• Location: NDSU Ag & Bio Sys, Room 201
• Time: Monday, 12:00noon-12:50pm
• Organizer:Artem Novozhilov

Minglian Lin (graduate student, Math, NDSU): Mathematical analysis of optimal portfolio on finite horizons for a stochastic volatility market model

Abstract: In this presentation, we consider the portfolio optimization problem in a financial market under a general utility function. Empirical results suggest that if a significant market fluctuation occurs, invested wealth tends to have a notable change from its current value. We consider an incomplete stochastic volatility market model that is driven by both a Brownian motion and a jump process. We obtain a closed-form formula for an approximation to the optimal portfolio in a small-time horizon. This is obtained by finding the associated Hamilton--Jacobi--Bellman integro-differential equation and then approximating the value function by constructing appropriate supersolution and subsolution. It is shown that the true value function can be obtained by sandwiching the constructed super-solution and subsolution. We also prove the accuracy of the approximation formulas. 

Nikita Barabanov (Math, NDSU): Antenna array acoustic signal recognition

Abstract: We consider the problem of finding the directions of arrival and the magnitudes of several acoustic signals in the presence of a noise using antenna array. The mathematical problem amounts to multidimensional minimization of a certain cost functional. The famous superresolution methods (synphase summation, Capon, Borgotti, Bienvenue, MUSIC) are considered and analyzed. To compare different methods, we introduce a special functional that is similar to the standard signal/noise ratio. The values of this functional show the “hardness” the problem instead of considering the vector of the standard signal/noise ratio, the number of measurements and the number of antenna receivers. Several advantages of the most recent method of minimization of residual power will be explained and illustrated on a number of examples.

Nikita Barabanov (Math, NDSU): Introduction to the Wavelet Analysis

Abstract: What are the wavelets? Bases in Hilbert spaces. Riesz systems. Systems with integer shifts. Orhonormal systems. Multiresolution analysis (MRA). Refinement equations, scaling functions, and masks. Criteria for MRA. Wavelet functions and spaces. Representations of functions in L_{2}. Haar wavelets. Battle-Remarie wavelets. Meyer wavelets. Wavelets with compact support. Daubeshies wavelets.

Nikita Barabanov (Math, NDSU): Artificial Neural Networks and Wavelet Analysis

Abstract: The following topic will be considered: models of neurons, Hebbian rule, architecture of ANN, principal component analysis, supervised ANN: back propagation, unsupervised learning: Kohonen ANN, Hopfield ANNs. Wavelet transform versus Fourier transform, Wavelets with compact support.

Nikita Barabanov (Math, NDSU): Methods of Optimization (Part I)

Abstract: Optimization is a very common task in finding the best parameters of models, optimal decision making or optimal control. The course “Methods of Optimization” includes the classical concepts, the problem settings, the methods, and the main results in the areas of finite and infinite dimensional optimization. The following topics will be covered: Introduction: classification of problems, relations to other chapters of applied mathematics Necessary background: line minimization, gradient methods, steepest descent, Newton’s method, Quasi-Newton’s methods. Conditional optimization. Elements of convex analysis; main concepts, Moro-Rokafellar and Dubovitsky-Milutin formulas. Theories of linear programming, convex programming, dynamical programming: Simplex method, Kuhn-Tucker theorem, Bellman equations. Optimization in infinite dimensional spaces: Frechet derivatives, strong differentiability, Lusternik theorem, theorems about right inverse operator and closed image, Banach theorem about open mappings. Elements of Calculus of Variations: Boltz problem, Euler-Lagrange equations, special cases, isoperimetric problem, conditions of the first and the second orders, Lagrange problem. Euler-Poisson equations. Maximum principle: problem statement, needle variations, reduction to finite dimensional case, Pontryagin-Boltyansky maximum principle, examples. Optimal estimation and optimal filtration: problem setting, Kolmogorov-Wiener and Kalman-Bucy filters, applications. All results are new.

