The paper solves a complex portfolio selection problem with nonlinear wealth equations.
problem Continuous time mean-variance portfolio selection with nonlinear wealth equations.
method Invoking the HJB equation and providing an explicit viscosity solution.
result Explicit efficient portfolio strategy and efficient frontier obtained.
Continuous-time mean-variance portfolio selection model with nonlinear wealth equations and bankruptcy prohibition is investigated by the dual method. A necessary and sufficient condition which the optimal terminal wealth satisfies is obtained through a terminal perturbation technique. It is also shown that the optimal…
We briefly review results on nonlinear kinetic equation of Boltzmann type which describe the evolution of wealth in a simple agents market. The mathematical structure of the underlying kinetic equations allows to use well-known techniques of wide use in kinetic theory of rarefied gases to obtain information on the proc…
An equation for the evolution of the distribution of wealth in a population of economic agents making binary transactions with a constant total amount of "money" has recently been proposed by one of us (RLR). This equation takes the form of an iterated nonlinear map of the distribution of wealth. The equilibrium distri…
We introduce and discuss a nonlinear kinetic equation of Boltzmann type which describes the influence of knowledge in the evolution of wealth in a system of agents which interact through the binary trades introduced in Cordier, Pareschi, Toscani, J. Stat. Phys. 2005. The trades, which include both saving propensity and…
The study applies wealth thermalization hypothesis to social networks and explains inequality.
problem Explains inequality in human society through wealth thermalization hypothesis.
method Uses Random Matrix Theory and social networks with nonlinear perturbation.
result Shows that wealth distribution follows Rayleigh-Jeans distribution, leading to inequality.
We introduce and discuss a nonlinear kinetic equation of Boltzmann type which describes the evolution of wealth in a pure gambling process, where the entire sum of wealths of two agents is up for gambling, and randomly shared between the agents. For this equation the analytical form of the steady states is found for va…
The paper analyzes portfolio selection with non-linear wealth dynamics and random coefficients.
problem Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients.
method Solves an auxiliary stochastic control problem to construct a candidate portfolio, verifies optimality using convex duality, and provides the efficient frontier.
result Obtains the efficient frontier in closed form, showing people prefer riskless assets over classical linear markets.
The paper optimizes wealth with concave coefficients in a recursive utility maximization problem.
problem Optimizing wealth with concave coefficients in a recursive utility maximization problem.
method Equivalent backward formulation, Fenchel-Legendre transform, convex duality method.
result Derives the optimal terminal wealth for investors with ambiguity aversion.
The logistic equation describes wealth condensation in a WAA-enhanced asset exchange model.
problem Analyzing wealth condensation in asset exchange models with wealth advantage.
method Introduced a continuous wealth advantage bias in the YSM, derived a logistic equation for the condensed wealth.
result Condensation of wealth follows a logistic equation in time.
Study chaotic dynamics in social stratification models leading to thermalization and turbulence.
problem Understanding social stratification dynamics through chaotic nonlinear systems.
method Modeling social network links with oscillators and energies, studying Hamiltonian evolution and nonlinear interactions.
result Chaotic dynamics leads to dynamical thermalization and Kolmogorov-Zakharov turbulence, with implications for wealth inequality.
We reformulate wealth taxation using Fokker-Planck equations to ensure tax neutrality.
problem Ensuring tax neutrality in wealth taxation frameworks.
method Reformulating the neutral wealth tax framework using stochastic dynamics and statistical physics, specifically Fokker-Planck equations.
result The framework clarifies when wealth taxation is a benign rescaling of dynamics and when it introduces new physics.
Fokker-Planck model shows debts are absorbed over time in wealth distribution.
problem Modeling wealth distribution with agents having debts.
method Fokker-Planck equation with variable diffusion coefficient.
result Debts are absorbed over time, leading to a positive wealth distribution.
An important class of economic models involve agents whose wealth changes due to transactions with other agents. Several authors have pointed out an analogy with kinetic theory, which describes molecules whose momentum and energy changes due to interactions with other molecules. We pursue this analogy and derive a Bolt…
Develops SPT with price impact, deriving formulas for wealth and arbitrage conditions.
problem Tackles price impact in high-dimensional markets.
method Incorporates nonlinear price impact and impact decay models.
result Derives master formula for trading strategies and wealth dynamics.
