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## Regression-based complexity reduction of the nested Monte Carlo methods

In this paper we propose a novel dual regression-based approach for pricing American options. This approach reduces the complexity of the nested Monte Carlo method and has especially simple form for time discretized diffusion processes. We analyse the complexity of the proposed approach both in the case of fixed and increasing number of exercise dates. The method is illustrated by several numerical examples.

Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradient estimates. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. In particular, the control variates allow us to optimize black-box functions over permutations using stochastic gradient descent. To illustrate the approach, we consider a variety of causal structure learning tasks for continuous and discrete data. We show that our method outperforms competitive relaxation-based optimization methods and is also applicable to non-differentiable score functions.

The Binder cumulant at the phase transition of Ising models on square lattices with ferromagnetic couplings between nearest neighbors and with competing antiferromagnetic couplings between next-nearest neighbors, along only one diagonal, is determined using Monte Carlo techniques. In the phase diagram a disorder line occurs separating regions with monotonically decaying and with oscillatory spin-spin correlations. Findings on the variation of the critical cumulant with the ratio of the two interaction strengths are compared to related recent results based on renormalization-group calculations.

We present a comparative study of several algorithms for an in-plane random walk with a variable step. The goal is to check the efficiency of the algorithm in case where the random walk terminates at some boundary. We recently found that a finite step of the random walk produces a bias in the hitting probability and this bias vanishes in the limit of an infinitesimal step. Therefore, it is important to know how a change in the step size of the random walk influences the performance of simulations. We propose an algorithm with the most effective procedure for the step-length-change protocol.

*Russian option market (presented by the only segment of Russian Trading System called Futures and Options on RTS — FORTS) has yet a very short history and considered underdeveloped. Intuitively the market seems ineffective in the sense that option prices allow long running arbitrage with no demand reaction leading to price adjustment as could be expected. In this paper basing on risk-neutral pricing we propose a method to calculate options fair prices and show the degree of market effectiveness in the sense of whether the arbitrage opportunities tend to drive market to an arbitrage-free equilibrium or not. The dynamics of underlying assets log returns is described as infinitely divisible Levy processes and mean-correcting Monte-Carlo simulation of risk-neutral market trajectories is applied to calculate option fair prices. Results show systematical ineffectiveness of FORTS.**Russian option market (presented by the only segment of Russian Trading System called Futures and Options on RTS — FORTS) has yet a very short history and considered underdeveloped. Intuitively the market seems ineffective in the sense that option prices allow long running arbitrage with no demand reaction leading to price adjustment as could be expected. In this paper basing on risk-neutral pricing we propose a method to calculate options fair prices and show the degree of market effectiveness in the sense of whether the arbitrage opportunities tend to drive market to an arbitrage-free equilibrium or not. The dynamics of underlying assets log returns is described as infinitely divisible Levy processes and mean-correcting Monte-Carlo simulation of risk-neutral market trajectories is applied to calculate option fair prices. Results show systematical ineffectiveness of FORTS.*

Risk Management approach is an essential part of the project. Large industries and particular companies incorporate RM Culture. Statistics shows, that companies with Project Management (PM) Structure reduce cost ineffectiveness up to 20%. In oil and gas industry PM Risk Analysis (PRMA) has been widely used for the last years. Various models and procedures have been developed to manage projects of different scale. Nonetheless, Offshore Projects (OP) complexity, high uncertainty of technical, financial, market and government factors, as well as different sea conditions, still makes sense to improve general PRMA models according to the oil and gas OP features. Traditional RM tools and techniques are not appropriate to cope with complex projects in the Arctic. Companies will have to modify risk assessment process or look for new methods. The paper suggests OPRMM, where the attempt to implement PM tools and techniques together with mathematical modeling and expert assessment is made and institutional factors are included. Practically, it is founded on the comparison between offshore field development in the Barents Sea and the Kara Sea. The reason for research is debates around future Arctic oil and gas projects and their commercial potential. Several large projects with participation of major international companies in the Barents Sea and the prospectivity of the Kara Sea Projects in conditions of technology difficulties are under discussion and have not reached the investment project phase yet. OPRMM starts with identifying the key factors, which could affect offshore field development. Inside the investment regime modified real option value (ROV) model for OP is developed: stop option and scale transformation option. Basing on the binominal trees and Monte Carlo Simulation it is possible to see the perspectives of the OP at an early stage in the conditions of high uncertainty. Incorporating the ROV model into investment regime allows operator to choose the territory to explore. The research shows, that offshore projects in the Arctic offshore is not only under the pressure of internal corporative factors, but also under influence of external institutional factors. New tools and approaches will be required in Arctic projects where no one wants to be looking in the wrong place.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.