This paper applies AMP theory to improve learning tasks.
problem Improving learning efficiency by optimizing task-specific models.
method Uses aggregated Markov processes to reduce model complexity and enhance learning.
result Demonstrates how AMP theory can be effectively applied to stochastic learning.
New learnability criteria for non-iid processes equivalent to online learning.
problem Statistical learning under non-iid stochastic processes is underdeveloped.
method Defined two learnability notions and showed their equivalence to online learning.
result Learnability criteria for non-iid processes are equivalent to online learning.
Extends Feller theory to non-locally compact spaces for stochastic equations.
problem Stochastic partial differential equations and fractional processes.
method Extended Feller processes and proofs of folklore results.
result No condition of generalized Feller semigroups can be dropped.
A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization o…
Paper develops SINNOs for approximating stochastic processes.
problem Approximating stochastic processes with neural networks.
method Developed stochastic interpolation neural network operators (SINNOs) with random coefficients.
result Established boundedness, interpolation accuracy, and approximation capabilities of SINNOs.
Time-subordinated Brownian motion models improve financial market stochastic distribution.
problem Improving stochastic distribution modeling in financial markets.
method Fourier theory and methodology for time-subordinated Brownian motion models, extending real domain to complex plane.
result Characterization and direct study of stochastic time-change from full process.
The paper solves investment problems with uncertain factors using game theory.
problem Optimal forward investment in an incomplete market with model uncertainty.
method Combining stochastic differential games and ergodic BSDE approach.
result Representation of robust forward performance processes in factor form.
The paper analyzes arbitrage theory in a fluctuating market of stochastic dimension.
problem Arbitrage opportunities in a market with time-varying asset numbers.
method Develops the fundamental theorem of asset pricing and optional decomposition theorem in a stochastic dimension market.
result Equivalence of conditions for no arbitrage and viability in a stochastic dimension market.
We combine general equilibrium theory and theorie generale of stochastic processes to derive structural results about equilibrium state prices.
We present a theory of homogeneous volatility bridge estimators for log-price stochastic processes. The main tool of our theory is the parsimonious encoding of the information contained in the open, high and low prices of incomplete bridge, corresponding to given log-price stochastic process, and in its close value, fo…
This research develops approximation theory for OOMs of infinite-dimensional processes.
problem Developing an approximation theory for OOMs of infinite-dimensional processes.
method Establishing an inner product structure and proving continuity of observable operators.
result A fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space is described.
Overview of non-stochastic-gradient SA algorithms in signal processing and ML.
problem Dealing with large data sets and uncertainties in signal processing and machine learning.
method General framework of SA algorithms using Lyapunov functions.
result Unified convergence properties of non-stochastic-gradient algorithms.
Analysis of SGD for Gaussian mixture classification using dynamical mean-field theory.
problem Learning dynamics of SGD for a neural network classifying Gaussian mixture.
method Applying dynamical mean-field theory to track SGD dynamics in high dimensions.
result Reveals how SGD navigates the non-convex loss landscape.
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.
Investigates spontaneous symmetry breaking in non-equilibrium systems.
problem Spontaneous symmetry breaking of ergodicity in non-equilibrium systems.
method Mathematical and effective field theory approaches to investigate symmetry breaking.
result Symmetry breaking phenomena observed in stochastic processes.
Method uses deep learning to estimate traffic intensity.
problem Estimating stochastic intensity of traffic processes.
method Deep neural networks for nonlinear filtering.
result Deep learning method accurately estimates traffic intensity.
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
The FSRM uses a multifractional process to capture price multifractality, revealing serial information for forecasting.
problem Capturing multifractal price dynamics for better forecasting.
method Developed a fractional stochastic regularity model based on multifractional processes and information theory.
result The serial information of the regularity process Ht can be theoretically determined, aiding in forecasting future price increments. This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares. This result is obtained by analyzing SGD as a stochastic process and by sharply characterizing the stationary covariance matrix of this process. The…
Paper introduces HRPCFD for efficient training of stochastic processes.
problem Discontinuities in stochastic processes over time.
method High Rank Path Development method and HRPCFD metric.
result Efficient algorithm for training HRPCFD from data.
Entropy-minimal measure calculated for a stochastic volatility model.
problem Calculating the entropy-minimal equivalent martingale measure in a stochastic volatility model.
method Revised related theory, calculated entropy-minimal measure.
result Entropy-minimal measure for the exponential Ornstein-Uhlenbeck model.
This paper extends subordinated models to include stochastic time changes, improving financial modeling.
problem Improving financial models to better capture market features like jump clustering and volatility persistence.
method Subordinated processes with Levy and stochastic arrival mechanisms.
result Strong consistency and asymptotic normality results for VG and VGSA processes under various stochastic arrival models.
