Special Mathematics Lecture
Contact:
Serge Richard (richard@math.nagoyau.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

SML official rule :
See here

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

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
1dimensional 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 BlackScholes 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 FeynmanKac 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 OrnsteinUhlenbeck 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 BlackScholes 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
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