Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy tailed jumps, and the time-fractional version codes heavy tailed waiting times. Thi…
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48 results for “random fractions”
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and re…
We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly-decaying autocorrelation functions. The relation between these correlated walks and the well-kno…
Motivated by applications to insurance mathematics, we prove some heavy-traffic limit theorems for processes which encompass the fractionally differentiated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution.
Motivated by applications to insurance mathematics, we prove some heavy-traffic limit theorems for process which encompass the fractionally integrated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution.
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…
Study models market volatility with persistent and temporary impacts.
problem Microstructure of rough volatility models driven by Poisson measures.
method Existence and uniqueness of solutions for stochastic path-dependent Volterra equations.
result Volatility process converges to fractional Heston model with spikes.
New neural operators model turbulence with memory and randomness.
problem Modeling turbulence in complex fluid dynamics with memory and randomness.
method Symmetrized activation functions, fractional derivatives, and stochastic noise.
result Theoretical guarantees for approximation quality in turbulent phenomena.
Study shows how certain stochastic models reach a steady state over time.
problem Understanding long-term behavior of stochastic volatility models.
method Novel coupling technique for Markov chains, applicable to random environments.
result Convergence to an invariant measure for multidimensional fractional models.
This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…
New formulas derived for variance gamma model option pricing.
problem Option pricing for the variance gamma model.
method Combining randomization method and fractional derivatives.
result Closed-form formulas for European options.
A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.
This overview article concerns the notion of fractional smoothness of random variables of the form , where is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance main…
BCD-Net improves PET image reconstruction in low-count scenarios.
problem Low-count PET imaging challenges due to high random fractions and low SNR.
method Modified BCD-Net architecture for iterative neural network-based PET image reconstruction.
result BCD-Net significantly improves CNR and RMSE of reconstructed images compared to traditional methods.
Randomized neural networks improve optimal stopping problems efficiently.
problem Approximating solutions to optimal stopping problems in high dimensions.
method Randomly generated neural networks with only the last layer trained.
result Our methods outperform state-of-the-art approaches in computation time and accuracy.
Random covers of surfaces have tangle-free monodromy and the Putman-Wieland property.
problem Understanding the properties of random covers of surfaces.
method Analyzing the fraction of degree covers of a surface with specific properties as increases.
result The fraction of degree covers of a surface with the Putman-Wieland property tends to 1 as approaches infinity.
Random forest hyperparameters affect variable selection in omics studies.
problem Impact of hyperparameters on variable selection in random forests.
method Two simulation studies using theoretical and empirical data.
result Hyperparameters influence variable selection more than the splitting strategy and sample fraction.
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 can be theoretically determined, aiding in forecasting future price increments.
Stochastic Maps, Wealth Distribution in Random Asset Exchange Models and the Marginal Utility of Relative Wealthcond-mat.stat-mech
We look at how asset exchange models can be mapped to random iterated function systems (IFS) giving new insights into the dynamics of wealth accumulation in such models. In particular, we focus on the "yard-sale" (winner gets a random fraction of the poorer players wealth) and the "theft-and-fraud" (winner gets a rando…
Fast simulates Volterra processes using RFF, focusing on S-fBM.
problem Efficiently simulate Volterra processes for fractional Brownian motion.
method Random Fourier Features (RFF) approximation of kernel, spectral representation, Hamiltonian Monte Carlo sampling.
result Quantitative guarantees for RFF approximation, competitive in terms of efficiency and error.
A random forest is a popular tool for estimating probabilities in machine learning classification tasks. However, the means by which this is accomplished is unprincipled: one simply counts the fraction of trees in a forest that vote for a certain class. In this paper, we forge a connection between random forests and ke…
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.
We study the effect of altruism in two simple asset exchange models: the yard sale model (winner gets a random fraction of the poorer player's wealth) and the theft and fraud model (winner gets a random fraction of the loser's wealth). We also introduce in these models the concept of bargaining efficiency, which makes …
Single gradient step finds adversarial examples in random neural networks.
problem Finding adversarial examples in neural networks with random architectures.
method Gradient descent approach applied to random undercomplete and overcomplete two-layers neural networks.
result A single gradient step is sufficient to find adversarial examples in random neural networks.
It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting …
We complement the theory of tick-by-tick dynamics of financial markets based on a Continuous-Time Random Walk (CTRW) model recently proposed by Scalas et al., and we point out its consistency with the behaviour observed in the waiting-time distribution for BUND future prices traded at LIFFE, London.
