Alexei Chekhlov

Drawdown Measure in Portfolio Optimization

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 Value-at-Risk. Some recommendations on how to select the optimal risk functionals for getting practically stable portfolios have been provided. A real-life asset-allocation problem has been solved using the proposed measures. For this particular example, the optimal portfolios for cases of Maximal Drawdown, Average Drawdown, and several intermediate cases between these two have been found.

The effect of small-scale forcing on large-scale structures in two-dimensional flows

Abstract The effect of small-scale forcing on large-scale structures in β-plane two-dimensional (2D) turbulence is studied using long-term direct numerical simulations (DNS). We find that nonlinear effects remain strong at all times and for all scales and establish an inverse energy cascade that extends to the largest scales available in the system. The large-scale flow develops strong spectral anisotropy: k − 5 3 Kolmogorov scaling holds for almost all φ, φ = arctan ( k y k x ) except in the small vicinity of kx = 0, where Rhiness k−5 scaling prevails. Due to the k−5 scaling, the spectral evolution of β-plane turbulence becomes extremely slow which, perhaps, explains why this scaling law has never before been observed in DNS. Simulations with different values of β indicate that the β-effect diminishes at small scales where the flow is nearly isotropic. Thus, for simulations of β-plane turbulence forced at small scales sufficiently removed from the scales where β-effect is strong, large eddy simulation (LES) can be used. A subgrid scale (SGS) parameterization for such LES must account for the small-scale forcing that is not explicitly resolved and correctly accommodate two inviscid conservation laws, viz. energy and enstrophy. This requirement gives rise to a new anisotropic stabilized negative viscosity (SNV) SGS representation which is discussed in the context of LES of isotropic 2D turbulence.

Large scale drag representation in simulations of two-dimensional turbulence

Numerical simulations of isotropic, homogeneous, forced and dissipative two-dimensional (2D) turbulence in the energy transfer subrange are complicated by the inverse cascade that continuously propagates energy to the large scale modes. To avoid energy condensation in the lowest modes, an energy sink, or a large scale drag is usually introduced. With a few exceptions, simulations with different formulations of the large scale drag reveal the development of strong coherent vortices and steepening of energy and enstrophy spectra that lead to erosion and eventual destruction of Kolmogorov–Batchelor–Kraichnan (KBK) statistical laws. Being attributed to the intrinsic anomalous fluctuations independent of the large scale drag formulation, these coherent vortices have prompted conjectures that KBK 2D turbulence in the energy subrange is irreproducible in long term simulations. Here, we advance a different point of view, according to which the emergence of coherent vortices is triggered by the inverse energy casc...

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlovet al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplified SGS representation was designed, referred to as the stabilized negative viscosity (SNV) representation, which was based on two algebraic terms only, negative Laplacian and positive biharmonic ones. It was found that the SNV scheme performed in a fashion very similar to the full equation and it was argued that this scheme and its derivatives should be applied for SGS representation in LES of quasi-2D flows.

Direct numerical simulation tests of eddy viscosity in two dimensions

Two‐parametric eddy viscosity (TPEV) and other spectral characteristics of two‐dimensional (2‐D) turbulence in the energy transfer subrange are calculated from direct numerical simulation (DNS) with 5122 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 agreement between all three results is observed.