoppong problems Explain numerical methods of optimal stopping problems.
Class 13 Numerical solutions of optimal stoppong problems Explain numerical methods of optimal stopping problems.
Class 14 The pricing problems in finance Explain the pricing problems in finance.
Class 15 Approximation of American option prices Explain approximation methods of American option prices.

None.

### Reference books, course materials, etc.

T. Rolski, H. Schmidli, V. Schmidt & J. Teugels著『Stochastic Processes for Insurance and Finance』 Wiley
D.P. Bertsekas, Dynamic Programming and Optimal Control I, II, Athena Scientific

Several reports.

### Related courses

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

None required.