Explicit Martingale Representations for Brownian Functionals and Applications to Option Hedging

Abstract

Using Clark-Ocone formula, explicit martingale representations for path-dependent Brownian functionals are computed. As direct consequences, explicit martingale representations of the extrema of geometric Brownian motion and explicit hedging portfolios of path-dependent options are obtained.     

Volatility Targeting Using Delayed Diffusions

ABSTRACT

A target volatility strategy (TVS) is a risky asset-riskless bond dynamic portfolio allocation which makes use of the risky asset historical volatility as an allocation rule with the aim of maintaining the instantaneous volatility of the investment constant at a target level. In a market with stochastic volatility, we consider a diffusion model for the value of a target volatility fund (TVF) which employs a system of stochastic delayed differential equations (SDDEs) involving the asset realized variance. First we prove that under some technical assumptions, contingent claim valuation on a TVF is approximately of Black-Scholes type, which is consistent with and supports the standing market practice. In second place, we develop a computational framework using recent results on Markovian approximations of SDDEs systems, which we then implement in the Heston variance model using an ad hoc Euler scheme. Our framework allows for efficient numerical valuation of derivatives on TVFs, whose typical purpose is the assessment of the guarantee costs of such funds for insurers.     

A complete representation theorem for G-martingales

In this article, we establish a complete representation theorem for G-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second-order term, which corresponds to the second-order derivative in Markovian case. The main ingredient of the article is a new norm for that second-order term, which is based on an operator introduced by Song.     

Topological centers of module actions induced by unitary representations

In this paper, we study the topological centers of bilinear maps induced by unitary representations, giving a characterization when the center is minimal and also conditions which guarantee that the center is maximal. Various examples whose topological centers are maximal, minimal or neither will be given. We will also investigate the topological centers of sub-representations, direct sums and tensor products.     

Generalized semi-inner products with applications to regularized learning

We propose a definition of generalized semi-inner products (g.s.i.p.). By relating them to duality mappings from a normed vector space to its dual space, a characterization for all g.s.i.p. satisfying this definition is obtained. We then study the Riesz representation of continuous linear functionals via g.s.i.p. As applications, we establish a representer theorem and characterization equation for the minimizer of a regularized learning from finite or infinite samples in Banach spaces of functions.     

Iterative solutions to the extended Sylvester-conjugate matrix equations

This paper is concerned with iterative solutions to a class of complex matrix equations. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of complex matrix equations. The range of the convergence factor is given to guarantee that the proposed algorithm is convergent for arbitrary initial matrix by applying a real representation of a complex matrix as a tool. By using some properties of the real representation, a sufficient convergence condition that is easier to compute is also given by original coefficient matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

由吴方法计算零维系统的有理单元表示

本文提出一个计算零维系统的有理单元表示的新算法.无需进行Grbner基运算,我们的算法仅运用了著名的吴方法.基于吴方法,我们的算法在Maple平台上被编制成一个通用程序RUR-Wu,可快速地计算出零维系统的有理单元表示.作为一个应用,本文提出了一个有效方法,用来计算某些多项式的整体最小值.此外,本文给出了几个实例,用来表明算法的效率.     

H~2×R中给定平均曲率曲面的Weierstrass表示

定义了H~2×R中曲面的法高斯映照,给出了给定平均曲率曲面的Weierstrass表示,证明了法高斯映照满足一个二阶偏微分方程,并且该方程是所得到的Weierstrass表示的完全可积条件.     

High dimensional model representation for stochastic finite element analysis

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.