| 1. |
- Musonda, John, 1981-
(författare)
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Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main object studied in this thesis is the multi-parametric family of unital associative complex algebras generated by the element $Q$ and the finite or infinite set $\{S_j\}_{j\in J}$ of elements satisfying the commutation relations $S_jQ=\sigma_j(Q)S_j$, where $\sigma_j$ is a polynomial for all $j\in J$. A concrete representation is given by the operators $Q_x(f)(x)=xf(x)$ and $\alpha_{\sigma_j}(f)(x)=f(\sigma_j(x))$ acting on polynomials or other suitable functions. The main goal is to reorder arbitrary elements in this family and some of its generalizations, and to study properties of operators in some representing operator algebras, including their connections to orthogonal polynomials. For $J=\{1\}$ and $\sigma(x)=x+1$, the above commutation relations reduce to the famous classical Heisenberg--Lie commutation relation $SQ-QS=S$. Reordering an element in $S$ and $Q$ means to bring it, using the commutation relation, into a form where all elements $Q$ stand either to the left or to the right. For example, $SQ^2=Q^2S+2QS+S$. In general, one can use the commutation relation $SQ-QS=S$ successively and transform for any positive integer $n$ the element $SQ^n$ into a form where all elements $Q$ stand to the left. The coefficients which appear upon reordering in this case are the binomial coefficients. General reordering formulas for arbitrary elements in noncommutative algebras defined by commutation relations are important in many research directions, open problems and applications of the algebras and their operator representations. In investigation of the structure, representation theory and applications of noncommutative algebras, an important role is played by the explicit description of suitable normal forms for noncommutative expressions or functions of generators. Further investigation of the operator representations of the commutation relations by difference type operators on Hilbert function spaces leads to interesting connections to functional analysis and orthogonal polynomials. This thesis consists of two main parts. The first part is devoted to the multi-parametric family of algebras introduced above. General reordering formulas for arbitrary elements in this family are derived, generalizing some well-known results. As an example of an application of the formulas, centralizers and centers are computed. Some operator representations of the above algebras are also described, including considering them in the context of twisted derivations. The second part of this thesis is devoted to a special representation of these algebras by difference operators associated with action by shifts on the complex plane. It is shown that there are three systems of orthogonal polynomials of the class of Meixner--Pollaczek polynomials that are connected by these operators. Boundedness properties of two singular integral operators of convolution type connected to these difference operators are investigated in the Hilbert spaces related to these systems of orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on the $L^2$-spaces and estimates of the norms are obtained. This investigation is also extended to $L^p$-spaces on the real line where it is proved again that the two operators are bounded.
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| 2. |
- Abola, Benard, 1971-, et al.
(författare)
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PageRank in evolving tree graphs
- 2018
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Ingår i: Stochastic Processes and Applications. - Cham : Springer. - 9783030028244 ; , s. 375-390
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Bokkapitel (refereegranskat)abstract
- In this article, we study how PageRank can be updated in an evolving tree graph. We are interested in finding how ranks of the graph can be updated simultaneously and effectively using previous ranks without resorting to iterative methods such as the Jacobi or Power method. We demonstrate and discuss how PageRank can be updated when a leaf is added to a tree, at least one leaf is added to a vertex with at least one outgoing edge, an edge added to vertices at the same level and forward edge is added in a tree graph. The results of this paper provide new insights and applications of standard partitioning of vertices of the graph into levels using breadth-first search algorithm. Then, one determines PageRanks as the expected numbers of random walk starting from any vertex in the graph. We noted that time complexity of the proposed method is linear, which is quite good. Also, it is important to point out that the types of vertex play essential role in updating of PageRank.
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| 3. |
- Betuel, Canhanga, et al.
