Turnpike Property and Convergence Rate for an Investment Model with General Utility Functions

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.Comment: 29 page

Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.Comment: 18 page

Turnpike Property and Convergence Rate for an Investment Model with General Utility Functions

Abstract In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions

Optimal Liquidation in a Finite Time Regime Switching Model with Permanent and Temporary Pricing Impact

Abstract. In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method

Characterization and Anti-Inflammatory Potential of an Exopolysaccharide from Submerged Mycelial Culture of Schizophyllum commune

Background and Purpose: Mushroom polysaccharides have attracted attention in food and pharmacology fields because of their many biological activities. The structure characterization and anti-inflammatory activity of exopolysaccharide from Schizophyllum commune were evaluated in present study.Methods: An exopolysaccharide from a submerged mycelial fermentation of S. commune was obtained using DEAE-52 cellulose and Sephadex G-150 chromatography. The molecular weight (MW), monosaccharide compositions, chemical compositions, methylation analysis, circular dichroism studies, Fourier transform infrared spectroscopy, nuclear magnetic resonance (NMR) spectra, scanning electron microscopy (SEM), and atomic force microscopy were investigated.Results: It was a homogeneous protein-bound heteropolysaccharide with MW of 2,900 kDa. The exopolysaccharide contained a β-(1→3) glycosidic backbone, (1→4)- and (1→6)- glycosidic side chain, and high amount of glucose. The anti-inflammatory activity of exopolysaccharide was assessed by inhibiting the production of nitric oxide (NO), inducible nitric oxide synthase (iNOS), and 5- lipoxygenase (5-LOX) from macrophages. This exopolysaccharide significantly (p < 0.05) inhibited lipopolysaccharides-induced iNOS expression levels in the cells in a dose-dependent manner.Conclusion: It indicated significant anti-inflammatory effects, which showed that exopolysaccharide might be exploited as an effective anti-inflammatory agent for application in NO-related disorders such as inflammation and cancer