Derives an option-pricing formula for fractional markets with skew and smile.
problem Developing a pricing formula for financial options with skew and smile.
method Employed the Lévy-Khintchine theorem and fractional Gaussian noise to generalize the Black-Scholes-Merton formula.
result An exponentially convergent option-pricing formula for fractional markets.
Study large deviations in fractional volatility models with non-Gaussian volatility.
problem Large deviations in fractional volatility models with non-Gaussian volatility.
method Established a small-noise large deviation principle for log-price.
result Logarithmic call price asymptotics for large strikes in a special case.
Fractional porous media equations yield q-Gaussian solutions for stock price returns.
problem Modeling stock price returns using fractional porous media equations.
method Analyzed three types of fractional extensions of the porous media equation.
result Local and non-local fractional extensions fit S&P 500 data better than classical models.
Researchers define a limit for fractional Brownian motion as Hurst parameter approaches zero.
problem Defining a limit for fractional Brownian motion with zero Hurst parameter.
method Developed a Gaussian random distribution and log-correlated random field as limits.
result Fractional Brownian motion converges to a Gaussian random distribution when Hurst parameter approaches zero.
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Beth…
Machine learning discovers fractional differential equations from data.
problem Discovering hidden physics in data-driven models of fractional differential equations.
method Gaussian Process regression with modified physics-informed kernel for space-fractional equations.
result Optimized machine learning of fractional differential equations, including heavy-tailed systems.
The study examines hedging strategies in financial models with transaction costs.
problem Hedging strategies under transaction costs in financial models.
method Analysis of Gaussian Volterra processes and conditional-mean hedging.
result Derivation of a formula for prediction laws in Gaussian Volterra processes.
Composite likelihood inference of fractional Gaussian processes with sequentially optimal subset selection
problem Estimating parameters in time series
method Composite likelihood method
result The method reduces computational cost
Study large deviation principle for fractional stochastic volatility models.
problem Large deviation principle for Volterra type fractional stochastic volatility models.
method Prove a small-noise large deviation principle under weaker conditions.
result Derive large deviation principle in small-time regime.
Unified analysis of Gaussian Process Thompson Sampling without discretization.
problem Sequential decision-making over continuous action spaces.
method Frequentist regret analysis based on fractional Gaussian process posteriors.
result Unified discretization-free regret bound for various kernel classes.
The so-called level crossing analysis has been used to investigate the empirical data set. But there is a lack of interpretation for what is reflected by the level crossing results. The fractional Gaussian noise as a well-defined stochastic series could be a suitable benchmark to make the level crossing findings more s…
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. As an extension of Welling and Teh (2001), we define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals and derive an …
Study shows stock price is a martingale if volatility's driving Brownian motion is negatively correlated with the stock.
problem Determining the martingale property of stock prices in fractional stochastic volatility models.
method Analyzed a class of fractional stochastic volatility models, including the rough Bergomi model, focusing on the correlation between stock and volatility.
result The stock price is a true martingale if and only if the correlation between the driving Brownian motions of the stock and the volatility is nonpositive.
Extends fractional Lp uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional Lp settings. method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
New method uses fractional posteriors for semiparametric inference with improved uncertainty quantification.
problem Semiparametric inference with nonparametric priors and fractional posteriors.
method Established a general Bernstein--von Mises theorem for fractional posterior distributions, proposed shifted-and-rescaled credible sets.
result Fractional posterior credible sets provide reliable uncertainty quantification but have inflated size; shifted-and-rescaled set is an efficient confidence set.
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…
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.
Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.
problem Characterizing geodesic flows on surfaces with fractional-linear integrals.
method Proving the dimension of fractional-linear integrals and giving a geometric criterion.
result The dimension of fractional-linear integrals is either 3 or 5, corresponding to constant curvature.
Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementatio…
Study fractional perimeter asymptotics on Riemannian manifolds as s approaches 0.
problem Asymptotics of fractional perimeter on Riemannian manifolds.
method Analysis of fractional Laplacian and existence of bounded harmonic functions.
result Asymptotics of fractional s-perimeter on all complete manifolds. The study examines the behavior of Gaussian processes' minimums and overshoots.
problem Understanding the behavior of Gaussian processes' minimums and overshoots.
method Analyzing conditional distributions and subsequential limits of minimizers.
result The scaled overshoot converges to an exponential random variable with mean σ_*^2.
We consider the problem of estimating the mean and covariance of a distribution from iid samples in Rn, in the presence of an η fraction of malicious noise; this is in contrast to much recent work where the noise itself is assumed to be from a distribution of known type. The agnostic problem includes many…
The paper introduces a new stochastic volatility model with long-term memory and jumps.
problem Developing a model for variance and volatility swaps with long-term memory and jumps.
method Fractional Barndorff-Nielsen and Shephard model incorporating long-term memory and jumps.
result Arbitrage-free prices for variance and volatility swaps derived for the new model.
