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…
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48 results for “Ergodic Averages”
We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesaro averages of spherical averages in a Fuchsian group.
We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.
We study how resetting affects geometric Brownian motion, showing it becomes stationary but remains non-ergodic.
problem Effects of stochastic resetting on geometric Brownian motion.
method Analysis of geometric Brownian motion under stochastic resetting.
result Resetting makes geometric Brownian motion stationary but non-ergodic.
The paper tackles learning to control systems with unknown parameters using Brownian noise.
problem Learning to control systems with unknown parameters.
method Proposes algorithms based on moving empirical averages and integrates statistical methods with stochastic control theory.
result Achieves a logarithmic expected regret rate.
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…
Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate finite sets of points at a predefined resolution gamma > 0. We establish a connection between the g…
The paper establishes CLTs for Markov chains and improves sampling algorithms for heavy-tailed distributions.
problem Establishing central limit theorems for ergodic averages of Markov chains.
method Drift conditions to provide necessary and sufficient conditions for CLTs, including lower bounds on convergence rates.
result Sharp conditions and convergence rates for various MCMC algorithms on heavy-tailed targets.
In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the…
Broken ergodicity and memory in the minority gamecond-mat.dis-nn
We study the dynamics of the `batch' minority game with market-impact correction using generating functional techniques to carry out the quenched disorder average. We find that the assumption of weak long-term memory, which one usually makes in order to calculate ergodic stationary states, breaks down when the persiste…
We analyze the generalization and robustness of the batched weighted average algorithm for V-geometrically ergodic Markov data. This algorithm is a good alternative to the empirical risk minimization algorithm when the latter suffers from overfitting or when optimizing the empirical risk is hard. For the generalization…
Study on mean field games with singular controls and their applications.
problem Optimal productivity expansion in dynamic oligopolies.
method Existence and uniqueness of mean field equilibria through nonlinear equations, Abelian limit for discounted and ergodic games.
result Valid connection between discounted and ergodic games, approximation of Nash equilibria.
In this work a novel method to quantify spectral ergodicity for random matrices is presented. The new methodology combines approaches rooted in the metrics of Thirumalai-Mountain (TM) and Kullbach-Leibler (KL) divergence. The method is applied to a general study of deep and recurrent neural networks via the analysis of…
Model-free reinforcement learning is known to be memory and computation efficient and more amendable to large scale problems. In this paper, two model-free algorithms are introduced for learning infinite-horizon average-reward Markov Decision Processes (MDPs). The first algorithm reduces the problem to the discounted-r…
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…
Study the averaging principle for non-autonomous slow-fast systems and apply it to financial local stochastic volatility models.
problem Understanding the behavior of non-autonomous slow-fast systems of stochastic differential equations.
method Prove the averaging principle under specific conditions and apply it to a financial model.
result Prices of derivatives converge to those calculated using the limit model under a risk-neutral measure.
We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite bracketing numbers, satisfy uniform laws of averages under strong dependence, and exhibit uniform mixing. Our results ar…
New algorithm learns optimal policy for average reward MDPs with sample complexity matching lower bound.
problem Learning optimal policy for average reward in uniformly ergodic MDPs.
method Developed an estimator with sample complexity of O(|S||A|t_{mix}ε^{-2}).
result First algorithm to match lower bound of existing literature.
In this paper we study the smooth convex-concave saddle point problem. Specifically, we analyze the last iterate convergence properties of the Extragradient (EG) algorithm. It is well known that the ergodic (averaged) iterates of EG converge at a rate of (Nemirovski, 2004). In this paper, we show that the last…
Simulation of conditional expectations under fast mean-reverting stochastic volatility modelsmath.NA
The paper studies efficient simulation methods for financial firm values under fast mean-reverting volatility.
problem Estimating the probability of firm default under fast mean-reverting stochastic volatility models.
method Approximations using ergodic averages and central limit theorem corrections for efficient simulation.
result Accuracy of approximations assessed through numerical simulation and payoff function estimation.
Optimizes liquidations in decentralized finance to manage credit risk.
problem Managing and liquidating positions in decentralized finance exchanges.
method Formulated as an ergodic optimal control problem, derived closed-form solutions for optimal liquidation strategies.
result Closed-form solutions balance immediate executions with price impacts and long-term rewards.
We generalise the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension two foliation endowed with an invariant transverse measure. We prove that the aver…
Voluntary insurance contracts constitute a puzzle because they increase the expectation value of one party's wealth, whereas both parties must sign for such contracts to exist. Classically, the puzzle is resolved by introducing non-linear utility functions, which encode asymmetric risk preferences; or by assuming the p…
Many problems in machine learning and game theory can be formulated as saddle-point problems, for which various first-order methods have been developed and proven efficient in practice. Under the general convex-concave assumption, most first-order methods only guarantee an ergodic convergence rate, that is, the uniform…
New algorithm identifies best policy in MDPs faster.
problem Identifying the best policy in Markov Decision Processes.
method Problem-dependent lower bound and first algorithm with instance-specific sample complexity.
result First algorithm with reduced exploration rate for faster convergence.
