不等式的序轴标根法在微积分上的应用

不等式的序轴标根法,蕴涵着丰富的数学知识、逻辑推理和技巧,应用比较广泛。本文在不等式的序轴标根法的基本原理的基础上,讨论了其在微积分上的应用。     

水电站调节保证计算中导叶关闭规律探讨

调节保证计算是水电站设计中的重要内容之一。水电站运行的过程中,为了最大限度的减少水压上升值与转速上升值,通过优化导叶折线关闭方式来解决过渡过程中水电站的运行安全问题是最经济的有效措施,而且技术上也容易实现。目前,普遍采用的两段或三段关闭规律。但是,理论上来讲,不管是两段还是三段关闭规律,都不是最优关闭规律。笔者从微积分的思想出发,提出了根据水轮机工作参数的变化实时地改变导叶关闭方式的非固定模式的导叶关闭方式,能更有效地提高水电站运行的稳定性和安全性。     

PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest,   r > 0  , and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark–Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach.     

Elliptical tempered stable distribution

Elliptical distributions are useful for modelling multivariate data, multivariate normal and Student t distributions being two special classes. In this paper, we provide a definition for the elliptical tempered stable (ETS) distribution based on its characteristic function, which involves a unique spectral measure. This definition provides a framework for creating a connection between the infinite divisible distribution (in particular the ETS distribution) with fractional calculus. In addition, a definition for the ETS copula is discussed. A simulation study shows the accuracy of this definition, in comparison to the normal copula for measuring the dependency of data. An empirical study of stock market index returns for 20 countries shows the usefulness of the theoretical results.     

First-Order Schemes in the Numerical Quantization Method

The numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first-order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are proposed.     

A note on the wise girls puzzle

Summary. This article analyzes the two wise girls puzzle, which is a simpler variant of the so-called three wise men puzzle, with some proof-theoretic tools. We formulate the puzzle in an epistemic logic. Our chief assumption is that the reasoning ability of each player of the puzzle is equivalent to what is described by the epistemic logic. We will interpret the behaviors of the players in the puzzle in terms of unprovability of certain statements. The proof-theoretic tools we employ are consequences of a meta-theorem, known as the cut elimination theorem. Received: August 1, 2000; revised version: May 15, 2001     

Equilibrium Models With Singular Asset Prices

General equilibrium models in which economic agents have finite marginal utility from consumption at the origin lead to financial assets having continuous prices with singular components. In particular, there is no bona fide "interest rate" in such models, although asset prices can be determined by equilibrium considerations (and uniquely, up to the formation of mutual funds). the singularly continuous processes in question charge precisely the set of time points at which some agent "drops out" of the economy, or "comes back" into it, between intervals of zero consumption. Not surprisingly, these processes are governed by local time.     

Optimal Malliavin Weighting Function for the Computation of the Greeks

This paper reexamines the Malliavin weighting functions introduced by Fournié et al. (1999) as a new method for efficient and fast computations of the Greeks. Reexpressing the weighting function generator in terms of its Skorohod integrand, we show that these weighting functions have to satisfy necessary and sufficient conditions expressed as conditional expectations. We then derive the weighting function with the smallest total variance. This is of particular interest as it bridges the method of Malliavin weights and the one of likelihood ratio, as introduced by Broadie and Glasserman (1996) . The likelihood ratio is precisely the weighting function with the smallest total variance. We finally examine when to use the Malliavin method and when to prefer finite difference.     

Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999) , as well as Ma and Zhang (2000) , using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.