数理ファイナンスセミナー

Math-Fi seminar on 27 Feb.

2020.02.07 Fri up
  • Date: 2020/02/27 (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:30-18:00
  • Speaker: Pierre Bras (École normale supérieure Paris)
  • Title: Bayesian inference and Metropolis-Hastings algorithms (provisional title)
  • Abstract:
Markov chains Monte Carlo (MCMC) methods are simulation methods for sampling from a probability distribution from which direct sampling is difficult, and are particulary used in bayesian learning. The Metropolis-Hastings algorithm is one of the most popular. I will present the algorithm, then prove convergence results, and present the adaptation of the algorithm to stochastic optimization problems. Please come and join us.
 

Math-Fi seminar on 13 Feb.

2020.02.07 Fri up
  • Date: 2020/02/13 (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:30-18:00
  • Speaker: Pierre Bras (École normale supérieure Paris)
  • Title: Stochastic optimization and neural networks
  • Abstract:
Stochastic algorithms aim to solve optimization problems by randomly exploring the state space. They appear in machine learning and in financial mathematics, and are the main tool used for calibrating neural networks. I will demonstrate some convergence properties, and present the gradient descent in the framework of neural networks.
 

Math-Fi seminar on 17 Jan.

2020.01.14 Tue up
  • Date: 2020/01/17 (Fri.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:30-18:40


  • First Speaker: Yushi Hamaguchi (Kyoto University)
  • Time: 16:30-17:30
  • Title: Time-inconsistent consumption-investment problems in incomplete markets
  • Abstract:
​ 投資家の時間選好を表す割引関数が古典的な指数割引関数でない場合、Bellmanの最適性原理が成立せず、効用最大化問題は時間非整合となることが知られている。つまり、現時点で見たときの将来の利得に関する最適戦略が、後の時点で見ると最適戦略とはならない。近年、このような時間非整合的な最適化問題が、確率制御理論、数理ファイナンス、経済学などで注目されている。本講演では、非マルコフかつ非完備マーケットの設定において、一般の割引関数の下での投資家の消費・投資戦略に関する効用最大化問題を考える。この問題において、時間非整合的な「最適戦略」に取って代わる時間整合的な解概念である「ナッシュ均衡戦略」の定義を紹介し、そのFBSDEを用いた特徴づけ、および時間整合的な効用最大化問題との対応について得られた結果を説明する。
 
  • Second Speaker: Yuki Ueda (Hokkaido University) 
  • Time: 17:40-18:40
  • Title: Introduction to free probability theory and infinitely divisible distributions
  • Abstract:
​ 自由群に付随した非可換量の考察から生まれた自由確率論は、古典確率論との関連深い結果を多く含む。本講演では、自由確率論の初歩から始め、古典確率論と特に関連があるものの一つである、分布の自由無限分解可能性について解説する。

Math-Fi seminar on 9 Jan.

2020.01.07 Tue up
  • Date: 2020/01/09 (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:00-19:00


  • First Speaker: Katsushi Nakajima (Ritsumeikan Asia Pacific University)
  • Time: 16:00-17:30
  • Title: TBA

  • Second Speaker: Koya Sakakibara (Kyoto University)
  • Time: 17:30-19:00
  • Title: Numerical analysis of interface problem
  • Abstract:
​ The interface appears in several problems, such as fluid dynamics between two different liquids. To study its evolution and dynamics forms the basis of the research in natural science. Although there are several mathematical studies for interfacial phenomena, they are, in general, so difficult, and numerical study becomes an essential tool in this field.
 In this talk, I will talk about the numerical analysis of interface problem, and especially consider two issues: The Hele-Shaw problem and grain boundary. The Hele-Shaw problem describes the motion of viscous fluid in a quasi-two-dimensional space, which started from a short paper by Henry Selby Hele-Shaw (1854–1941). It is now recognized as a basic mathematical model to study the fingering phenomena (also known as the Saffman–Taylor instability), and several researchers have studied this problem; however, there are still several open questions. A problem on the grain boundary appears in the field of material science. A grain boundary is an interface between two grains, or crystallites, in a polycrystalline material. It is the two-dimensional defect in the crystal structure. The study of grain boundaries and their effects on the mechanical, electrical, and other properties of materials forms an essential topic in material science. My study aims to understand the mechanism of grain boundaries from mathematical and numerical points of view.
 In the first half of this talk, I will explain this problem and construct some efficient numerical scheme based on the method of fundamental solutions and the asymptotic uniform distribution method. I will also briefly survey the geometric numerical integration, which aims to construct a numerical scheme which inherits properties of the original problem in some discrete sense. In the second half of this talk, I will move on to the problem on grain boundaries and consider manifold-valued total variation flows. I will introduce spatially discretized total variation flow and construct a numerical scheme using the exponential map of the manifold. I will also present an energy dissipation property and convergence result.

Math-Fi seminar on 10 Oct.

