歷史演講 PAST TALKS

Abstract: We consider a financial model with a large trader and insider characteristics. This trader has some influence on the dynamics of prices. Moreover, the information of the insider is the final price plus a blurring noise that disappears as the final time approaches. Under such a setting, we try to obtain the explicit solution of a risk-sensitive portfolio optimization problem with finite time horizon.

摘要如附件

In 1966, P. M. Cohn introduced the concept of a generalized Euclidean ring, inspired by the property that any invertible matrix over a Euclidean ring can be row-reduced to the identity matrix. In 1984, Dennis, Magurn and Vaserstein proved that the integral group ring ZCm of finite cyclic group Cm is generalized Euclidean. It is well known that a Euclidean ring is quasi-Euclidean and a quasi-Euclidean ring is generalized Euclidean. In this thesis, we construct a finitely generated nonprincipal ideal of ZG for nontrivial abelian group G to show that ZG is neither Euclidean nor quasi-Euclidean. Moreover, we give an upper bound for the number of generators of an ideal in ZCm. The case m being a power of a prime is treated more seriously. In the final chapter, following the Wedderburn-Artin Theorem, we give a more accessible proof than Bass’to show that a semilocal ring has stable rank one, hence it is a generalized Euclidean ring.

Abstract "The" integration by part in Wiener space is based on a representation of Heisenberg algebra, an infinite dimensional Lie algebra. We seek for another possibility by constructing a representation of infinite dimensional Lie algebra on Wiener space. I start from looking at established formulas in a different way.

Abstract： We give a brief report on recent progress on phase retrieval and compare our proposed method with the state-of-the-art. We discuss the potential of extending our method to other imaging problems. This is a joint work with Pengwen Chen (NCHU) and Gi-Ren Liu (NCKU).

Abstract： We will discuss ideas and adopting proper models in investigating high frequency gamma brainwaves.

Abstract Deep learning is an effective approach to solve image recognition problems. People like to think intuitively from the trading chart. This study used the characteristics of deep learning to train computers how to imitate people’s thinking from the trading chart. We have three steps as follows: 1. Before training, we need to pre-process our input data from quantitative data to images. 2. We use Convolutional-Neural-Network (CNN), which is a kind of the deep learning, to train our trading model. 3. We evaluate the models by the accuracy of the classification. With this approach, a trading model is obtained to help make trading strategies. The main application is designed to help clients automatically obtain personalized trading strategies.

Abstract： In this talk, we shall discuss subellipticity and asymptotic eigenvalue distribution of second-order partial differential operators with non-negavtive principal symbols, especially for sum of square of vector fields.

Abstract An important problem in the theory and application of statistical methods to many areas of science and humanities is the estimation of means of functions with respect to some well-defined target distribution. If the target distribution is a probability distribution or at least a totally finite measure, at least two methods are well known: the i.i.d. Monte Carlo and Markov chain Monte Carlo. In our work, we discuss Monte Carlo methods to statistically estimate the integrals of a class of functions with respect to some distributions that may not be finite, based on regenerative processes. To establish the asymptotically consistent estimator, it is necessary to impose appropriate moment conditions on the function with respect to the target distribution. This in turn leads to important, nontrivial questions regarding the local times of such processes. The present talk focuses on regenerative stochastic processes in continuous time such as Brownian motion.

Abstract： Our brain is made up of billions of neurons which send electrical to communicate with each other. The combination of parts of neurons sending signals at once produces an enormous amount of brain waves. It is found that different brain wave frequencies correspond to different tasks ranging from less active sleep to more active awake phases. Gamma waves are the fastest of the brain wave frequencies and signify the highest state of focus possible. They are associated with peak concentration and the brain’s optimal frequency for cognitive functioning. We will focus on the discussion of model choices and the network properties in producing gamma frequency.

Abstract： In this paper, we consider a generalized model of 2×2 Keller-Segel system with nonlinear chemical gradient and small cell diffusion. The existence of the traveling pulses for such equations is established by the methods of geometric singular perturbation (GSP in short) and trapping regions from dynamical systems theory. By the technique of GSP, we show that the necessary condition for the existence of traveling pulses is that their limiting proles with vanishing diffusion can only consist of the slow flows on the critical manifold of the corresponding algebraic-differential system. We also consider the linear stability of these pulses by the spectral analysis of the linearized operators.