In this paper, we study the problems of principal Generalized Eigenvector computation and Canonical Correlation Analysis in the stochastic setting. We propose a simple and efficient algorithm, Gen-Oja, for these problems. We prove the global convergence of our algorithm, borrowing ideas from the theory of fast-mixing M…
Abstract: Surveying connections between ML and Control Theory.
problem Addressing the intersection of Machine Learning and Control Theory.
method Develops connections through reinforcement learning, supervised learning, deep learning, and stochastic gradient descent.
result Machine Learning and Control Theory are interconnected, with ML solving large control problems and Control Theory providing tools for ML.
A novel VAE called πVAE models complex data with stochastic processes.
problem Efficient inference of stochastic processes for Bayesian deep learning with big data.
method πVAE is a finitely exchangeable, Kolmogorov consistent, continuous stochastic process VAE.
result πVAE achieves state-of-the-art performance in tasks like spatial interpolation.
In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the co…
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 sampling method uses stochastic interpolants and FBSDEs.
problem Sampling from high-dimensional distributions with unnormalized densities.
method Stochastic interpolants and FBSDEs to define and solve diffusion process.
result Effective sampling from challenging distributions.
New method for risk quantification using quantile processes and measure distortions.
problem Risk quantification and valuation in financial markets.
method Develops a novel stochastic valuation principle based on probability measure distortions induced by quantile processes.
result Introduces a system of subjective probability measures that indexes a stochastic valuation principle susceptible to probability measure distortions.
Unified approach for data-driven control of stochastic processes.
problem Developing practical strategies for stochastic control problems with unknown dynamics.
method Reduction to rate-optimal estimators of invariant distribution risk.
result Data-driven strategies can achieve better performance than known methods.
Paper models Pavlov's classical conditioning using stochastic processes and Langevin equations.
problem Lack of modeling for Pavlov's classical conditioning.
method Modeling neural and synaptic dynamics via Langevin equations.
result Pavlov's mechanism spontaneously leads to synaptic weights similar to Hebb's.
Generative models learn latent process to match target distributions.
problem Training flow-matching models with auxiliary stochastic dynamics.
method Introduces latent process generator matching, treating generative state as a deterministic image of a Markov process.
result Learn generator of a stochastic process with same marginal distributions.
In this work we present a new approach on studying dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have research the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generaliz…
Algorithm ensures demographic parity in regression without sensitive attribute data.
problem Performing regression with demographic parity constraints.
method Post-processing algorithm using accurate estimates and sensitive attribute predictor.
result Generates predictions meeting demographic parity constraint.
Matrix H-theory models stock market fluctuations using hierarchical multivariate distributions.
problem Understanding collective behavior in stock market fluctuations.
method Matrix H-theory framework for multivariate stochastic processes with hierarchical structure.
result Matrix H-theory effectively describes stock market fluctuations using Meijer G-functions.
We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small ste…
In this short note, we will strengthen the classic Doob's Lp inequality for sub-martingale processes. Because this inequality is of fundamental importance to the theory of stochastic process, we believe this generalization will find many interesting applications.
Develops likelihood-based methods for trawl processes, improving forecasting accuracy.
problem Statistical modeling of trawl processes with heavy tails and long memory.
method Composite likelihood estimation as a stochastic optimization problem, using gradient descent methods.
result New gradient estimators with significantly reduced variance for trawl processes.
In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presen…
Quantum algorithm speeds up financial option pricing.
problem Optimizing stopping times in stochastic processes for finance.
method Combines quantum computing techniques with LSM for optimal stopping.
result Achieves nearly quadratic speedup in runtime.
Study on stochastic covariant derivatives in curved space-time.
problem Analyzing covariant derivatives in curved space-time under stochastic processes.
method Using Itô-Wiener processes and stochastic calculus, including Besov spaces, Schrödinger operators, and white noise.
result Developed a framework for stochastic geodesics and white noise in fractoid spaces.
We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We intr…
We are interested in understanding stability (almost sure boundedness) of stochastic approximation algorithms (SAs) driven by a `controlled Markov' process. Analyzing this class of algorithms is important, since many reinforcement learning (RL) algorithms can be cast as SAs driven by a `controlled Markov' process. In t…
This paper investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by \emph{general}, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a u…
We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued …
Stochastic encoders outperform deterministic ones in 'perfect perceptual quality'.
problem Understanding when stochastic encoders outperform deterministic ones.
method Provided a toy example to illustrate performance.
result Stochastic encoders can significantly outperform deterministic ones in 'perfect perceptual quality'.
The Continuous-Time Random Walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this article we will show how the random combination of two different unbiased CTRWs can give raise to a process with clear drift, if one of them is a CTRW with memory. If one identifies the other one as no…
Current economic theories miss most of economic dynamics.
problem Accuracy of economic theories and policies depend on economic variables and processes.
method Identify and analyze overlooked economic variables and processes.
result Many economic variables and processes not accounted for in current theories.