Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions

This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the timeinconsistent stopping control problems under general multi-dimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair $(\hat{u},C)$ of controlstopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a non-exponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general non-constant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount

Equilibria for Time-inconsistent Singular Control Problems

We study a time-inconsistent singular control problem originating from irreversible reinsurance decisions with non-exponential discount. Novel definitions of weak and strong equilibria for time-inconsistent singular control problems are introduced. For the problem with non-exponential discount, both sufficient and necessary conditions are derived, providing a thorough mathematical characterization of both equilibria. In particular, the weak equilibrium can be characterized by an extended HJB system, which is a coupled system of non-local parabolic equations with free boundaries. Finally, by showing the existence of classical solutions to the extended HJB system, the existence of weak equilibrium is established under some additional assumptions

Consumption-investment decisions with endogenous reference point and drawdown constraint

We propose a consumption-investment decision model where past consumption peak $h$ plays a crucial role. There are two important consumption levels: the lowest constrained level and a reference level, at which the risk aversion in terms of consumption rate is changed. We solve this stochastic control problem and derive the optimal value function, optimal consumption plan, and optimal investment strategy in semi-explicit forms. We find five important thresholds of wealth, all as functions of $h$, and most of them are implicit nonlinear functions. As can be seen from numerical results, this intuitive and simple model has significant economic implications and there are at least three important predictions: the marginal propensity to consume out of wealth is generally decreasing in wealth but can be increasing with an intermediate level of wealth; the implied relative risk aversion is a smile in wealth; the welfare of the poor is more vulnerable to wealth shocks than the wealthy

Dynamic portfolio selection for nonlinear law-dependent preferences

This paper addresses the portfolio selection problem for nonlinear law-dependent preferences in continuous time, which inherently exhibit time inconsistency. Employing the method of stochastic maximum principle, we establish verification theorems for equilibrium strategies, accommodating both random market coefficients and incomplete markets. We derive the first-order condition (FOC) for the equilibrium strategies, using a notion of functional derivatives with respect to probability distributions. Then, with the help of the FOC we obtain the equilibrium strategies in closed form for two classes of implicitly defined preferences: CRRA and CARA betweenness preferences, with deterministic market coefficients. Finally, to show applications of our theoretical results to problems with random market coefficients, we examine the weighted utility. We reveal that the equilibrium strategy can be described by a coupled system of Quadratic Backward Stochastic Differential Equations (QBSDEs). The well-posedness of this system is generally open but is established under the special structures of our problem

Dynamic portfolio selection under generalized disappointment aversion

This paper addresses the continuous-time portfolio selection problem under generalized disappointment aversion (GDA). The implicit definition of the certainty equivalent within GDA preferences introduces time inconsistency to this problem. We provide the sufficient and necessary condition for a strategy to be an equilibrium by a fully nonlinear integral equation. Investigating the existence and uniqueness of the solution to the integral equation, we establish the existence and uniqueness of the equilibrium. Our findings indicate that under disappointment aversion preferences, non-participation in the stock market is the unique equilibrium. The semi-analytical equilibrium strategies obtained under the constant relative risk aversion utility functions reveal that, under GDA preferences, the investment proportion in the stock market consistently remains smaller than the investment proportion under classical expected utility theory. The numerical analysis shows that the equilibrium strategy's monotonicity concerning the two parameters of GDA preference aligns with the monotonicity of the degree of risk aversion.Comment: 38 pages, 5 figure

A new exploration of the fl ipped classroom of systematic anatomy

in recent years, new teaching reform mode is common, and more and more are applied in teaching work. However, diff erent teaching reforms achieve diff erent goals. As long as the reform mode can achieve high-quality teaching eff ect, it is the pursuit of teachers. At present, we should consider whether diff erent reform models can be integrated with each other. After the integration, we may get the ideal teaching eff ect. Therefore, teachers should put forward more innovative teaching methods. We should make rational use of diff erent teaching reform modes to make teaching achievements to a higher level

