Optimal Order Convergence Implies Numerical Smoothness

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability

Optimal Order Scheduling for Deterministic Liquidity Patterns

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. The broker's aim is thus to minimize execution costs he incurs from the adverse impact of his trades on market prices. By contrast to the previous literature, see, e.g., Obizhaeva and Wang (2005), Predoiu, Shaikhet, and Shreve (2011), we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis and, under minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience adjusted market depth

Numerical methods for an optimal order execution problem

This paper deals with numerical solutions to an impulse control problem arising from optimal portfolio liquidation with bid-ask spread and market price impact penalizing speedy execution trades. The corresponding dynamic programming (DP) equation is a quasi-variational inequality (QVI) with solvency constraint satisfied by the value function in the sense of constrained viscosity solutions. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the DPQVI. The convergence of this discrete-time scheme is shown by viscosity solutions arguments. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Numerical results are presented by examining the behaviour of optimal liquidation strategies, and comparative performance analysis with respect to some benchmark execution strategies. We also illustrate our optimal liquidation algorithm on real data, and observe various interesting patterns of order execution strategies. Finally, we provide some numerical tests of sensitivity with respect to the bid/ask spread and market impact parameters

Optimal-order isogeometric collocation at Galerkin superconvergent points

In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in [1] and the variational collocation method presented in [2]. The focus is on smoothest B-splines/NURBS approximations, i.e, having global $C^{p-1}$ continuity for polynomial degree $p$. Within the framework of [2], we select as collocation points a subset of those considered in [1], which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behaviour with respect to [2], achieving optimal $L^2$-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in [1], where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.Comment: 21 pages, 20 figures (35 pdf images

Optimal order quantities with volume discounts before and after price increase

An inventory problem in which annual demand is normally distributed with known means and standard deviations is considered. A purchase price increase is imminent before the next order is placed. Volume discounts are also given in accordance to the size of the order. A model to compute an optimal order quantity and an optimal delivery point is presented. This model can also account for any price change that may occur from time to time

Extracting Several Resource Deposits of Unknown Size: Optimal Order

Oil companies often announce revised estimates of their reserves. This indicates that stock uncertainty is a prevalent feature of natural resource industries. In this paper we consider the multi-deposit case where resource extraction produces information about the size of reserves. We show that the optimal order of extracting resource deposits depends both on the informational characteristics of the extraction process and on the extraction costs. Differences in extraction costs, a key consideration highlighted in Solow and Wan (1976), must be balanced against the relative value of information generated by the extraction of various deposits. Our model supplies an explanation of why high cost deposits are sometimes extracted when lower cost deposits have not been exhausted. Les compagnies pétrolières révisent souvent les chiffres de leurs réserves, ce qui indique que l’incertitude concernant les stocks est prévalente. Nous considérons le cas où l’extraction donne des informations sur la taille des réserves. Nous prouvons que l’ordre optimal d’exploitation des stocks dépend des propriétés du processus d’extraction concernant la révélation d’information et des coûts. La différence des coûts, qui est une considération importante dans Solow and Wan (1976), doit être balancée contre la valeur informative des réserves. Notre modèle fournit une explication du fait que les réserves plus coûteuses sont parfois exploitées avant l’épuisement des réserves moins coûteuses.order of extraction, value of information, uncertainty, ordre d’extraction, valeur de l’information, incertitude

Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework

Assuming geometric Brownian motion as unaffected price process $S^0$, Gatheral & Schied (2011) derived a strategy for optimal order execution that reacts in a sensible manner on market changes but can still be computed in closed form. Here we will investigate the robustness of this strategy with respect to misspecification of the law of $S^0$. We prove the surprising result that the strategy remains optimal whenever $S^0$ is a square-integrable martingale. We then analyze the optimization criterion of Gatheral & Schied (2011) in the case in which $S^0$ is any square-integrable semimartingale and we give a closed-form solution to this problem. As a corollary, we find an explicit solution to the problem of minimizing the expected liquidation costs when the unaffected price process is a square-integrable semimartingale. The solutions to our problems are found by stochastically solving a finite-fuel control problem without assumptions of Markovianity

Formation of optimal-order necklace modes in one-dimensional random photonic superlattices

We study the appearance of resonantly coupled optical modes, optical necklaces, in Anderson localized one-dimensional random superlattices through numerical calculations of the accumulated phase. The evolution of the optimal necklace order m* shows a gradual shift towards higher orders with increasing the sample size. We derive an empirical formula that predicts m* and discuss the situation when in a sample length L the number of degenerate in energy resonances exceeds the optimal one. We show how the \emph{extra} resonances are pushed out to the miniband edges of the necklace, thus reducing the order of the latter by multiples of two.Comment: 4 pages, 4 figure