Long memory stochastic volatility in option pricingcond-mat.other
The aim of this paper is to present a simple stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black-Scholes equation involving volatility with long-range dependence. We consider the option price as a sum of classical Black-Scholes pric…
New formula refutes random CSPs with fewer constraints.
problem Refuting random constraint satisfaction problems efficiently.
method Introduced a non-backtracking matrix and proved an Ihara-Bass formula.
result Efficiently refutes random CSPs with fewer constraints.
Linear models can be poisoned by shifting a fraction of one class's data, revealing scaling laws and weight alignment.
problem Understanding and quantifying data poisoning in linear models.
method Analysis of ridge least squares with an unpenalized intercept, using resolvent techniques and random matrix theory.
result Closed-form limits for the poisoned score, revealing scaling laws and weight alignment with the poisoning direction.
The study examines the chaos of fractional Brownian fields as Hurst parameter approaches zero.
problem Understanding the chaos of fractional Brownian fields as their Hurst parameter tends to zero.
method Defining normalizing kernels and using Berestycki's ``good points'' approach to derive the limiting measure of multiplicative chaos.
result The limiting measure of multiplicative chaos converges to a log-correlated Gaussian field as the Hurst parameter approaches zero.
New spectral algorithm estimates random graph parameters robustly against corrupted nodes.
problem Estimating the parameter of an Erdős-Rényi random graph with adversarial corruption.
method Spectral algorithm designed for computational efficiency, with an inefficient but information-theoretic alternative.
result Achieves optimal error rate up to logarithmic factors, matching statistical lower bounds.
Study financial contagion and risk in sparse networks with directed edges.
problem Analyzing systemic risk in sparse financial networks with balance-sheet interactions.
method Linear fraction of institutions with zero out-degree, sender-truncated subgraph G_sh, adversarial and random systemic events, explicit fan-in accumulation bound.
result Maximal forward reachability in G_sh is O(log n) with high probability in the subcritical regime, and multi-hit defaults are negligible in the supercritical regime.
ROCKET speeds up time series classification without sacrificing accuracy.
problem High computational complexity and intractability of existing time series classification methods.
method Simple linear classifiers using random convolutional kernels.
result Achieves state-of-the-art accuracy with significantly reduced computational expense.
Assessing the performance of a learned model is a crucial part of machine learning. However, in some domains only positive and unlabeled examples are available, which prohibits the use of most standard evaluation metrics. We propose an approach to estimate any metric based on contingency tables, including ROC and PR cu…
The performance of the Self-Organizing Map (SOM) algorithm is dependent on the initial weights of the map. The different initialization methods can broadly be classified into random and data analysis based initialization approach. In this paper, the performance of random initialization (RI) approach is compared to that…
Gradient descent amplifies random features in neural networks to useful ones.
problem Generalization in neural networks trained on corrupted data.
method Characterization of feature-learning process in two-layer ReLU networks trained by gradient descent.
result Gradient descent amplifies random features to useful ones, achieving near optimal generalization error.
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…
Portfolio Optimization under Fast Mean-reverting and Rough Fractional Stochastic Environmentq-fin.MF
Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price volatility: both fast-time scale on the order of days and slow-scale on the order of…
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…
Modeling financial networks to predict systemic crises.
problem Predicting systemic financial crises in complex networks.
method Developed inhomogeneous random financial networks (IRFNs) to model bank interactions.
result Found a condition for a locally tree-like independence (LTI) property, leading to fixed point equations for system equilibrium.
New method solves optimal stopping problems using rough path signatures.
problem Optimal stopping problems in finance and other fields.
method Using rough path signatures and deep neural networks.
result Solves optimal stopping problems efficiently under minimal assumptions.
Estimation of functions of variables is considered using ridge combinations of the form where the activation function is a function with bounded value and derivative. These include single-hidden layer neural networks, polynomials, …
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.
Introduces fractional k-dimensional measure bridging fractional length and area.
problem Defining fractional measures for dimensions between 0 and n-1.
method Introduces a parameterized fractional measure that converges to Hausdorff measure.
result Fractional measure converges to Hausdorff measure with a known constant factor.
The problem of hierarchical clustering items from pairwise similarities is found across various scientific disciplines, from biology to networking. Often, applications of clustering techniques are limited by the cost of obtaining similarities between pairs of items. While prior work has been developed to reconstruct cl…
The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the fractional tangent bundle on a manifold, using a method of Radu Miron. The fractional Lei…
This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when the natura…
New method for robust regression with near-optimal performance even with high corruption rates.
problem Robust linear regression with response variable corruptions.
method Adaptive hard thresholding for consistent estimation.
result Near-optimal consistent estimation of the true regression vector with fraction of corruptions.