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Of all publications in the section: 10
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Article
Shnourkoff P. V., Novikov D. A. Working papers by Cornell University. Series math "arxiv.org". 2018. No. arXiv:1811.10993 [q-fin.GN]. P. 1-15.

The paper proposes a new stochastic intervention control model conducted in various commodity and stock markets. The essence of the phenomenon of intervention is described in accordance with current economic theory. A review of papers on intervention research has been made. A general construction of the stochastic intervention model was developed as a Markov process with discrete time, controlled at the time it hits the boundary of a given subset of a set of states. Thus, the problem of optimal control of interventions is reduced to a theoretical problem of control by the specified process or the problem of tuning. A general solution of the tuning problem for a model with discrete time is obtained. It is proved that the optimal control in such a problem is deterministic and is determined by the global maximum point of the function of two discrete variables, for which an explicit analytical representation is obtained. It is noted that the solution of the stochastic tuning problem can be used as a basis for solving control problems of various technical systems in which there is a need to maintain some main parameter in a given set of its values.

Added: Feb 26, 2019
Article
Dunin-Barkowski P., Popolitov A., Shadrin S. et al. Working papers by Cornell University. Series math "arxiv.org". 2017. Vol. 1712. No. 08614. P. 1-38.
We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mari\~no spectral curve for the colored HOMFLY-PT polynomials of torus knots.   This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data.
Added: Jan 2, 2018
Article
Gayfullin S., Gaifullin A. A. Working papers by Cornell University. Series math "arxiv.org". 2013.
Added: Nov 15, 2013
Article
Polyakov N. L. Working papers by Cornell University. Series math "arxiv.org". 2018. P. 1-22.

We propose a classification of symmetric conservative clones with a finite carrier. For the study, we use the functional Galois connection (Inv_Q,Pol_Q), which is a natural modification of the connection (Inv,Pol) based on the preservation relation between functions f on a set A (of all finite arities) and sets of functions hAQ for an arbitrary set Q.

Added: Oct 9, 2018
Article
Moreno-Franco H. A., Kelbert M. Working papers by Cornell University. Series math "arxiv.org". 2018.

We guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and an integro-differential operator whose Lévy measure has bounded variation. This type of equation arises in singular stochastic control problems where the state process is a jump-diffusion with infinite activity and finite variation jumps. By means of ε-penalized controls we show that the value function associated to this class of problems agrees with the solution to our HJB equation.

Added: Jan 25, 2017
Article
Nikolai L. Poliakov, Saveliev D. I. Working papers by Cornell University. Series math "arxiv.org". 2018. P. 1-46.

There exist two known canonical concepts of ultrafilter extensions of first-order models; one comes from modal logic and universal algebra, another one from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification; the ultrafilter extensions generalize this fact to discrete spaces endowed with an arbitrary first-order structure. Results of such type are referred to as extension theorems. We offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order models in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and an appropriate semantic for generalized models of this form. We establish necessary and sufficient conditions under which generalized models are the canonical ultrafilter extensions of some ordinary models, and provide their topological characterization. Then we provide even a wider concept of ultrafilter interpretations together with their semantic based on limits of ultrafilters, and show that it absorbs the former concept as well as the ordinary concept of models. We establish necessary and sufficient conditions under which generalized models in the wide sense are those in the narrow sense, or are the ultrafilter extensions of some ordinary models, and establish extension theorems.

Added: Feb 18, 2019
Article
P. V. Shnurkov. Working papers by Cornell University. Series math "arxiv.org". 2017. No. 1709.03442v1. P. 1-16.

In this paper, a general stochastic model with controls applied at the moments when the random process hits the boundary of a given subset of the state set is proposed and studied. The general concept of the model is formulated and its possible applications in technical and economic systems are described. Two versions of the general stochastic model, the version based on the use of a continuous-time semi-Markov process with embedded absorbing Markov chain and the version based on the use of a discrete-time Markov process with absorbing states, are analyzed. New representations of the stationary cost index of the control quality are obtained for both versions. It is shown that this index can be represented as a linear-fractional integral functional of two discrete probability distributions determining the control strategy. The results obtained by the author of this paper about an extremum of such functionals were used to prove that, in both versions of the model, the control is deterministic and is determined by the extremum points of functions of two discrete arguments for which the explicit analytic representations are obtained. The perspectives of the further development of this stochastic model are outlined.

Added: Dec 13, 2017
Article
Hernández-Hernández D., Pérez J., Moreno-Franco H. A. Working papers by Cornell University. Series math "arxiv.org". 2018.

In this work we study the optimal execution problem with multiplicative price impact in algorithm trading, when an agent holds an initial position of shares of a financial asset. The inter-selling-decision times are modelled by the arrival times of a Poisson process. The criterion to be optimised consists in maximising the expected net present value of gains of the agent, and it is proved that an optimal strategy has a barrier form, depending only on the number of shares left and the level of asset price.  

Added: May 2, 2017
Article
Atanov A., Volokhova A., Ashukha A. et al. Working papers by Cornell University. Series math "arxiv.org". 2019.
Added: Jul 11, 2019
Article
P. V. Shnurkov, K. A. Adamova. Working papers by Cornell University. Series math "arxiv.org". 2019. No. arXiv:1906.05824v1. P. 1-14.

The paper is devoted to the study of the unconditional extremal problem for a fractional linearintegral functional defined on a set of probability distributions. In contrast to results proved earlier,the integrands of the integral expressions in the numeratorand the denominator in the problem underconsideration depend on a real optimization parameter vector. Thus, the optimization problem isstudied on the Cartesian product of a set of probability distributions and a set of admissible values ofa real parameter vector. Three statements on the extremum ofa fractional linear integral functionalare proved. It is established that, in all the variants, the solution of the original problem is completelydetermined by the extremal properties of the test function of the linear-fractional integral functional;this function is the ratio of the integrands of the numeratorand the denominator. Possible applicationsof the results obtained to problems of optimal control of stochastic systems are described.

Added: Jun 17, 2019