Insurance speeds wealth growth by altering wealth dynamics.
problem Why do people voluntarily take insurance when it increases wealth inequality?
method We evaluated contracts by their effect on the time-average growth rate of wealth, assuming only knowledge of wealth dynamics.
result The puzzle of voluntary insurance contracts disappears when wealth changes are non-ergodic.
In modern portfolio theory, the balancing of expected returns on investments against uncertainties in those returns is aided by the use of utility functions. The Kelly criterion offers another approach, rooted in information theory, that always implies logarithmic utility. The two approaches seem incompatible, too loos…
Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A com…
The paper studies the discrete-time average of geometric Brownian motion and its application to Asian options pricing.
problem Understanding the pricing of Asian options with discrete-time averaging.
method Deriving asymptotics for the discrete-time average of geometric Brownian motion and analyzing its impact on Asian options pricing.
result Derives the asymptotics for the price of Asian options with discrete-time averaging in the Black-Scholes model.
Study confirms the Epps effect using different volume time averaging methods for JSE stocks.
problem Demonstrating the Epps effect in stock market data using various aggregation methods.
method Used two non-parametric covariance estimators (Malliavin and Mancino, Hayashi and Yoshida) and two volume time averaging methods (asset intrinsic and synchronised volume time).
result MM estimator more representative of trade time reality, confirming market phenomenology.
The paper investigates length averages in foliations, contrasting with time averages in dynamical systems.
problem Investigate the existence and non-existence of length averages in foliations.
method Generalize the existence problem of time averages in dynamical systems to foliations and introduce the concept of length averages.
result Length averages exist everywhere for codimension one orientable singular foliations without degenerate singularities on compact surfaces under a mild condition.
New method for unbiased regression reduces excess risk.
problem Least squares regression with optimal solution and Hessian matrix.
method Averaged stochastic gradient descent with time-average estimator.
result Unbiased estimator with O(1/k) expected excess risk.
We analyze the question whether sliding window time averages applied to stationary increment processes converge to a limit in probability. The question centers on averages, correlations, and densities constructed via time averages of the increment x(t,T)=x(t+T)-x(t)and the assumption is that the increment is distribute…
Derives time-averaged active inference from control principles.
problem Finite-horizon or discounted-surprise problems in active inference.
method Derives infinite-horizon, average-surprise active inference from optimal control principles.
result Unified objective functional for sensorimotor control.
Gambles are random variables that model possible changes in monetary wealth. Classic decision theory transforms money into utility through a utility function and defines the value of a gamble as the expectation value of utility changes. Utility functions aim to capture individual psychological characteristics, but thei…
Logarithmic regret for continuous-time reinforcement learning.
problem Continuous-time Markov decision processes with unknown transition probabilities and holding times.
method Upper confidence reinforcement learning, mean holding time estimation, stochastic comparison of point processes.
result Logarithmic regret bound achieved in finite time.
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
Paper proposes a mean-field gradient descent for zero-sum games, proving convergence to Nash equilibrium.
problem Finding mixed Nash equilibria in zero-sum games with multiple players.
method Mean-field gradient descent dynamics with time-averaging, incorporating exponentially discounted gradients.
result Exponential convergence rate to mixed Nash equilibrium with respect to total variation metric.
We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatl…
The paper develops a new theory to understand deep learning optimization.
problem Understanding the dynamics of optimization in deep learning, especially in the edge of stability regime.
method Developed a central flow differential equation to describe the time-averaged trajectory of oscillatory optimizers.
result Central flows can predict long-term optimization trajectories with high numerical accuracy.
The definition of the covariant space-time averaging scheme for the objects (tensors, geometric objects, etc.) on differentiable metric manifolds with a volume n-form, which has been proposed for the formulation of macroscopic gravity, is analyzed. An overview of the space-time averaging procedure in Minkowski spacetim…
Enhances UPSA to reduce noise in financial data.
problem Noise in financial data affects UPSA's performance.
method Time-averaging optimal penalty weights and using Average Oracle correlation eigenvalues.
result Combining time-averaging and Average Oracle correlation eigenvalues improves UPSA's performance.
