Weak Solutions of Semilinear PDEs in Sobolev Spaces and Their Probabilistic Interpretation via the FBSDEs

Abstract

The article is devoted to representation of weak solutions (in Sobolev sense) of degenerate parabolic partial differential equations through forward-backward stochastic differential equations. Before, we prove a weak version of a norm equivalence result.     

A Generalized Itô-Ventzell Formula to Derive Forward Utility Models in a Jump Market

Our main topic in this article is the forward utility field, which is a quite a new concept introduced by Musiela and Zariphopoulou. Different from most article in this field discussing forward utility in a continuous market, we extend this concept to jump market case. We first provide a generalized Itô-Ventzell formula, which can be applied in a general jump semimartingale driven by Brownian motion and Poisson random measure. Then three special forward utility models are discussed by exploiting this generalized Itô-Ventzell formula.     

Adaptive Mixture Modeling Metropolis Methods for Bayesian Analysis of Nonlinear State-Space Models

We describe a strategy for Markov chain Monte Carlo analysis of nonlinear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis–Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the nonlinearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.     

Conjugate Direction Boosting

Boosting in the context of linear regression has become more attractive with the invention of least angle regression (LARS), where the connection between the lasso and forward stagewise fitting (boosting) has been established. Earlier it has been found that boosting is a functional gradient optimization. Instead of the gradient, we propose a conjugate direction method (CDBoost). As a result, we obtain a fast forward stepwise variable selection algorithm. The conjugate direction of CDBoost is analogous to the constrained gradient in boosting. Using this analogy, we generalize CDBoost to: (1) include small step sizes (shrinkage) which often improves prediction accuracy; and (2) the nonparametric setting with fitting methods such as trees or splines, where least angle regression and the lasso seem to be unfeasible. The step size in CDBoost has a tendency to govern the degree between L₀- and L₁-penalization. This makes CDBoost surprisingly flexible. We compare the different methods on simulated and real datasets. CDBoost achieves the best predictions mainly in complicated settings with correlated covariates, where it is difficult to determine the contribution of a given covariate to the response. The gain of CDBoost over boosting is especially high in sparse cases with high signal to noise ratio and few effective covariates.     

二维稳态辐射传输方程的有限差分求解法

针对扩散光学层析在小动物成像中的应用问题并基于混浊介质空间光子三维散射的实际物理效应,提出的二维稳态辐射传输方程的有限差分数值求解新方法.在此基础上,研究了不同的空间剖分网格和角度离散密度对求解准确度的影响,并通过将所提方法与蒙特卡洛模拟进行比对,验证方法的正确性.研究表明：在均匀组织体内,当离散角度达到一定数量时,由辐射传输方程的有限差分解获得的透射面和侧面的光子密度对空间网格大小并不敏感,而在反射面上光子密度计算则需要较密的空间网格才能够达到一定准确度.本研究为发展基于辐射传输方程的扩散光学层析理论奠定了基础.     

Ultrasound-assisted forward osmosis desalination using inorganic draw solutes

Internal concentration polarization (ICP) represents a serious challenge in forward osmosis (FO) desalination since it causes a significant decline in the water flux across the membrane. Mitigation of ICP is cumbersome since the phenomenon occurs within the membrane porous support layer and mitigation procedures such as inducing turbulence or changing the hydrodynamic conditions tend to be ineffective. In this study, the effect of 40 kHz ultrasound on FO desalination of synthesized brackish and seawater was investigated. The studied process utilizes two different inorganic draw solutes (magnesium and copper sulfate) that are available commercially, can generate high osmotic pressures, and can be easily separated from the product water. Different concentrations of the draw solutions were considered. Results show that the applied ultrasound was effective in partially mitigating the ICP effects and enhancing the water flux. Depending on the feed and the draw solution concentration, flux enhancements of up to 34.6% and 43.9% were observed with magnesium sulfate and copper sulfate draw solution, respectively. In addition, it was observed that the effect of ultrasound on flux enhancement was more evident at lower draw solution concentrations. Although water flux was enhanced, ultrasound resulted in an increased reverse draw solute flux across the membrane.     

Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.     

近红外光谱法快速测定新疆薰衣草精油主要组分

为建立快速测定新疆薰衣草精油中芳樟醇(linalool)、乙酸芳樟酯(linalyl acetate)含量的定量分析模型,采用近红外吸收光谱法(NIR)测定了165个精油样品,通过对近红外光谱吸收峰分析,在7 100～4 500 cm-1波数范围内化学信息量比较丰富且噪音低,可选择此波数段为分析区间。剔除8个异常样本后,通过聚类方法划分为105个校正集样本和52个验证集样本,结合气相色谱质谱(GC-MS)法测定的薰衣草精油linalool和linalyl acetate的含量,建立原始数据矩阵。对比不同的预处理方法对原始近红外光谱的过滤作用,其中正交信号变换(OSC)方法效果明显,对两种化合物的预测均方根误差(RMSEP) 分别为0.226和0.558,再用向前间隔偏最小二乘法(FiPLS)剔除与待测成分无关或呈非线性关系的波长点,最终得到8个间隔区间共160个波长点的数据子集。使用OSC-FiPLS优化处理的数据子集结合偏最小二乘法(PLS)建立新疆薰衣草精油中linalool和linalyl acetate的快速定量分析模型,在模型中二成分的隐变量数都为8。交互验证均方根误差(RMSECV)分别为0.170和0.416;预测均方根误差(RMSEP)分别为0.188和0.364。结果表明,经OSC和FiPLS预处理建立的PLS-NIR定量分析模型稳健性好,测定精度高,能快速测定薰衣草精油中linalool和linalyl acetate含量,并且具有良好的预测能力。可为新疆薰衣草精油主要成分的快速定量分析提供一种新的有效方法。     

Estimating error bounds for quaternary subdivision schemes

This paper deals with the error estimation techniques of quaternary subdivision schemes. The estimation is expressed in terms of initial control point sequences and constants. It is independent of subdivision process and parametrization therefore its evaluation is straightforward.     

On Lipschitz solutions for some forward–backward parabolic equations

We investigate the existence and properties of Lipschitz solutions for some forward–backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two important cases of forward–backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona–Malik model in image processing and the other that of Höllig's model related to the Clausius–Duhem inequality in the second law of thermodynamics.