Abstract： In this talk, we will introduce positive characteristic multiple zeta values (MZV's) initiated by Thakur in 2004. Our main result is to relate each MZV as certain coordinate of the logarithm of certain t-module at an algebraic point (joint work with Y. Mishiba). It generalizes the work of Anderson-Thakur in 1990.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

The research of Longstaff (1995) paved the way for a new wave of ``how marketability affect security values" study. Longstaff (1995) adopted the pricing formula of a lookback put derived just under Black-Scholes economy. His model has been widely used to study the value of marketability of a security, and has good empirical supports. However, it is puzzled why a such simple model can work so well, especially it ignores many practical features. In this paper, we address two of them: discrete monitoring and jump risk. We provide a general framework of approximating discrete monitoring options with jump risk, by extending the Keener's approach from diffusion models to jump diffusion models. To conclude, we find that the discrete monitoring and the jump risk each has significant impacts on the model. And it has an interesting phenomenon between the two effects. This study may be of importance in explaining the Longstaff's model is still reasonably good, as well as in proving empirical researches with a better understanding of how jump risk and discrete monitoring drive discount for marketability restriction.

摘要如附件

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract Being inspired by the work of Imkeller et al. (2001), which first develops a measured-valued version of Malliavin calculus to study the enlargement of filtration by an “exogenous” random variable on the Wiener space, in this article, we consider the enlargement of filtration on the Poisson space. We construct a measure-valued Malliavin calculus on the Poisson space with the Malliavin type calculus defined in Mensi and Privault (2003). Without resorting to the traditional Jacod’s condition (Jacod (1985)) nor its local version (Jeanblanc and Song (2012)), the measure-valued Poisson Malliavin calculus can be applied to compute the “information drift”, which follows the similar spirit in Imkeller et al. (2001). However, in contrast to the latter work, our result can be applied for a general L^2 -integrable “exogenous” random variable, while that in Imkeller et al. (2001) requires more restrictive Malliavin differentiability. Finally, we illustrate a concrete application of our theory for an example extracted from Mensi and Privault (2003).

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract： Across the world, governments are pushing computer programming on younger pupils. How is a mathematical education relevant to the increasing number of students who are writing code? This talk illustrates the relevance with some concepts (structures, functions) and habits (examples, reductions) that students pick up from the mathematics that they learn. These illustrations are drawn from language translation with vector spaces, deep learning in AlphaGo, the resources needed by any artificial intelligence, and the intractabilities and impossibilities that limit computation.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract： Existence and admissibility of delta-shock solutions is discussed for hyperbolic systems of conservation laws, with a focus on systems which do not admit classical Lax-admissible solution to certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive delta-shock wave solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued and the resulting distributions fit into a generalized concept of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of delta-shock waves in hyperbolic systems, J. Differential Equations 211 (2005), 333-381]. In this framework, it can be shown that every 2x2 system of conservation laws admits delta-shock solutions. As an example, singular solution of the classical shallow-water equations are investigated. It is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally leads to singular solutions featuring Dirac delta distributions.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： 馬航 MH370 的失聯，很長時間牽動著我們的心。為什麼會找不到？是不是搜索方法不對頭還是因為別的原因？這促發我回顧早年關於最優搜索問題的研究。於是想到要跟大家分享研究這個課題的樂趣，也作為壹個案例，談談何為創新，研究如何步步深入。

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract Let us consider the probability of falling below a target growth rate of the wealth process, or its ratio to a preset portfolio taken as a benchmark, up to time horizon T. Then, study the asymptotic behavior of minimizing probability as T tends to infinity. We formulate this problem as "large deviation control", in which we are concerned with asymptotic behavior of the minimizing probability regarding as an extension of the large deviation principle. More precisely, we aim to find the rate function of the asymptotics, its "effective domain" where the rate function does not vanish nor diverge, and then asymptotically optimal strategies which realize the asymptotics. We also discuss the problem under model uncertainty.

Abstract: In this short course, we consider the problem of how to recover a low rank matrix from its limited information, which arises from numerous practical applications in machine learning, imaging, signal processing, computer vision, etc. Typically, the low rank matrix to be recovered is of large scale and the number of linear measurements is small. We will start with some examples that can be formulated to the low-rank matrix reconstruction problem. Then we will introduce both convex and non-convex algorithms, as well as the theory on the guarantee of their convergence to the correct low-rank matrix. Organizer: I-Liang Chern (NTU)

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract In this talk we consider the ruin probability minimization for insurers. Hipp and Plum (2000) treated this problem with a Black- Scholes model. We extend this work to a linear Gaussian stochastic factor model. In our model, the mean returns and volatilities of the risky asset depend linearly on economic factor that are formulated as the solution of a linear stochastic differential equation. We formulate a stochastic control problem. Moreover, a Hamilton-Jacobi-Bellman (HJB) equation that is derived via a dynamic programming approach has an explicit solution found by solving quadratic equation. Hence, the optimal strategy can be constructed explicitly.

Abstract: We will report the recent progress on the global well-posedness of the small data Cauchy problem for the Boltzmann equation of some cut-off soft potential model. The scattering behavior of the solution is also found.