The so-called "Yard-Sale Model" of wealth distribution posits that wealth is transferred between economic agents as a result of transactions whose size is proportional to the wealth of the less wealthy agent. In recent work [B.M. Boghosian, "Kinetics of Wealth and the Pareto Law," {\it Phys. Rev. E} {\bf 89} (2014) 042…
Agent-based model for wealth distribution with negative wealth.
problem Modeling wealth distribution with negative wealth and validating against empirical data.
method Agent-based model, Fokker-Planck equation, numerical solution, inverse problem solving.
result Agreement with empirical data of an average error less than 0.16% over 27 years.
Study extends wealth tax neutrality framework to heterogeneous investors.
problem Analyzing wealth tax neutrality in populations with varying return-generating ability.
method Extended Fokker-Planck framework to heterogeneous investors, deriving extended Fokker-Planck equation.
result Proportional wealth tax no longer neutral due to varying return-generating ability, leading to different real incidence and wealth distribution changes.
Wealth redistribution through Fokker-Planck equation controls preserves Gini coefficient.
problem Preserving Gini coefficient through proportional wealth tax.
method Formulating optimal redistribution as a control problem for Fokker-Planck equation.
result Progressive taxes redistribute within policy-relevant timescales.
A simplified model shows how wealth distribution can be derived from random exchanges.
problem Understanding wealth inequality and its distribution over time.
method Stylized random exchange model, Markov chain, discrete and continuous stochastic processes, Boltzmann-type kinetic equations.
result Existence of equilibrium distribution in the stylized model.
Econophysics provides a strategy for understanding the potential mechanisms underlying the anomalous distribution of wealth found in real societies. We present a computational nonlinear stochastic model for the distribution of wealth that depends upon three parameters and two mechanisms: trade and investment. To avoid …
We study the model of interacting agents proposed by Chatterjee et al that allows agents to both save and exchange wealth. Closed equations for the wealth distribution are developed using a mean field approximation. We show that when all agents have the same fixed savings propensity, subject to certain well defined app…
Using the analogy with inelastic granular gasses we introduce a model for wealth exchange in society. The dynamics is governed by a kinetic equation, which allows for self-similar solutions. The scaling function has a power-law tail, the exponent being given by a transcendental equation. In the limit of continuous trad…
Investigates portfolio selection among competitive agents with mean-variance preferences.
problem Optimizing portfolios with multi-agent competition and relative wealth comparison.
method Reformulated as a constrained, non-homogeneous stochastic linear-quadratic control problem; derived optimal feedback strategies; used decoupling techniques and fixed-point theory to solve nonlinear BSDEs.
result Characterized three scenarios based on market and competition parameters: unique Nash equilibrium, no Nash equilibrium, or infinitely many Nash equilibria.
An insurer optimizes investment in a market with bank account and risky assets, considering nonlinear economic factors.
problem Optimizing investment strategy for an insurer in a market with bank account and risky assets.
method Adapting dynamic programming approach, deriving Hamilton--Jacobi--Bellman (HJB) equation, proving unique solvability, and solving coupled FBSDEs.
result Derives the optimal investment strategy for an insurer.
Model shows how investment traps wealth across generations.
problem Existence of wealth traps between social strata.
method Developed a model linking investment and intergenerational wealth.
result Proved a `rat race' theorem showing investment traps wealth.
The paper analyzes optimal retirement timing considering age-dependent mortality risk.
problem Optimal retirement timing under age-dependent mortality risk.
method Formulated as a stochastic control and optimal stopping problem, transformed into a finite time horizon, three-dimensional degenerate optimal stopping problem.
result Existence of an optimal retirement boundary, characterized as a unique solution to a nonlinear integral equation.
In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to stu…
Study on optimal portfolio selection with varying borrowing and saving rates in continuous-time markets.
problem Optimal portfolio selection in markets with different borrowing and saving rates.
method Hamilton-Jacobi-Bellman equation, partial differential equation, verification argument.
result Existence and smoothness of the value function, identification of trading regions and strategies.
Study competitive agents' optimal consumption and investment strategies with relative performance criteria.
problem Optimizing consumption and investment strategies for multiple agents with relative performance considerations.
method Derived a closed-form solution for an n-player game and mean field game, analyzing the impact of risk tolerance and competitiveness parameters. result Unique equilibria found, showing nonlinear and non-monotone dependence on agents' risk tolerance and competitiveness parameters.