Peters (2011a) defined an optimal leverage which maximizes the time-average growth rate of an investment held at constant leverage. It was hypothesized that this optimal leverage is attracted to 1, such that, e.g., leveraging an investment in the market portfolio cannot yield long-term outperformance. This places a str…
A new MFG framework for evolving clusters from Gaussian mixtures.
problem Evolutionary clustering of time-dependent Gaussian mixtures.
method Control-theoretic framework based on Mean Field Games (MFG) with coupled HJB and Fokker-Planck systems.
result MFG dynamics recover classical EM algorithm trajectories with mass conservation.
We give a new proof of the representation of implied volatility as a time-average of weighted expectations of local or stochastic volatility. With this proof we clarify the question of existence of 'forward implied variance' in the original derivation of Gatheral, who introduced this representation in his book 'The Vol…
Stylized facts of empirical assets log-returns Z include the existence of (semi) heavy tailed distributions fZ(z) and a non-linear spectrum of Hurst exponents τ(β). Empirical data considered are daily prices of 10 large indices from 01/01/1990 to 12/31/2004. We propose a stylized model of price dynamics which is…
A resolution of the St. Petersburg paradox is presented. In contrast to the standard resolution, utility is not required. Instead, the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoulli's resolution, which uses logarithmic utility, but is …
The paper introduces Robust Correlated Equilibrium for games with time-varying costs and proposes an algorithm to achieve it.
problem Games with time-varying costs and disturbances.
method Proposes Robust Correlated Equilibrium and a decentralized algorithm to learn optimal strategies.
result The algorithm converges to the Robust Correlated Equilibrium, showing no regret for each controller.
New algorithms sample random graph homomorphisms for network analysis.
problem Sampling random graph homomorphisms from a graph into a large network.
method Proposed two MCMC algorithms with bounds on mixing times and concentration.
result Network observables are stable under renormalized cut distance.
The study analyzes how stochastic recursive algorithms converge to Markov chains.
problem Understanding convergence of stochastic recursive algorithms to Markov chains.
method Analyzes iterated random operators and contraction operators over Polish spaces.
result The distribution of random sequences converges to the invariant distribution of the Markov chain.
We study the price dynamics of stocks traded in the NASDAQ market by considering the statistical properties of an ensemble of stocks traded simultaneously. For each trading day of our database, we study the ensemble return distribution by extracting its first two central moments. According to previous results obtained …
Study short maturity Asian options under CEV model, presenting an analytical approximation.
problem Analyzing short maturity behavior of Asian options in CEV model.
method Presented an analytical approximation for Asian options prices under CEV model.
result Good numerical agreement with Monte Carlo simulations and benchmark test cases.
This paper analyzes popular time-nonseparable utility functions that describe "habit formation" consumer preferences comparing current consumption with the time averaged past consumption of the same individual and "catching up with the Joneses" (CuJ) models comparing individual consumption with a cross-sectional averag…
We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model…
Lower bounds found for tracking continuous dynamics models.
problem Minimizing deviation and tracking efforts for continuous dynamics.
method Relates to time-average control of Brownian motion via linear programming.
result Existence of asymptotic lower bounds for tracking problems.
AdaBoost is one of the most popular ML algorithms. It is simple to implement and often found very effective by practitioners, while still being mathematically elegant and theoretically sound. AdaBoost's interesting behavior in practice still puzzles the ML community. We address the algorithm's stability and establish m…
Two algorithms reduce regret in non-stationary MAB problems.
problem Non-stationary stochastic multiarmed bandit problems.
method LM-DSEE and SW-UCB# algorithms for abruptly-changing and slowly-varying environments.
result Expected cumulative regret is upper bounded by sublinear functions of time.
