Special Mathematics Lecture

Contact:
Serge Richard (richard@math.nagoya-u.ac.jp), Rm. 247 in Sci. Bldg. A

Introduction to stochastic calculus (Fall 2023)

Registration code : 0063621
Schedule : Wednesday (18.30 - 20.00) in room 207 of Science building A

• Syllabus : See here

• SML official rule : See here

• NU EMI : Link to the website

• Class dates :
  October 4, 11, 18, 25
  November 1, 8, 15, 22, 29
  December 6, 13, 20
  January 10, 17
  

• Program :
  Mathematical Background
  Gaussian processes
  Brownian motion
  Stochastic integrals
  Itô processes and stochastic differential equations
  Markov processes
  Applications to finance
  

• Lecture notes:
  The cumulative notes
  

• Weekly summaries : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

• Study sessions : Will be organized on an individual basis by some students.

• For the evaluation, you need to submit the solutions of some exercises and/or the proofs of some statements. These submissions can take place at any time during the semester. If you have any question, contact me or Vic Austen.

• Works submitted by the students :
  Complements and unions imply intersections, by Yat Ming Luk
  Complements and unions imply intersections: II, by Ai Yamada
  Three pairwise i.i.d. random variables that are not i.i.d., by Rasmus Skadborg
  Simple properties of a measure, by Hadiko Rifqi Aufa Sholih
  Simple properties of a measure, by Uyanga Khoroldagva
  On the power set of a finite set, by Hevidu Samarakoon
  On independence, by Ziyu Liu
  On probabilities and continuity of probabilities, by Yuu Hiramatsu
  Power set of a set of N elements, by Masumi Okamoto
  On densities and conditional expectation, by Rasmus Skadborg
  Proof of Markov's inequality, by Tran Le Phuong Quynh
  Example of a Brownian motion, by Ai Yamada
  Gaussian vector of standard Gaussian distributions, by Uyanga Khoroldagva
  The conditional expectation: a bounded operator in L^p-spaces, by Tue Tai Nguyen
  Why normal distribution's integral is 1?, by Qiuling Low
  1-dimensional Brownian processes are Gaussian processes, by Rafi Rizqy Firdaus
  Expectations for absolutely continuous and discrete random variables, by Rafi Muflih Abdur
  Moments of Brownian motion, by Sirawich Saranakomkoop
  Brownian martingales, by Li Guoming
  On Itô integral, by Rafi Rizqy Firdaus
  Examples of martingales, by Al Kafi Muhtasin
  Martingales, by Tetta Watari
  Sum of independent Gaussian random variables, by Tetta Watari
  Langevin equation, by Tue Tai Nguyen
  Markov property of Brownian motion, by Hadiko Rifqi Aufa Sholih
  Explicit calculations of the Greeks, by Rafi Rizqy Firdaus
  Normalization of the Gaussian distribution, by Yuu Hiramatsu
  Applications of Itô lemma, by Yuu Hiramatsu
  Applications of Itô lemma, by Ngo Gia Linh
  Some properties of the time evolution operator, by Rasmus Skadborg
  Gambler's ruin problem with Brownian motion without drift, by Tran Le Phuong Quynh
  Cantelli's inequality, an improved version of Chebyshev's inequality, by Zhou Yifan
  Solution of the Black-Scholes model, by Ngo Gia Linh and Rafi Rizqy Firdaus
  Gaussian conditioning, by Uyanga Khoroldagva
  Numerical methods for simulating 1D Brownian motion and solving the Langevin equation, by Tue Tai Nguyen
  The gambler's ruin problem, by Zhou Yifan
  On homogeneous Markov property, by Ngo Gia Linh
  On Gaussian vectors, by Oleh Dmytruk
  The discrete time stochastic integral is a martingale, by Tetta Watari
  The quadratic variation of Itô process and a solution to an Itô equation, by Tetta Watari
  On Feynman-Kac formula with terminal value, by Ngo Gia Linh and Rafi Rizqy Firdaus
  About the covariance matrix, by Quan Nguyen Minh
  The Brownian process is a Gaussian process, by Quan Nguyen Minh
  Conditional expectation, by Quan Nguyen Minh
  On a few classical probability distributions, by Rafi Muflih Abdur
  Summary on Ornstein-Uhlenbeck process, by Rafi Muflih Abdur
  On how to define a measure, by Hideto Tsubouchi
  On Gaussian random variables and the Gaussian integral, by Pratham Dhomne
  On conditional expectation and L^p-spaces, by Pratham Dhomne
  On martingales, by Pratham Dhomne
  Moment generating function for the univariate Gaussian random variable, by Pratham Dhomne
  On the Black-Scholes model, by Au Yik Hau
  Uncorrelated but dependent random variables, and normalization, by Kondo Shinya
  On quadratic variation of Brownian motion, by Kondo Shinya
  Moments of Brownian motion, by Ashuurradnaa Purevnyam
  

• References : (electronic version available upon request)
  [A] J.-L. Arguin, A first course in stochastic calculus
  [B] P. Baldi, Stochastic calculus, an introduction through theory and exercises
  [D] R. Durrett, Stochastic calculus, a practical introduction
  [E] L.C. Evans, An introduction to stochastic differential equations
  [K] F. Klebaner, Introduction to stochastic calculus with applications
  [Ku] H.-H. Kuo, Introduction to stochastic integration
  [M] T. Mikosch, Elementary stochastic calculus with finance in mind
  [SP] R. Schilling; L. Partzsch, Brownian Motion: an introduction to stochastic processes