The evolution of prices on ideal market is given by geometrical Brownian motion, where Gaussian white noise describes fluctuations. We study the effect of correlations introduced by a color noise.
Improved privacy-preserving statistical estimates with customizable noise reduction.
problem Balancing privacy and accuracy in statistical estimation.
method Introducing the Brownian mechanism, which adds Gaussian noise to a sequence of estimates, gradually reducing it based on the practitioner's needs.
result The Brownian mechanism produces more accurate estimates while maintaining strong privacy guarantees, outperforming existing methods.
The study tackles rough noise in high-frequency financial data using fractional Brownian motion.
problem Impediments to analyzing high-frequency financial data due to noise.
method Assuming an efficient price process as a continuous Itô semimartingale, the study derives consistent estimators and confidence intervals for roughness parameters and volatilities.
result The rough noise model explains divergence rates in volatility signature plots over time and between assets.
Upper and lower bounds on regret for noisy optimization of Brownian motion.
problem Optimizing a one-dimensional Brownian motion with noisy observations.
method Upper bound uses confidence bounds and Markov property; lower bound uses hypothesis testing reduction.
result Upper and lower bounds are tight up to a factor of O((logT)1.5). The paper proposes estimators for bid-ask spreads with and without serial dependence.
problem Estimating bid-ask spreads in financial markets with and without serial dependence.
method The authors propose moment-based estimators for bid-ask spreads, considering both geometric Brownian motion and geometric fractional Brownian motion for price dynamics, and Ornstein-Uhlenbeck process for microstructure noise.
result The estimators are consistent and asymptotically normal, and perform well compared to existing approaches on simulated data.
We develop a variational framework for SDEs driven by fractional noise.
problem Capturing long-term dependencies in SDEs driven by fractional noise.
method Markov approximation of fractional Brownian motion, variational inference, neural networks.
result Efficient variational inference of posterior path measures for neural-SDEs.
Cryptocurrency time-series predictability is low, resembling Brownian noise.
problem Low predictability of cryptocurrency exchange rates.
method Complexity and model predictions of Litecoin, Binance Coin, Bitcoin, Ethereum, and XRP exchange rates.
result Simpler models outperform complex ones in cryptocurrency forecasting.
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index H∈(0,1). This process has sta…
The Epps effect helps distinguish between continuous and discrete financial tick data.
problem Determining whether financial tick data represents continuous or discrete events.
method Deriving and correcting the Epps effect, proposing experiments to discriminate between models.
result Tick data is better represented as discrete events rather than continuous Brownian diffusions.
Quantum probability theory constructs Martingales for non-Brownian financial models.
problem Constructing Martingales for financial models using fractional Brownian motion.
method Quantum probability theory and Wick product.
result Quantum probability framework allows for Martingale construction without Brownian integrals.
Modeling financial markets with memory using fractional calculus and Brownian motion.
problem Capturing memory effects in financial markets using stochastic models.
method Fractional Langevin equation with colored noise generated by fractional Brownian motion.
result Anomalous marginal glass phase observed in some regions of the system.
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.
New bounds for private matrix approximation using Gaussian noise and Dyson Brownian Motion.
problem Private approximation of symmetric matrices with Gaussian noise.
method Viewing Gaussian noise as Dyson Brownian Motion to track eigenvalue and eigenvector evolution.
result Improved bounds on Frobenius-distance utility for private matrix approximation.
NANSDE-Net models time series with memory using neural ARMA-type noise.
problem Modeling time series with long- or short-memory characteristics.
method Developed NANSDE-Net, a generative model that incorporates Neural Network-kernel ARMA-type noise.
result NANSDE-Net matches or outperforms existing models in reproducing long- and short-memory features of data.
Paper develops Euler scheme for fractional delay diff. eqs with additive noise.
problem Developing a consistent Euler-Maruyama scheme for fractional stochastic delay diff. eqs.
method Euler-Maruyama scheme for fractional Brownian motion with additive noise.
result Achieved convergence rate of H+1/2 for smooth delays when H>1/2.
High frequency based estimation methods for a semiparametric pure-jump subordinated Brownian motion exposed to a small additive microstructure noise are developed building on the two-scales realized variations approach originally developed by Zhang et. al. (2005) for the estimation of the integrated variance of a conti…
Study confirms rough volatility in financial data, independent of microstructure noise.
problem Characterizing volatility in financial markets, especially rough volatility.
method Used range-based volatility estimators to confirm findings from fractional behavior.
result Log-volatility behaves like fractional Brownian motion with an even lower Hurst exponent.
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
Complex network analysis reveals dominant stocks in financial stock returns correlations.
problem Inferring financial stock returns correlations from complex network analysis.
method Simulated geometric Brownian motion for stocks, complex network analysis, eigenvector centrality, clustering.
result Returns correlation matrix is dominated by stocks with high eigenvector centrality and clustering.
Continuous-time Kyle model shows privacy subsidy from noise-perturbed order flow.
problem Quantifying break-even fees for committed-AMM exchanges under privacy-aggregated information.
method Extended Nakamura's (2026) single-period result to continuous-time, observing order flow perturbed by Brownian noise.
result Cumulative privacy subsidy is identified as equivalent to Loss-Versus-Rebalancing in price observation gap.
Identifies most probable flows for Kunita SDEs in fluid dynamics.
problem Modeling stochastic processes with Eulerian noise and deterministic drifts.
method Equipping the domain with a Riemannian metric from the noise, solving the resulting PDEs.
result Most probable flows differ from deterministic flows, especially under noise.