We look at the meaning of 'relaxation' in the wealth exchange models that are recently proposed in Econophysics to interpret the wealth distributions. To quantify and characterise the process of relaxation, we define an appropriate quantity and evaluate that numerically for the systems of many agents. Also, the numeric…
We solve an optimal consumption problem with habit formation constraints.
problem Maximizing utility with habit formation constraints.
method Formulated and solved a deterministic optimal consumption problem.
result Optimal consumption policies derived explicitly.
Investment and insurance decisions are studied in a model with nonlinear portfolio frictions and background risk.
problem Investment and insurance decisions under a model with nonlinear portfolio frictions and background risk.
method Dynamic programming approach to find optimality conditions.
result Agent can choose to assume, partially assume, or purchase total insurance against adverse jumps in wealth.
We explore martingale and convex duality techniques to study optimal investment strategies that maximize expected risk-averse utility from consumption and terminal wealth. We consider a market model with jumps driven by (multivariate) marked point processes and so-called non-linear wealth dynamics which allows to take …
We provide an exact solution to the ideal-gas-like models studied in econophysics to understand the microscopic origin of Pareto-law. In these class of models the key ingredient necessary for having a self-organized scale-free steady-state distribution is the trading or collision rule where agents or particles save a d…
Investment strategies for rank-dependent utility agents are derived in a continuous-time market.
problem Time inconsistency in rank-dependent utility models.
method Study of consistent planners seeking intra-personal equilibrium strategies.
result Explicit final wealth profile replicating equilibrium strategies, with scaling function derived.
We establish when the two problems of minimizing a function of lifetime minimum wealth and of maximizing utility of lifetime consumption result in the same optimal investment strategy on a given open interval O in wealth space. To answer this question, we equate the two investment strategies and show that if the indi…
Optimal control strategies modify wealth inequality variance.
problem Minimizing wealth inequality variance in a market economy.
method Kinetic models and model predictive control for wealth dynamics.
result Control modifies Pareto index of wealth distribution.
Paper solves complex investment-consumption problem with numerical methods.
problem Optimal investment and consumption strategies with proportional transaction costs.
method Monte Carlo simulation and finite difference method for approximating gradients.
result Numerical results validate optimal trading strategies and properties.
Study pricing game options in imperfect markets with default.
problem Pricing game options in markets with default and imperfections.
method Extend Kifer's results to imperfect markets, introduce seller's price, prove equivalence to Dynkin game value function.
result Seller's price equals the value function of a generalized Dynkin game under nonlinear expectation.
Flow taxes and stock taxes preserve portfolio neutrality under specific conditions.
problem Analyzing the impact of different types of taxes on portfolio choice.
method Extending the neutrality result to a full system of ownership taxes, showing how each tax modifies the drift of the wealth process.
result The combined system of taxes preserves portfolio neutrality under three conditions, and the drift-shift symmetry generalizes to a drift-shift-and-rescale symmetry.
Firm growth process in the developing economies is known to produce divergence in their growth path giving rise to bimodality in the size distribution. Similar bimodality has been observed in wealth distribution as well. Here, we introduce a modified kinetic exchange model which can reproduce such features. In particul…
Study relaxes boundedness constraints in Ramsey consumption problem.
problem Optimal consumption in the stochastic Ramsey problem without boundedness constraints.
method Non-standard stochastic differential equation, probabilistic arguments, viscosity solutions.
result Value function is the unique classical solution to a nonlinear elliptic equation, leading to optimality of feedback consumption process.
We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed earlier by the authors. The model considers a system of rational agents interacting in a game theoretical framework. This evolution drives the dynamic of the agents in both wealth and economic configu…
Modeling consumption and investment decisions with reference point and drawdown constraints.
problem Modeling consumption and investment decisions with reference point and drawdown constraints.
method Solving a stochastic control problem to derive value function, optimal consumption plan, and investment strategy in semi-explicit forms.
result Five important thresholds of wealth, all as functions of h, and significant economic implications. Optimal wealth strategy derived for jump-diffusion models with liabilities.
problem Maximizing utility in jump-diffusion models with random liabilities.
method Forward Backward SDEs system for optimal strategy.
result Explicit results for pure jump model and exponential utilities.
Model optimal growth strategy in a market with short-lived assets.
problem Investment market with short-lived assets and endogenous prices.
method Formulate stochastic equation for wealth processes and prove existence of optimal strategy.
result Existence of a submartingale strategy ensuring investor's wealth growth asymptotically.
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.