2019.10.07 Mon up
  • Date: 10 Oct. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:30-18:00
  • Speaker: Dan Crisan (Imperial College London)
  • Title: second lecture 

Math-Fi seminar on 3 Oct.

2019.09.30 Mon up
  • Date: 3 Oct. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:30-18:00
  • Speaker: Dan Crisan (Imperial College London)
  • Title: Modelling multi-period carbon markets using singular forward backward SDEs
  • Abstract:
I will introduce a model the evolution of emissions and the price of emissions allowances in a carbon market such as the European Union Emissions Trading System (EU ETSP). The model accounts for multiple trading periods (and phases) and multiple times at which compliance can occur. At the end of each trading period, the participating firms must surrender allowances for the emissions made during that period, but any excess allowances can be used for compliance in the following periods. We show that the multi-period allowance pricing problem is well-posed for various mechanisms linking the trading periods (such as banking, borrowing and withdrawals). The results are based on the analysis of a forward-backward stochastic differential equation with the following special characteristics: i. the forward and backward components are coupled,  ii. the final condition is singular and iii. the forward component of the model is degenerate. I will also introduce an infinite period model, that is, a model for carbon market with a sequence of compliance times and no end date. I will show that, under appropriate conditions, the value function for the multi-period pricing problem converges, as the number of periods increases, to a value function for this infinite period model. This is joint work with Jean-Francois Chassagneux (Paris Diderot) and Hinesh Chotai (Citybank).
 

Math-Fi seminar on 12 Sep.

2019.09.09 Mon up
  • Date: 12 Sep. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:30-18:00
  • Speaker: José Manuel Corcuera (Universitat de Barcelona)
  • Title: Contingent Convertibles (final lecture)
 

Math-Fi seminar on 29 Aug.

2019.08.05 Mon up
  • Date: 29 Aug. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:00-18:00
  • Speaker: Wei Zhu/Maoqi Hu (University of Liverpool)
  • Title: Introduction to Ruin Theory (final lecture)
  • Abstract:
Risk theory in general and ruin probabilities in particular are traditionally considered as part of insurance mathematics, and has been an active area of research from the days of Lundberg all the way up to today. One of the central topics in the risk theory literature is deriving the probability of ruin in the collective risk model. The classical risk model and renewal risk models will be focused in this course, where the claim number processes are assumed to be Poisson counting processes and any general renewal counting processes, respectively.
 
The first part of this course is about the classical risk model. Different approaches to derive the ruin probability will be shown and explained. The natural extension of the classical risk model leads to the renewal risk model. Very general assumptions on interarrival times are possible for the renewal risk model, which includes the classical risk model, Erlang risk model and fractional Poisson risk model. A new family of differential operators are defined in order to construct the fractional integro-differential equations for ruin probabilities in such renewal risk models. Through the characteristic equation approach, specific fractional differential equations for the ruin probabilities can be solved explicitly, allowing for the analysis of the ruin probabilities

Math-Fi seminar on 22 Aug.

2019.08.05 Mon up
  • Date: 22 Aug. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:00-18:00
  • Speaker: Wei Zhu/Maoqi Hu (University of Liverpool)
  • Title: Some Aspects about Life Insurance and Life Expectancy in China (second lecture)
  • Abstract: 
​Life insurance has undergone enormous change in the last two to three decades. Given the changes occurring in the interconnected worlds of finance and life insurance, we believe that this is a good time to recast the mathematics of life contingent risk to be better adapted to the products, science and technology that are relevant to current and future actuaries. Pension, a part of the life insurance, is an importance insurance in our life. However, most countries face the challenge in pension. Population aging and life expectancy both of these will influence the pension system. The course will first introduce the background and give some basial theories about life insurance and pension. Then the course will discuss the influence factors of life expectancy and also use Spatial Statistical tools to assess mortality differences in China.
 

Math-Fi seminar on 9 Aug.

2019.08.05 Mon up
  • Date: 9 Aug. (Fri.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time: 16:00-18:00
  • Speaker: Wei Zhu/Maoqi Hu (University of Liverpool)
  • Title: Introduction to Fractional Calculus (first lecture)
  • Abstract:
It is surprisingly, but most scientists and engineers remain unaware of fractional calculus; it is not being taught in universities and colleges; and others remain sceptical of this field. There are several reasons for that: several of the definitions proposed for fractional derivatives were inconsistent, meaning they worked in some cases but not in others. The mathematics involved appeared very different from that of integer order. There were almost no practical applications of this field. During the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics. Some of the areas where fractional calculus has made a profound impact.
This course starts from scratch and provides with the background necessary for the understanding of the fractional calculus. It will start from the birth of the fractional calculus. Different approaches defining fractional integral and derivatives will be presented and discussed. Useful and practical properties of each specific fractional derivative will be shown as well. In the end of this two-hour course, there will be some geometric interpretation of fractional derivative to help students have a better understanding of this world.