(författare)
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Calibration of Multiscale Two-Factor Stochastic Volatility Models: A Second-Order Asymptotic Expansion Approach
- 2018
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Konferensbidrag (refereegranskat)abstract
- The development of financial markets imposes more complex models on the option pricing problems. On the previous papers by the authors, we consider a model under which the underlying asset is driven by two independent Heston-type stochastic volatility processes of multiscale (fast and slow) mean-reverting rates and we compute an approximate solution for the option pricing problem, using asymptotic expansion method. In the present paper, we aim to calibrate the model using the market prices of options on Euro Stoxx 50 index and an equity stock in the European market. Our approach is to use the market implied volatility surface for calibrating directly a set of new parameters required in our second-order asymptotic expansion pricing formula for European options. This secondorder asymptotic expansion formula provides a better approximation formula for European option prices than the first-order formula, as explained in an earlier work of the authors.
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| 4. |
- Biganda, Pitos, 1981-, et al.
(författare)
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Traditional and lazy pageranks for a line of nodes connected with complete graphs
- 2018
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Ingår i: Stochastic Processes and Applications. - Cham : Springer. - 9783030028244 ; , s. 391-412
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Bokkapitel (refereegranskat)abstract
- PageRank was initially defined by S. Brin and L. Page for the purpose of measuring the importance of web pages (nodes) based on the structure of links between them. Due to existence of diverse methods of random walk on the graph, variants of PageRank now exists. They include traditional (or normal) PageRank due to normal random walk and Lazy PageRank due to lazy random walk on a graph. In this article, we establish how the two variants of PageRank changes when complete graphs are connected to a line of nodes whose links between the nodes are in one direction. Explicit formulae for the two variants of PageRank are presented. We have noted that the ranks on a line graph are the same except their numerical values which differ. Further, we have observed that both normal random walk and lazy random walk on complete graphs spend almost the same time at each node.
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| 5. |
- Lundengård, Karl, 1987-, et al.
(författare)
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Optimization of the Determinant of the Vandermonde Matrix and Related Matrices
- 2018
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Ingår i: Methodology and Computing in Applied Probability. - : Springer. - 1387-5841 .- 1573-7713. ; 20:4, s. 1417-1428
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Tidskriftsartikel (refereegranskat)abstract
- The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Grobner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.
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| 6. |
- Malyarenko, Anatoliy, 1957-, et al.
(författare)
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Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities
- 2018
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Ingår i: Proceedings : 5th Stochastic Modeling Techniques and Data Analysis International Conference withDemographics Workshop (SMTDA2018). - ISAST: International Society for the Advancement of Science and Technology. ; , s. 409-422
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Konferensbidrag (refereegranskat)abstract
- We consider a market model with four correlated factors and two stochastic volatilities, one of which is rapid-changing, while another one is slow-changing in time. An advanced Monte Carlo methods based on the theory of cubature in Wiener space, is used to find the no-arbitrage price of the European call option in the above model.
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| 7. |
- Muhumuza, Asaph Keikara, 1975-, et al.
(författare)
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The Generalized Vandermonde Interpolation Polynomial Based on Divided Differences
- 2018
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Konferensbidrag (refereegranskat)abstract
- In this article, we will construct the divided differences interpolation polynomial based on the generalized Vandermonde determinant approach. Some results regarding the appropriateness for this method for curve-fitting and approximation will be discussed. The proposed interpolation technique will be tested by construction of approximative models based on experimental data.
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| 8. |
- Silvestrov, Sergei, Professor, 1970-, et al.
(författare)
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Dmitrii S. Silvestrov
- 2018
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Ingår i: Stochastic Processes and Applications. - Cham : Springer. - 9783030028244 ; , s. 1-4
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Bokkapitel (refereegranskat)abstract
- This chapter presents short biographical notes about Professor Dmitri S. Silvestrov.
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| 9. |
- Silvestrov, Sergei, Professor, 1970-, et al.
(författare)
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Preface
- 2018
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Ingår i: Stochastic Processes and Applications. - : Springer. - 9783030028244 - 9783030028251 ; , s. vii-x
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Bokkapitel (övrigt vetenskapligt/konstnärligt)
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| 10. |
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