Stochastic Control for Optimal Execution:Fast Approximation Solution Scheme Under Nested Mean-semi Deviation and Conditional Value at Risk

When executing a large order of stocks in a market,one important factor in forming the optimal trading strategy is to consider the price impact of large-volume trading activity.Minimizing a risk measure of the implementation shortfall,i.e.,the difference between the value of a trader’s initial equity position and the sum of cash flow he receives from his trading process,is essentially a stochastic control problem.In this study,we investigate such a practical problem under a dynamic coherent risk measure in a market in which the stock price dynamics has a feature of momentum effect.We develop a fast approximation solution scheme,which is critical in highfrequency trading.We demonstrate some prominent features of our derived solution algorithm in providing useful guidance for real implementation.

Skin formation in drying a film of soft matter solutions：Application of solute based Lagrangian scheme

When a film of soft matter solutions is being dried, a skin layer often forms at its surface, which is a gel-like elastic phase made of concentrated soft matter solutions. We study the dynamics of this process by using the solute based Lagrangian scheme which was proposed by us recently. In this scheme, the process of the gelation（i.e., the change from sol to gel） can be naturally incorporated in the diffusion equation. Effects of the elasticity of the skin phase, the evaporation rate of the solvents, and the initial concentration of the solutions are discussed. Moreover, the condition for the skin formation is provided.

REMARKS ON BOUNDS ON THE DISCREPANCY OF APPROXIMATE SOLUTIONS CONSTRUCTED BY GODUNOV'S SCHEME

The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.

An improved wind retrieval algorithm for the HY-2A scatterometer

Since January 2012,the National Satellite Ocean Application Service has released operational wind products from the HY-2A scatterometer(HY2-SCAT),using the maximum-likelihood estimation(MLE) method with a median filter. However,the quality of the winds retrieved from HY2-SCAT depends on the sub-satellite cross-track location,and poor azimuth separation in the nadir region causes particularly low-quality wind products in this region. However,an improved scheme,i.e.,a multiple solution scheme(MSS) with a two-dimensional variational analysis method(2DVAR),has been proposed by the Royal Netherlands Meteorological Institute to overcome such problems. The present study used the MSS in combination with a 2DVAR technique to retrieve wind data from HY2-SCAT observations. The parameter of the empirical probability function,used to indicate the probability of each ambiguous solution being the "true" wind,was estimated based on HY2-SCAT data,and the 2DVAR method used to remove ambiguity in the wind direction. A comparison between MSS and ECMWF winds showed larger deviations at both low wind speeds(below 4 m/s) and high wind speeds(above 17 m/s),whereas the wind direction exhibited lower bias and good stability,even at high wind speeds greater than 24 m/s. The two HY2-SCAT wind data sets,retrieved by the standard MLE and the MSS procedures were compared with buoy observations. The RMS error of wind speed and direction were 1.3 m/s and 17.4°,and 1.3 m/s and 24.0° for the MSS and MLE wind data,respectively,indicating that MSS wind data had better agreement with the buoy data. Furthermore,the distributions of wind fields for a case study of typhoon Soulik were compared,which showed that MSS winds were spatially more consistent and meteorologically better balanced than MLE winds.

Solving Different Types of Differential Equations Using Modified and New Modified Adomian Decomposition Methods

The Modified Adomian Decomposition Method (MADM) is presented. A number of problems are solved to show the efficiency of the method. Further, a new solution scheme for solving boundary value problems with Neumann conditions is proposed. The scheme is based on the modified Adomian decomposition method and the inverse linear operator theorem. Several differential equations with Neumann boundary conditions are solved to demonstrate the high accuracy and efficiency of the proposed scheme.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

Quasi-periodic Solutions for the Derivative Nonlinear Schrdinger Equation with Finitely Differentiable Nonlinearities

The authors are concerned with a class of derivative nonlinear Schr¨odinger equation iu_t + u_(xx) + i?f(u, ū, ωt)u_x=0,(t, x) ∈ R × [0, π],subject to Dirichlet boundary condition, where the nonlinearity f(z1, z2, ?) is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.