Abstract:
The brief introduction examines a problem in stochastic analysis identified by A. N. Shiryaev, which led the author of this paper to two research directions. The first direction is the study of the properties of martingale spaces close to $L_1$ or to $L_\infty$. For martingale spaces L_p̄ with mixed norms (where the components of an infinite-dimensional vector $\bar{p}$ tend to $1$), a generalization of Pelczynski's well-known theorem on the absence of an unconditional basis in this space will be given. A partial solution to Professor E. M. Semenov's problem on the coincidence of $L_{\bar{p}}$ with $L_\infty$ will also be presented. The second direction will present results on the interpolation of arbitrage financial markets. It will be shown how this technique is related to Haar interpolation of signed martingale measures. The talk will present the author's results from his two years at MSU.
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