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.
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.
Poisson processes, compound Poisson processes, risk processes, ruin probability, optimal stopping problems, pricing, American options.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
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 | |
---|---|---|
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.
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.
None required.