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…
In this paper we present detailed simulation results on the wealth distribution model with quenched saving propensities. Unlike other wealth distribution models where the saving propensities are either zero or constant, this model is not found to be ergodic and self-averaging. The wealth distribution statistics with a …
Ergodic Measures in Stratamath.DS
Study counts ergodic measures in surface lamination strata.
problem Counting ergodic measures in surface lamination strata.
method Determined through analysis of geodesic laminations.
result Number of ergodic measures identified in each stratum.
A Model-free Learning Algorithm for Infinite-horizon Average-reward MDPs with Near-optimal Regretcs.LG
A new model-free algorithm achieves near-optimal regret for infinite-horizon MDPs.
problem Model-free reinforcement learning for infinite-horizon average-reward MDPs.
method Exploration Enhanced Q-learning (EE-QL) for weakly communicating MDPs.
result Achieves regret bound for general weakly communicating MDPs.
Actor-Critic method achieves optimal regret for unichain MDPs.
problem Scalable regret analysis for infinite-horizon average-reward MDPs.
method NAC-B, a Natural Actor-Critic with batching.
result Order-optimal regret of in infinite-horizon average-reward MDPs.
Restricted Boltzmann Machines (RBM) have attracted a lot of attention of late, as one the principle building blocks of deep networks. Training RBMs remains problematic however, because of the intractibility of their partition function. The maximum likelihood gradient requires a very robust sampler which can accurately …
We construct a Poincaré section for the horocycle flow on the modular surface , and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…
New progress on frame flow ergodicity for nearly pinched manifolds.
problem Ergodicity of frame flow on negatively-curved manifolds.
method New ideas leading to ergodicity for nearly 0.25-pinched manifolds.
result Achieved progress towards Brin's conjecture.
Recent results on ergodic theory for Riemann surface laminations and foliations.
problem Ergodic theorems for laminations and foliations on Riemann surfaces.
method Leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles, Lyapunov exponents.
result Definition and study of canonical Lyapunov exponents for singular holomorphic foliations.
The paper improves importance sampling and MCMC methods for complex distributions.
problem Improving sampling efficiency for distributions with atoms or heavy tails.
method Develops minimax optimal trial distributions and importance-tempered MCMC.
result Importance-tempered MCMC can be uniformly ergodic for certain distributions.
Formula connects foliated simplicial volume with group cost.
problem Calculating integral foliated simplicial volume.
method Ergodic decomposition formula for simplicial volume.
result Integration formula linking foliated simplicial volume and group cost.
In this note we show that the Riemann moduli spaces equipped with the Weil--Petersson metric are quantum ergodic for . We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.
We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quan…
Strong stability of ergodic iterations proven without ergodic driving sequence.
problem Ensuring strong stability of ergodic iterations under non-ergodic driving sequences.
method Revisiting processes driven by stationary ergodic sequences, proving strong stability under mild conditions on recursive maps.
result Strong stability of iterations proven without ergodic driving sequence.
Study explores optimal strategies in games with multiple players and mean-field interactions.
problem Optimal strategies in games with multiple players and mean-field interactions.
method Exploration of three different notions of optimality, including mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria.
result Approximation of cooperative and competitive equilibria in large -player games by mean-field control and mean-field equilibria.
The paper studies the ergodicity of frame flow on even-dimensional manifolds.
problem Understanding the ergodicity of frame flow on even-dimensional manifolds.
method Analyzing pinching conditions to determine ergodicity.
result The frame flow is ergodic under specific pinching conditions for even-dimensional manifolds.
Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
Log-Ergodic Dynamics in Stochastic Monetary Velocity: Theoretical Insights and Economic Implicationsq-fin.GN
Log-ergodic model improves velocity of money prediction.
problem Improving velocity of money prediction for economic control.
method Log-ergodic processes to simulate monetary velocity.
result Log-ergodic model offers superior predictive power.
'Ergodicity economics' is criticized as pseudoscience.
problem Flawed conceptual basis of mainstream economic theory.
method Claims 'ergodicity economics' is more parsimonious and clearer.
result Peters' approach has not produced falsifiable implications.
Study improves reinforcement learning for stable long-term performance.
problem Distributionally robust average-reward reinforcement learning for stable long-term performance.
method Proposes two algorithms to achieve near-optimal sample complexity.
result Achieves a sample complexity of for estimating optimal policy and robust average reward.
We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second the…
New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.
problem Stabilizing quantum ergodicity for complex systems.
method Combines mixed quantization techniques with stable ergodicity results for partially hyperbolic systems.
result Establishes stable quantum ergodicity for spin Hamiltonians.
A new condition for the genericity of ergodic measures on non-positively curved Riemannian manifoldsmath.DS
The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to .
result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.