2017 Advanced topics in Analysis G1

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Academic unit or major
Graduate major in Mathematics
Ninomiya Syoiti 
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Course description and aims

This lecture and its sequel ``Advanced topics in analysis H1'' are aimed at those wishing to learn about the basic mathematical concepts of mathematical finance. The following notions, continuous time martingales, Brownian motion, stochastic integral, stochastic differential equations, and the Black-Scholes model are discussed in these two lectures. In this lecture we start from the introduction of stochastic processes, develop the theory of martingales, and introduce the definition of the Brownian motion.

Student learning outcomes

Students are expected that they understand the basic notions of continuous time martingales and Brownian motion.


Mathematical Finance, Martingale(discrete time)

Competencies that will be developed

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

Class flow

Blackboard and handouts

Course schedule/Required learning

  Course schedule Required learning
Class 1 Probability theory Details will be provided during each class session
Class 2 stochastic processes
Class 3 discrete time martingale
Class 4 Doob's inequality/Stopping time/Optional Sampling theorem
Class 5 Martingale
Class 6 Quadratic Variation
Class 7 Brownian Motion(1)
Class 8 Browninan Motion(2)


None in particular

Reference books, course materials, etc.

D. Williams, ``Probability with Martingales'', Cambridge
R. J. Elliott and P. E. Kopp, ``Mathematics of Financial Markets'', Springer
N. Ikeda, S. Watanabe, "Stochastic Differential Equations and Diffusion Processes", North Holland

Assessment criteria and methods

Based on reports.

Related courses

  • MTH.C361 : Probability Theory
  • MTH.C508 : Advanced topics in Analysis H1

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

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