Artem Novozhilov (Math, NDSU): Mathematical epidemiology, Covid-19, and some open questions in the theory of selection systems (Part I)

Abstract: I plan to include three interrelated topics in my talk. First, I will start with the discussion of the classical paper by Kermack and McKendrick (1927), which can be considered as the origin of modern epidemiology. We will talk about basic predictions of this model and introduce the key terminology. Switching the subject, secondly, I will move to the recent Covid-19 pandemic, and many (mostly failed) attempts to make mathematical predictions on the future course of the pandemic. Finally, I plan to look into recent progress (caused mostly by Covid modeling) in the so-called theory of selection systems, discuss one particular result (which was discovered independently by several research teams), and state a very interesting (at least in my admittedly biased opinion) open mathematical problem. 

The talk will be using some very basic facts from the theory of ODEs, probability theory, and dynamical system theory, but no special preparation is necessary. I will attempt to make the talk mostly self-contained. 

Indranil SenGupta (Math, NDSU): Introduction to financial mathematics (Part II)

Abstract: (see below)

Indranil SenGupta (Math, NDSU): Introduction to financial mathematics

Abstract: In this talk I will explore some basics concepts of financial mathematics. In particular, I will discuss concepts such as the time-value of money, arbitrage, hedging, options, and Black-Scholes equation. No significant math background is necessary for the first lecture. 

Nikita Barabanov (Math, NDSU): Einstein addition of vectors and matrices (Part III)

Abstract: (See below)

Nikita Barabanov (Math, NDSU): Einstein addition of vectors and matrices (Part II)

Abstract: We consider the binary operations arising in the theories of relativity and quantum mechanics. The addition of velocities known as Einstein addition can be represented as a composition of operations well known in differential geometry: parallel transport, logarithmic and exponential mappings. An analysis of properties of this operation shows that it forms a gyrocommutative gyrogroup. Recent investigations prove that the operations isomorphic to Einstein addition are the only smooth operations in the ball that are invariant with respect to rotation and have constant Gaussian curvature. Moreover, these are the only operations such that parallel transport of geodesics are also geodesics. The first part of the presentation will be devoted to these results. In the second part we consider a generalization of such operations to the case of rectangular matrices. This topic is related to the phenomenon of entanglement known in quantum mechanics. The corresponding operation is called Einstein addition for matrices. We will show how this operation may be represented as a composition of three operations mentioned above. Finally, we consider a nice gyrogroup of positive definite matrices that appears in the quantum theory. It turns out that this group is isomorphic to Einstein addition. Moreover, for n=2 a similar “normalized” group is isomorphic to Einstein addition in three-dimensional unit ball. We will discuss these and related results.

Nikita Barabanov (Math, NDSU): Einstein addition of vectors and matrices

Abstract: We consider the binary operations arising in the theories of relativity and quantum mechanics. The addition of velocities known as Einstein addition can be represented as a composition of operations well known in differential geometry: parallel transport, logarithmic and exponential mappings. An analysis of properties of this operation shows that it forms a gyrocommutative gyrogroup. Recent investigations prove that the operations isomorphic to Einstein addition are the only smooth operations in the ball that are invariant with respect to rotation and have constant Gaussian curvature. Moreover, these are the only operations such that parallel transport of geodesics are also geodesics. The first part of the presentation will be devoted to these results. In the second part we consider a generalization of such operations to the case of rectangular matrices. This topic is related to the phenomenon of entanglement known in quantum mechanics. The corresponding operation is called Einstein addition for matrices. We will show how this operation may be represented as a composition of three operations mentioned above. Finally, we consider a nice gyrogroup of positive definite matrices that appears in the quantum theory. It turns out that this group is isomorphic to Einstein addition. Moreover, for n=2 a similar “normalized” group is isomorphic to Einstein addition in three-dimensional unit ball. We will discuss these and related results.