The continuous-time pre-commitment KMM problem in incomplete markets

This paper studies the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (2005) in incomplete financial markets, which concerns with the portfolio selection under smooth ambiguity. The decision maker (DM) is uncertain about the dominated priors of the financial market, which are characterized by a second-order distribution (SOD). The KMM model separates risk attitudes and ambiguity attitudes apart and the aim of the DM is to maximize the two-fold utility of terminal wealth, which does not belong to the classical subjective utility maximization problem. By constructing the efficient frontier, the original KMM problem is first simplified as an one-fold expected utility problem on the second-order space. In order to solve the equivalent simplified problem, this paper imposes an assumption and introduces a new distorted Legendre transformation to establish the bipolar relation and the distorted duality theorem. Then, under a further assumption that the asymptotic elasticity of the ambiguous attitude is less than 1, the uniqueness and existence of the solution to the KMM problem are shown and we obtain the semi-explicit forms of the optimal terminal wealth and the optimal strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black-Scholes financial market, which show that DM with higher ambiguity aversion tends to be more concerned about extreme market conditions with larger bias. In the end of this work, numerical comparisons with the DMs ignoring ambiguity are revealed to illustrate the effects of ambiguity on the optimal strategies and value functions.Comment: 53 pages, 7 figure

Generalized Sparse Bayesian Learning and Application to Image Reconstruction

Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are tailored to specific problems, with particular forward models and priors. Here, we present a generalized approach to sparse Bayesian learning. It has the advantage that it can be used for various types of data acquisitions and prior information. Some preliminary results on image reconstruction/recovery indicate its potential use for denoising, deblurring, and magnetic resonance imaging

Fast Multiscale Functional Estimation in Optimal EMG Placement for Robotic Prosthesis Controllers

Electrocardiogram (EMG) signals play a significant role in decoding muscle contraction information for robotic hand prosthesis controllers. Widely applied decoders require large amount of EMG signals sensors, resulting in complicated calculations and unsatisfactory predictions. By the biomechanical process of single degree-of-freedom human hand movements, only several EMG signals are essential for accurate predictions. Recently, a novel predictor of hand movements adopts a multistage Sequential, Adaptive Functional Estimation (SAFE) method based on historical Functional Linear Model (FLM) to select important EMG signals and provide precise projections. However, SAFE repeatedly performs matrix-vector multiplications with a dense representation matrix of the integral operator for the FLM, which is computational expansive. Noting that with a properly chosen basis, the representation of the integral operator concentrates on a few bands of the basis, the goal of this study is to develop a fast Multiscale SAFE (MSAFE) method aiming at reducing computational costs while preserving (or even improving) the accuracy of the original SAFE method. Specifically, a multiscale piecewise polynomial basis is adopted to discretize the integral operator for the FLM, resulting in an approximately sparse representation matrix, and then the matrix is truncated to a sparse one. This approach not only accelerates computations but also improves robustness against noises. When applied to real hand movement data, MSAFE saves 85%$\sim$90% computing time compared with SAFE, while producing better sensor selection and comparable accuracy. In a simulation study, MSAFE shows stronger stability in sensor selection and prediction accuracy against correlated noise than SAFE

Reproducing Kernel Banach Spaces with the l1 Norm

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.Comment: 28 pages, an extra section was adde