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## Теоретико-игровая динамическая модель инсайдерских торгов с ненулевым спрэдом

The model of multistage insider trading between two market agents for one-type risky assets is considered. One of the players (insider) has private information about liquidation value of the asset. At each step of the bidding each player simultaneously proposes bid and ask prices for one share with fixed non-zero spread. The uninformed player uses the history of insider's moves to update his beliefs. For the bidding of unlimited duration we construct upper and lower bounds of the guaranteed insider's gain and the strategies of both players insuring these bounds. Insider's loses in the case of disclosure his private information are obtained.

We consider a discrete model of insider trading in terms of repeated games with incomplete information. The solution of the bidding game of beforehand unlimited duration was obtained by V. Domansky (2007). Insider's optimal strategy in the infinite stage game generates the simple random walk of posterior probabilities over the lattice l/m, l=0,...,m with absorption at the extreme points 0 and 1 and provides the expected gain 1/2 per step to insider. In this paper we calculate insider's profit in the game of any finite duration when he applies the strategy above. It is shown that this strategy is his epsilon-optimal strategy in n-stage game, where epsilon decreases exponentially. This means that the sequence of n-stage game values converges to the value of infinite game at least exponentially. The result obtained is interpreted as the loss of insider in the case of sudden disclosure of his private information. For the special case we compare obtained insider's profit with the exact game value (result of V. Kreps, 2009) and demonstrate that error term in the case of optimal insider's behaviour also decreases exponentially.

This article presents an engineering approach to estimating market resiliency based on analysis of the dynamics of a liquidity index. The method provides formal criteria for defining a “liquidity shock” on the market and can be used to obtain resiliency-related statistics for further research and estimation of this liquidity aspect. The developed algorithm uses the results of a spline approximation for observational data and allows a theoretical interpretation of the results. The method was applied to real data resulting in estimation of market resiliency for the given period.

Repeated bidding games were introduced by De Meyer and Saley (2002) to analyze the evolution of the price system at finance markets with asymmetric information. In the paper of De Meyer and Saley arbitrary bids are allowed. It is more realistic to assume that players may assign only discrete bids proportional to a minimal currency unit. This paper represents a survey of author's results on discrete bidding games with asymmetric information.

The article addresses the major problems of application of Russian laws regulating the insider trading issues. Analysis of the court and enforcement practice of application of the mentioned law shows that in practice it is very hard or almost not feasible to bring the insider to the liability for the insider trading. That can be explained by lack of effective legal mechanism needed both for the investigation of the “insider trading” cases carried out by the competent state authorities and for bringing insiders to respective liability.

The paper considers a game-theoretical model of bidding with asymmetric information. One player has the inside information on the liquidation price of risky asset. The model is formalized with the repeated game with incomplete information on the side of uninformed player. We consider the case of external stopping of the game at the random moment. Insider's expected profit in the game of random duration if she applies the strategy optimal in infinite-stage game is obtained. This result allows to calculate the loss of insider in case of sudden disclosure of his private information.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

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.