Least-Squares Approximation of Random Variables by Stochastic Integrals

This paper describes efficient algorithms for determining how buffer space should be allocated in a flow line. We analyze two problems: a primal problem, which minimizes total buffer space subject to a production rate constraint; and a dual problem, which maximizes production rate subject to a total buffer space constraint. The dual problem is solved by means of a gradient method, and the primal problem is solved using the dual solution. Numerical results are presented. Profit optimization problems are natural generalizations of the primal and dual problems, and we show how they can be solved using essentially the same algorithms.

In this paper, we consider the Lyapunov-type linear programming and its dual over symmetric cones. By introducing and characterizing the generalized inverse of Lyapunov operator in Euclidean Jordan algebras, we establish two kinds of Lyapunov-type Farkas’ lemmas to exhibit feasibilities of the corresponding primal and dual programming problems, respectively. As one of the main results, we show that the feasibilities of the primal and dual problems lead to the solvability of the primal problem and zero duality gap under some mild condition. In this case, we obtain that any solution to the pair of primal and dual problems is equivalent to the solution of the corresponding KKT system.

3 - Further Topics in Linear Programming

IG-OU processes are a subclass of the non-Gaussian processes of Ornstein–Uhlenbeck type, which are important models appearing in financial mathematics and elsewhere. The simulation of these processes is of interest for its applications in statistical inference. In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables—one has an inverse Gaussian distribution and the other has a compound Poisson distribution. And in distribution, the compound Poisson random variable is equal to a sum of Poisson-distributed number positive random variables, which are independent identically distributed and have a common specified density function. The exact simulation of the IG-OU processes, proceeding from time 0 and going in steps of time interval Δ, is achieved via the representation of the stochastic integral. Comparing to the approximate method, which is based on Rosinski’s infinite series representation of the same stochastic integral, by the quantile–quantile plots, the advantage of the exact simulation method is obvious. In addition, as an application, we provide an estimator of the intensity parameter of the IG-OU processes and validate its superiority to another estimator by our exact simulation method.

In this paper we study the problem of minimizing total weighted tardiness, a proxy for maximizing on-time delivery performance, on parallel nonidentical batch processing machines. We first formulate the (primal) problem as a nonlinear integer programming model. We then show that the primal problem can be solved exactly by solving a corresponding dual problem with a nonlinear relaxation. Since both the primal and the dual problems are NP-hard, we use genetic algorithms, based on random keys and multiple choice encodings, to heuristically solve them. We find that the genetic algorithms consistently outperform a standard mathematical programming package in terms of solution quality and computation time. We also compare the smaller problem instances to a breadth-first tree search algorithm that gives evidence of the quality of the solutions.

A neural net algorithm for multidimensional maximum entropy spectrum estimation

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter states the Monge–Kantorovich problem and provides the duality result in a fairly general setting. The primal problem is interpreted as the central planner's problem of determining the optimal assignment of workers to firms, while the dual problem is interpreted as the invisible hand's problem of obtaining a system of decentralized equilibrium prices. In general, the primal problem always has a solution (which means that an optimal assignment of workers to jobs exists), but the dual does not: the optimal assignment cannot always be decentralized by a system of prices. However, the cases where the dual problem does not have a solution are rather pathological, and in all of the cases considered in the rest of the book, both the primal and the dual problems have solutions.

This paper investigates whether Standards and Poors’ 100 stock index (henceforth, S&P100) is a mean-variance efficient portfolio. In other words, the paper examines if there is any other portfolio allocation rule that provides better risk-return scenario than one that is achieved under value-weighted strategy used in S&P100 index. Our results show that the realized risk of the S&P100 index is not the lowest for the return at any other portfolio combinations using all stocks in index. There are other portfolio combinations that provide lower risk for the same realized rate of return. Similarly, the realized return of S&P100 index is not the highest for the standard deviations at any other portfolio combinations. Based on these results, we conclude that the S&P100 index may not be considered to be a mean-variance efficient portfolio.

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. This section presents the formulation of the primal problem, the definition and properties of the perturbation function, the definition of stable primal problem, and the existence conditions of optimal multiplier vectors.

Abstract. In this paper, some omissions in Mishra and Lai [13], havebeen pointed out and their corrective measures have been discussed briefly. 1. IntroductionA pair of primal and dual problems in mathematical programming is calledsymmetric if the dual of the dual is the primal problem. Dorn [6] introducedthe concept of symmetric duality in quadratic programming. His results wereextended to nonlinear convex programming problems by Dantzig et al. [4] andlater by Bazaraa and Goode [3] over arbitrary cones.Mangasarian [11] introduced the concept of second-order duality for non-linear problems. Since then, many authors [1, 2, 7, 9, 15, 16] have workedon second-order symmetric duality. Mishra and Lai [13] studied Mond-Weirtype second-order multiobjective symmetric duality for the following pair ofproblems:(P) K−minimize f(x,y)−12p T ∇ yy f(x,y)psubject to −∇ y (λ T f)(x,y) −∇ yy (λ T f)(x,y) ∈ C ∗2 ,y T [∇ y (λ T f)(x,y) +∇ yy (λ T f)(x,y)] > 0,λ ∈ K ∗ , x ∈ C 1 .(D) K−maximize f(u,v)−12q

We show first how the joint semi-quantum and quantum operators of a finite family of random variables having finite moments of all orders can be recovered from their joint number operator. We then characterize the polynomially symmetric and polynomially factorizable random variables in terms of their joint number operator. Finally, we present the quantum decomposition of the number and quantum operators of the random variables whose orthogonal polynomials are the Gegenbauer polynomials.

A novel portfolio selection model in a hybrid uncertain environment

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196---1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.