Abstract: We will present the large time behavior of the linearized Landau equation in hard and modulated soft potentials. Moreover, the soothing effect will be discussed in this talk. This is a joint work with Haitao Wang and Yu-Chu Lin.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: This is the second part of a series of lectures on the projection method for viscous incompressible flows. The underlying idea of projection method for solving viscous incompressible flow problems was first introduced independently by Chorin and Temam around 1968. This approach is based on applying the Helmholtz-Hodge decomposition to the incompressible Navier-Stokes equations. The most attractive feature of projection method is that one only needs to solve a sequence of decoupled elliptic equations for the velocity and the pressure at each time step. Therefore, it is very efficient for large-scale numerical simulations. In this talk, we will give a brief review of two variants of the projection method, namely the velocity-correction scheme and the consistent splitting scheme. Reference: J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, 195 (2006), pp. 6011-6045. Organizer: 楊肅煜 教授 (國立中央大學數學系)

Abstract This talk focuses on the problem of modeling for small world effect on complex networks. We will consider the supercritical Poisson continuous percolation on d-dimensional torus T^d_n with volume n^d. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph G_n. Firstly, we show that the diameter of G_n grows at most polynomially fast in ln n and we call it the Poisson Geometry Small World. Secondly, we show that the random walk on G_n possesses the rapid mixing property, namely, the random walk mixes in time at most polynomially large in ln n.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: We introduce the mathematical models for high frequency trading proposed by Cartea, Jaimungal and Penalva. In addition, we also discuss another models driven by jump diffusions proposed by Gueant, Lehale, Fernandez-Tapia. The value functions and the corresponding optimal quotes for the trading strategies in both models are obtained using the Hamiltonian-Jocabi-Bellman equations. The explicit solutions are given by the proper ansatz.

Abstract: 藉由奧斯卡獲獎電影「模仿遊戲」，二次世界大戰的密碼戰與電腦科學之父Alan Turing 的傳奇故事，得以呈現在觀眾面前。由於電腦普及與網路發達，現代資訊戰無時無刻不上演，遠比二次世界大戰期間激烈。我們將從二次大戰德軍密碼機Enigma的破譯談起，比較與現代密碼攻防之差異，並介紹數學與資訊安全的關聯。 歡迎大學生踴躍參加 如欲參加演講，煩請回EMail，以利統計人數。 聯絡人: 助理 詹茹婷 (03)422-7151轉65124；juting@math.ncu.edu.tw

Abstract: In this talk, we discuss some properties of the quantum Zakharov system. First, we talk about the semi-classical limit of the QZ system which is the classical Zakharov system as the quantum parameter goes to zero. One of the difficulties of the convergence is that for the same initial data the solutions of the two systems lie in spaces with different regularities. Thus we need to measure the discrepancy of the solutions to the two systems under a norm with lower regularity. We improve the known result. Then we will discuss the existence and stability of the standing waves of the QZ system. There are several ways to ensure the existence of the ground states. Two ways are based on the minimization method with concentration compactness and with Lagrange multiplier, while the other one is a perturbation of the ground state of the cubic nonlinear Schr\"odinger equation. For the 1D case we can prove that the standing wave is of orbital stability.

Abstract: A branching process is a mathematical model which has been commonly used to describe the evolution of a population in various research fields such as genealogy, physics, ecology, epidemiology, finance, etc. One way to investigate the population is to look forward to its future. But, when a population grows so old, it is always interesting to know what happened to it in the past. The coalescence problem provides a way to understand the structure of the population and the ancestry of the individuals in it. Here, we will consider branching processes with different settings and, in each process, We pick two individuals from those who are alive at the current time by simple random sampling without replacement and trace their lines of descent backward in time till they meet for the first time. We call the common ancestor of these chosen individuals at the coalescent time their most recent common ancestor. The coalescence problem is to investigate the limit behaviors of some characteristics of this most recent common ancestor such as its death time and its generation number. Moreover, we will also apply the results from the coalescence problem to branching random walks.

Abstract: The underlying idea of projection method for solving viscous incompressible flow problems was first introduced independently by Chorin and Temam around 1968. This approach is based on applying the Helmholtz-Hodge decomposition to the incompressible Navier-Stokes equations. The most attractive feature of projection method is that one only needs to solve a sequence of decoupled elliptic equations for the velocity and the pressure at each time step. Therefore, it is very efficient for large scale numerical simulations. In this talk, we will give a brief review of the projection methods, especially focus on the pressure-correction schemes. Reference: J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, 195 (2006), pp. 6011-6045.

Abstract: We will talk about why time-frequency analysis is important for dealing with non-stationary signals, particularly those arise in physiological and other natural sciences. In addition to surveying the difficulty encountered in applying traditional signal processing methods, a few tools that have been developed to overcome the problems will be introduced, including the concept of sparse decomposition. Finally, we will present some new perspectives and recent progress regarding these issues, including an introduction to some of the speaker's own work.