2017 Advanced topics in Analysis H1

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

In this lecture, the theory of Ito integral (stochastic integral) is introduced. We start from the definition of the Ito integral and the Ito formula is introduced. Then we develop some important aspects of stochastic differential equations. As an application of Girsanov's theory and martingale representation, the Black-Scholes formula is derived.

Student learning outcomes

Students are expected that they understand the notion of stochastic integral, are able to use the Ito formula properly, and understand the relations between the basic notion of the theory of mathmatical finance and stochastic integrals.


Stochastic Integral, Ito formula, Stochastic Differential Equation, Mathematical Finance

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 Stochastic Integral/Ito formula Details will be provided during each class session
Class 2 Representation theorem
Class 3 Stochastic Differential Equation/Solution
Class 4 Properties of solutions to stochastic differential equations/Storong markov property
Class 5 Stochastic Differential Equation(2)/Exponential martingale/Girsanov's theorem
Class 6 Feynmann-Kac formula/Heat Equation
Class 7 Application to mathematical finance(1)
Class 8 Application to mathematical finance(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.C507 : Advanced topics in Analysis G1
  • MTH.C361 : Probability Theory

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

Knowledge about basic continuous time stochastic processes (Advanced topics in Analysis G1)

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