To charge or not to charge?

Discussions on the success or failure of OR practice tend to focus on specific case studies which often highlight major dilemmas that face OR professionals: Model robustness, simplicity vs complexity, optimising vs satisficing, conflict between criteria, who is the ‘client’?, and so on. But above all, the following issues predominate: Should OR be concerned with tactical or strategic problems? Where should it be placed in the organisational structure and who should it report to? These issues inevitably raise the questions of accountability, coupled with that of charging for OR services. It is this issue in particular, whether to charge or not to charge for OR services, that the paper seeks to explore.     

Set-Indexed Itô Calculus Along Paths

Abstract

Set-indexed stochastic analysis and set-indexed stochastic calculus are faced here with a new approach of dimension's reduction. We introduce a new tool (main flow) in order to deal with one-parameter calculus in set-indexed framework. We prove an Itô formula for any Brownian functional where the Brownian component is not a martingale on the whole set of indices but induces such a martingale. As first extensions, we provide definitions of bracket and local time in set-indexed context.     

To run or not?: Some dynamic programming models in cricket

In cricket, particularly near the end of an innings, batsmen of different abilities need to manage the rate at which they score runs. Either batsman can choose to bat aggressively or defensively, which alters their chances of scoring runs or being dismissed. Since they change ends when they score a run and at the end of an over, by scoring an odd or even number of runs the two batsmen also determine which of them will face the next ball. It may be worthwhile to refuse a run to keep the slower or lower scoring batsman from the strike. Some dynamic programming models are developed which could be used to maximise the total number of runs scored.     

Some Results of Backward Itô Formula

Abstract

We use the notion of backward integration, with respect to a general Lévy process, to treat, in a simpler and unifying way, various classical topics as: Girsanov theorem, first order partial differential equations, the Liouville (or Lyapunov) equations and the stochastic characteristic method.     

The Peano phenomenon for Itô equations

The weak convergence of the measures generated by the solutions of stochastic Itô equations with low diffusion is studied, as the diffusion tends to zero. It is proved that the limiting measure in the presence of the Peano phenomenon for a relevant ordinary differential equation is concentrated on its extreme solutions with definite weights. The formulas for their calculation are given.     

To lay out or not to lay out?

The Quadratic Assignment Problem (QAP) is known as one of the most difficult problems within combinatorial optimization. It is used to model many practical problems including different layout problems. The main topic of this paper is to provide methods to check whether a particular instance of the QAP is a layout problem. An instance is a layout problem if the distances of the objects can be reconstructed on the plane and/or in the 3-dimensional space. A new mixed integer programming model is suggested for the case if the distances of the objects are supposed to be rectilinear distances. If the distances are Euclidean distances then the use of the well-known Multi-Dimensional Scaling (MDS) method of statistics is suggested for reconstruction purposes. The well-known difficulty of QAP makes it a popular and suitable experimental field for many algorithmic ideas including artificial intelligence methods. These types of results are published sometimes as layout problems. The methods of reconstruction can be used to decide whether the topic of a paper is layout or only general QAP. The issue what the OR community should expect from AI based algorithms, is also addressed.     

Itô's formula for linear fractional PDEs

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itô-type formula for the process X.     

Itô maps and analysis on path spaces

We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross–Sobolev spaces of differentiable functions and proving their intertwining with solution maps, , of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given , is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.     

A Differentiation Theory for Itô's Calculus

Abstract

A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic variation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications, some of which we address in an upcoming article.     

Linear information for approximation of the Itô integrals

We study optimal approximation of stochastic integrals in the Itô sense when linear information, consisting of certain integrals of trajectories of Brownian motion, is available. Upper bounds on the nth minimal error, where n is the fixed cardinality of information, are obtained by the Wagner–Platen algorithm and are O(n ^???3/2) or O(n ^???2), depending on considered class of integrands. We also show that Ω(n ^???2) is a lower bound which holds even for very smooth integrands.     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

Diffusion maps tailored to arbitrary non-degenerate Itô processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.     

Change-point inference on volatility in noisy Itô semimartingales

This work is concerned with tests on structural breaks in the spot volatility process of a general Itô semimartingale based on discrete observations contaminated with i.i.d. microstructure noise. We construct a consistent test building up on infill asymptotic results for certain functionals of spectral spot volatility estimates. A weak limit theorem is established under the null hypothesis relying on extreme value theory. We prove consistency of the test and of an associated estimator for the change point. A simulation study illustrates the finite-sample performance of the method and efficiency gains compared to a skip-sampling approach.     

Solutions of nonlinear impulsive Itô functional-differential equations: Stability by the linear approximation

We study the p-stability (2 ≤ p < ∞) of solutions of nonlinear impulsive Itô functional-differential equations. To this end, we use the stability theory developed for deterministic functional-differential equations. The moment stability of solutions of nonlinear impulsive Itô functional-differential equations is studied with the use of the problem on the admissibility of a pair of spaces for linear impulsive Itô functional-differential equations. We prove assertions similar to traditional theorems on stability by the first approximation.