Based on the Molecular Mechanism of CiteSpace Sanqi Total Saponin and the Prevention and Treatment of Renal Fibrosis, Hot Spot Visualization Analysis

This study aims to study the current situation, hot spots and development trends of Sanqi total saponins in the prevention and treatment of renal fibrosis through a visual map, in order to provide a direction and trend for the research of traditional Chinese medicine on the prevention and treatment of renal fibrosis. Method: By using CiteSpace visualization software, the relevant literature on the use of Sanqi total saponins in the CNKI database for the treatment of renal fibrosis was visually analyzed, including posting trends, authors, institutions and keywords, and the corresponding knowledge map was generated. Results: From 2005 to 2022, a total of 1,533 documents were included in the research results. From 2008 to 2017, an average of 99 articles were published every year, accounting for 67.5% of the total number of articles published. Among them, the number of articles published in 2010 and 2013 reached 114, the highest in previous years. At the same time, Yang Min is the author who has published the most articles, with a total of 29 papers. Beijing University of Traditional Chinese Medicine is the main research institution in this field. Through cluster analysis of research hotspots, ten clusters were obtained, including rats, left ventricular reconstruction, oxidative stress, drug interaction, astragaloside, apoptosis, matrix metalloproteinase-9,oleoside, lipopolysaccharide, atherosclerosis, etc. Future research focuses include oxidative stress, autophagy process, Sanqi total saponins, inflammatory response, cell proliferation and apoptosis, and pharmacokinetics. Conclusion: The knowledge map reveals the research status and research hotspots of Sanqi total saponins in the prevention and treatment of renal fibrosis, including oxidative stress, inflammation, cell proliferation and apoptosis, which will become the focus of research on the prevention and treatment of renal fibrosis in traditional Chinese medicine in the future. However, the collaboration between research authors and scientific research institutions lacks closeness, lacks core authors, and has not yet formed an influential research organization. The depth and breadth of research need to be further enhanced

Potential of Core-Collapse Supernova Neutrino Detection at JUNO

JUNO is an underground neutrino observatory under construction in Jiangmen, China. It uses 20kton liquid scintillator as target, which enables it to detect supernova burst neutrinos of a large statistics for the next galactic core-collapse supernova (CCSN) and also pre-supernova neutrinos from the nearby CCSN progenitors. All flavors of supernova burst neutrinos can be detected by JUNO via several interaction channels, including inverse beta decay, elastic scattering on electron and proton, interactions on C12 nuclei, etc. This retains the possibility for JUNO to reconstruct the energy spectra of supernova burst neutrinos of all flavors. The real time monitoring systems based on FPGA and DAQ are under development in JUNO, which allow prompt alert and trigger-less data acquisition of CCSN events. The alert performances of both monitoring systems have been thoroughly studied using simulations. Moreover, once a CCSN is tagged, the system can give fast characterizations, such as directionality and light curve

Detection of the Diffuse Supernova Neutrino Background with JUNO

As an underground multi-purpose neutrino detector with 20 kton liquid scintillator, Jiangmen Underground Neutrino Observatory (JUNO) is competitive with and complementary to the water-Cherenkov detectors on the search for the diffuse supernova neutrino background (DSNB). Typical supernova models predict 2-4 events per year within the optimal observation window in the JUNO detector. The dominant background is from the neutral-current (NC) interaction of atmospheric neutrinos with 12C nuclei, which surpasses the DSNB by more than one order of magnitude. We evaluated the systematic uncertainty of NC background from the spread of a variety of data-driven models and further developed a method to determine NC background within 15\% with {\it{in}} {\it{situ}} measurements after ten years of running. Besides, the NC-like backgrounds can be effectively suppressed by the intrinsic pulse-shape discrimination (PSD) capabilities of liquid scintillators. In this talk, I will present in detail the improvements on NC background uncertainty evaluation, PSD discriminator development, and finally, the potential of DSNB sensitivity in JUNO