Alexei Chekhlov

This book describes recently developed mathematical models, methodologies, and case studies in diverse areas, including stock market analysis, portfolio optimization, classification techniques in economics, supply chain optimization, development of e-commerce applications, etc. It will be of interest to both theoreticians and practitioners working in economics and finance.

This thesis focuses upon the large-scale and long -time statistical properties of several forced-dissipative fluid dynamical systems, including: (i) the one-dimensional Burgers equation, (ii) a model one-dimensional equation without Galilean invariance, and the two-dimensional Navier -Stokes system with (vi) and without (iii) effects of differential rotation in the beta-plane approximation. It is shown that a certain large-scale forcing in the Burgers equation results in statistical properties which are remarkably close to those of three-dimensional fully developed turbulence. The corresponding probability distribution function of velocity differences possesses nontrivial algebraic tails due to the effects of shock waves, thus leading to a biscaling behavior of the velocity structure functions. A phenomenological theory describing the experimental findings is proposed. Experimental results are compared with predictions of the one-loop renormaliaed perturbation expansion. It is demonstrated that cubic non-linearity in a one-dimensional Burgers-like system which violates Galilean invariance allows efficient analytical treatment using the Renormalization Group (RG) and the epsilon-expansion methods, unlike its Burgers counterpart. The corresponding fixed-point critical behavior is studied in detail using a finite-step RG transformation. For the two-dimensional Navier-Stokes system, it is shown that a two-parametric eddy viscosity in the inverse energy transfer regime is in excellent agreement with predictions based upon the RG theory, as well as other closure models. This result yields a new strategy of large -eddy simulation of two-dimensional turbulent flows which was successfully tested in a wide range of flow parameters. Effects of differential rotation are shown to strongly alter the large-scale properties of forced two-dimensional turbulence. The directional energy spectrum at very long times is found to be essentially anisotropic with two scaling laws; one similar to Kolmogorov k^{-5/3} -law and the other to Rhines k^{ -5}-law. Practical applications of our results are discussed.

Numerical simulation results of basic exactly solvable fluid flows using the previously proposed by H. Chen Lattice Boltzmann Method (LBM) formulated on a general curvilinear coordinate system are presented. As was noted in the theoretical work of H. Chen, such curvilinear Lattice Boltzmann Method preserves a fundamental one-to-one exact advection feature in producing minimal numerical diffusion, as the Cartesian lattice Boltzmann model. As we numerically show, the new model converges to exact solutions of basic fluid flows with the increase of grid resolution in the presence of both natural curvilinear geometry and/or grid non-uniform contraction, both for near equilibrium and non-equilibrium LBM parameter choices.

Two-parametric eddy viscosity (TPEV) and other spectral characteristics of two-dimensional (2D) turbulence in the energy transfer sub-range are calculated from direct numerical simulation (DNS) with 512 2 resolution. The DNS-based TPEV is compared with those calculated from the test field model (TFM) and from the renormalization group (RG) theory. Very good agree

A new one-parameter family of risk measures called Conditional Drawdown (CDD) has been proposed. These measures of risk are functionals of the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter α, in the case of a single sample path, drawdown functional is defined as the mean of the worst (1 - α) * 100% drawdowns. The CDD measure generalizes the notion of the drawdown functional to a multi-scenario case and can be considered as a generalization of deviation measure to a dynamic case. The CDD measure includes the Maximal Drawdown and Average Drawdown as its limiting cases. Mathematical properties of the CDD measure have been studied and efficient optimization techniques for CDD computation and solving asset-allocation problems with a CDD measure have been developed. The CDD family of risk functionals is similar to Conditional Value-at-Risk (CVaR), which is also called Mean Shortfall, Mean Excess Loss, or Tail Val...