New insights show stochastic initialization prevents token clustering in deep Transformers.
problem Understanding token dynamics in deep stochastic Transformers.
method Analysis of deep Transformers with random initialization noise, proving convergence to an interacting-particle system on the sphere.
result Initialization noise prevents token clustering, leading to antipodal formations.
Python package for estimating Hurst exponent in fBm.
problem Estimating Hurst exponent in fractional Brownian motion.
method Whittle's likelihood method applied to fractional Gaussian noise.
result Implementation achieves state-of-the-art accuracy and speed.
New method generates synthetic time series paths with more flexibility.
problem Restrictions in generating synthetic paths using Brownian reference.
method Introduces Triangular-Reference Schrödinger Bridges (TR-SBTS) for time series generation.
result Generates synthetic paths with more flexibility in stochastic volatility and correlated noise.
We study the mean escape time in a market model with stochastic volatility. The process followed by the volatility is the Cox Ingersoll and Ross process which is widely used to model stock price fluctuations. The market model can be considered as a generalization of the Heston model, where the geometric Brownian motion…
In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. W…
This paper studies an optimal trading problem that incorporates the trader's market view on the terminal asset price distribution and uninformative noise embedded in the asset price dynamics. We model the underlying asset price evolution by an exponential randomized Brownian bridge (rBb) and consider various prior dist…
We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results…
Model predicts stock prices using GAN and RoI Pooling.
problem Predicting stock prices influenced by macroeconomic factors.
method Markov Decision Process, GAN, RoI Pooling.
result Identifies macroeconomic factors' influence on stock prices.
Derives effective continuous dynamics for adaptive SGD methods.
problem Analyzing noise in adaptive SGD methods.
method Stochastic modified equations framework and Malladi's scaling rules.
result Sampling-induced noise in SGD limits to independent Brownian motions.
Geometry-aware noise improves model generalization on complex manifolds.
problem Improving model generalization on highly curved data manifolds.
method Add geometry-aware noise to input space, projecting Gaussian noise onto tangent space of manifold and mapping it via geodesic curve.
result Geometry-aware noise leads to improved generalization and robustness on highly curved manifolds.
New algorithm tackles high-dimensional simulation optimization, converging efficiently.
problem High-dimensional simulation optimization challenges.
method Sparse grid experimental design combined with kernel ridge regression using Brownian field kernel, followed by expected improvement strategy.
result Established upper bounds on convergence rate, demonstrating superior performance in practice.
This work extends Tweedie's formulae to non-Gaussian processes for better diffusion model generation.
problem Limited exploration of non-Gaussian diffusion models and corresponding Tweedie's formulae.
method Extended Tweedie's formulae to geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes.
result Demonstrated potential of non-Gaussian models in image and financial time series generation.
Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced w…
This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager's information set plus noise. The noise makes…
Alternative finance models from physics for non-equilibrium systems.
problem Inequities of classical finance models in physics-based perspective.
method Physics-based insights for non-equilibrium finance models.
result Alternative models for non-equilibrium finance systems.
Proposes a method to estimate SDE noise from a single trajectory.
problem Estimating SDE noise from a single data trajectory without ergodicity or stationarity.
method Combining Taylor expansions, Girsanov transformations, and drift function's initial value for drift and noise estimation.
result First SSISDE algorithm capable of identifying SDE dynamics from a single trajectory.
New analysis shows SGD prefers wide minima due to heavy-tailed noise.
problem Analyzing the behavior of SGD in deep learning settings.
method Invoking the generalized CLT to model SGD as driven by a Lévy motion.
result Established a connection between SGD convergence and tail-index of noise.
The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the classical central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic diff…
Generative model for Lévy area improves SDE simulation accuracy.
problem Simulating Lévy areas for high-order SDEs is challenging due to non-Gaussian nature and lack of fast sampling algorithms.
method LévyGAN, a deep-learning model with a GNN-inspired architecture, generates approximate samples of Lévy area.
result LévyGAN matches all joint and conditional odd moments exactly and achieves state-of-the-art performance in 4D Brownian motion.
Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, by practitioners in business and economic policy making, as a conditional volatility forecasting models. Furthermore ARCH models still are attracting an interest of the researchers. In this contribution we consider the well known GARCH(1…
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
This study shows how DDPM can be represented by the OU process.
problem Designing optimal noise schedules for DDPM.
method Formal equivalence between DDPM and OU process, heuristic designs based on Fisher Information.
result Fisher-Information-motivated schedule corresponds to cosine noise schedule.
Study finds roughness in volatility despite diffusive instantaneous volatility.
problem Determining the roughness of volatility in financial assets.
method Non-parametric method based on normalized p-th variation for estimating roughness of sample paths. result Realized volatility exhibits rough behavior with a significantly smaller Hurst exponent than instantaneous volatility.
Study refracted skew Brownian motion, find densities and asymptotics.
problem Modeling and analyzing refracted skew Brownian motion.
method Perturbation approach to find potential densities, transition density, and asymptotic behaviors.
result Expressions and asymptotic behaviors of refracted skew Brownian motion.
New perspective on SGD reveals short-range memory effects in deep learning.
problem Understanding the efficacy of stochastic gradient descent (SGD) in deep learning.
method Proposed that SGD is a discretization of an SDE driven by fractional Brownian motion (FBM).
result SGD stays longer in flat minima, favoring generalization.
New model uses generalized fractional Brownian motion for stock price prediction.
problem Traditional models fail to accurately predict stock price fluctuations.
method Introduces generalized fractional Brownian motion as a new stochastic process for price modeling.
result Validates the new model for option pricing and risk assessment.