Study finds a small correction to Asian option volatility.
problem Implied volatility of Asian options at short maturity.
method Large deviations property and asymptotic expansion for the Hartman-Watson distribution.
result Subleading correction to Asian option volatility is derived.
Unified Growth Theory contradicted by USSR economic data.
problem Unified Growth Theory fails to explain USSR economic growth.
method Analysis of historical USSR economic data.
result USSR economic growth was hyperbolic, not stagnant.
Abstract Coxeter groups have growth rates that are Perron numbers.
problem Understanding growth rates of Coxeter groups.
method Defined a class of Coxeter groups, ∞--spanned, and analyzed their growth rates. result For ∞--spanned Coxeter groups, geodesic growth rate strictly dominates word growth rate and appears to be a Perron number. Linear GDP growth explained by AI performance models.
problem Missing explanation for linear GDP growth since 1950.
method Compare AI performance models to GDP growth patterns.
result Linear growth in AI performance may explain GDP growth.
World GDP growth is predicted to become unsustainable.
problem Predicting the future of world economic growth.
method Analysis of World Bank data on GDP growth rates.
result World economic growth is predicted to become unsustainable.
Unified Growth Theory contradicted by Asian economic data.
problem Unified Growth Theory fails to explain Asian economic growth.
method Analysis of historical economic growth data in Asia.
result Asian economic growth was hyperbolic, not stagnant.
Study short maturity Asian options in local volatility models.
problem Analyzing Asian options with short maturities under local volatility.
method Derive asymptotics for out-of-the-money, in-the-money, and at-the-money cases; solve non-trivial variational problem; present analytical approximation.
result Good numerical agreement with Monte Carlo simulations and Black-Scholes model for practical parameters.
Unified Growth Theory contradicted by African economic data.
problem Contradiction between Unified Growth Theory and African economic growth data.
method Analysis of Galor's data for African countries.
result Data contradicts Unified Growth Theory's claims about economic growth regimes.
Two new algorithms reduce group regret in abruptly changing multi-player bandit problems.
problem Reducing group regret in multi-player bandit problems in environments that change suddenly.
method Design of two novel algorithms: RR-SW-UCB# and SW-DLP.
result Expected cumulative group regret converges to zero over time.
Unified Growth Theory contradicted by Latin American economic data.
problem Unified Growth Theory fails to explain Latin American economic growth.
method Analysis of historical economic growth data from Maddison.
result Unified Growth Theory is inconsistent with Latin American data.
Analyzes historical economic growth trends using hyperbolic distributions.
problem Understanding the natural tendency of historical economic growth.
method Data analysis of world and regional economic growth using hyperbolic distributions.
result Historical economic growth follows hyperbolic distributions with specific parameters.
Study measures economic growth sources in Iran's mining sector using neoclassical growth accounting.
problem Determining the share of economic growth sources in Iran's mining sector.
method Neoclassical growth accounting approach, using production function and Solow residual equation.
result Average annual growth rate of TFP was 2.94% over 30 years.
Simple math predicts growth trends.
problem Complex growth projections are hard to understand.
method Direct or indirect analysis of growth rates.
result Simple assumptions lead to understandable growth predictions.
Unified Growth Theory debunked: economic growth is insecure and unsustainable.
problem The mystery of the great divergence in income per capita.
method Analysis of economic data to show that growth trajectories are increasing vertically over time.
result Unified Growth Theory is incorrect and promotes misleading concepts.
Calculates the systolic growth of nilpotent Lie groups, providing new insights.
problem Understanding the growth of systolic volume in nilpotent Lie groups.
method Expressed systolic growth in terms of discrete subrings, developed methods for lower bounds.
result First computations of systolic growth for non-equivalent volume growth cases.
The Unified Growth Theory is a puzzling collection of myths based on illusions created by hyperbolic distributions. Some of these myths are discussed. The examination of data shows that the three stages of growth (Malthusian Regime, Post-Malthusian Regime and Modern Growth Regime) did not exist and that Industrial Revo…