Study approximates weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
problem Approximating weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
method Used Euler type scheme with integrated kernels to study weak convergence rate.
result Obtained weak convergence rate of min(3α−1,1) for discretised rough Ornstein-Uhlenbeck process and stochastic rough volatility model. A new method calculates fractional moments using the moment-generating function.
problem Computing fractional moments from probability densities.
method Integral framework based on moment-generating function.
result Exact integral expressions for various types of moments.
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 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 framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
Researchers derive an analytic expression for Gaussian stochastic volatility models.
problem Analyzing rich autocorrelation structures and persistence in financial markets.
method Two different analytic derivations of the joint characteristic function.
result First analytic formulae for option pricing in rough volatility models.
Develops a framework to apply Kelly criterion to stock markets using probability distributions.
problem Applying Kelly criterion to stock market investments with varying probability distributions.
method Calculates Kelly fractions for stocks using an arbitrary probability distribution, involving only first and second moments.
result Agrees with existing results for geometric Brownian motion and can be applied to other distributions.
New rough stochastic volatility models using log-modulated fractional Brownian motion.
problem Analyzing rough stochastic volatility models over the range 0≤H<1/2. method Introducing log-modulated fractional Brownian motion (log-fBm) to handle H=0 and analyze over the full range. result Obtained skew asymptotics of log(1/T)−pTH−1/2 as To0 for H≥0, no flattening of skew as Ho0. A new distribution family extends the α-stable distribution with a degree of freedom parameter.
problem Lack of moments in the α-stable distribution. method Wright function framework to combine and extend distribution families.
result Generalized α-stable distribution with valid moments. The paper introduces a new method to detect rough volatility and market states using fractional derivatives.
problem Testing self-similarity in fractional processes from a single observed trajectory is difficult under long-range dependence.
method The paper introduces a regime-adaptive KS/GL--KS framework based on the discrete Grünwald--Letnikov (GL) fractional derivative.
result The method detects rough volatility and persistent, anti-persistent, or efficient market states in financial applications.
New method learns fractional order of PDEs from flocking particle simulations.
problem Deriving effective nonlocal influence functions from discrete agent-based models.
method Agent-based model, fractional PDEs, Gaussian process regression, Bayesian optimization.
result Learned Euler equations accurately predict flocking behavior.
Develops a new framework for drawdown risk beyond Gaussian assumptions.
problem Understanding drawdowns in systematic trading strategies.
method Monte-Carlo simulation, non-Gaussian extensions, fractional Brownian motion.
result Drawdowns and related measures vary differently under non-Gaussian assumptions.
New robust regression method works with fewer data points than previous methods.
problem Adversary can corrupt most of the data, making traditional regression models unreliable.
method Developed a Huber loss estimator for robust linear regression with nearly linear sample size and inverse-polynomial inlier fraction.
result The Huber loss estimator is consistent for nearly linear sample size and inverse-polynomial inlier fraction.
This paper provides an algorithm for simulating improper (or noncircular) complex-valued stationary Gaussian processes. The technique utilizes recently developed methods for multivariate Gaussian processes from the circulant embedding literature. The method can be performed in O(nlog2n) operations, where…
Method predicts LFSM increments from past observations using codifference.
problem Forecasting LFSM increments from discrete-time observations.
method Uses codifference for serial dependence, with conditional expectation or projection for α>1 or α<2. result Method shows promising performance in forecasting volatilities, capturing kurtosis and serial dependence.
Study analyzes FLMC for non-convex optimization with finite-time bounds.
problem Non-convex optimization challenges in machine learning.
method Fractional Langevin Monte Carlo (FLMC) with α-stable noise.
result Finite-time bounds for expected suboptimality of FLMC.
The paper models cryptocurrency price and volatility with jumps and fractional volatility.
problem Empirical evidence shows jumps in cryptocurrency price and volatility.
method Fractional stochastic volatility model with jumps and short-term volatility dependency.
result Fractional stochastic volatility models outperform other models in pricing and hedging cryptocurrency options.
Adaptive Gaussian process models for efficient Bayesian inference.
problem Expensive forward models in Bayesian inference.
method Fully Bayesian approach with adaptive training designs maximizing expected improvement.
result Adaptive designs lead to more accurate posterior estimation at lower cost.
Algorithm estimates mixtures of arbitrary Gaussians robustly in presence of corruptions.
problem Estimating mixtures of arbitrary Gaussians in the presence of a constant fraction of arbitrary corruptions.
method Polynomial-time algorithm using partial clustering and tensor decomposition.
result Resolves the main open problem in several previous works on algorithmic robust statistics.
This work leverages recent advances in probabilistic machine learning to discover conservation laws expressed by parametric linear equations. Such equations involve, but are not limited to, ordinary and partial differential, integro-differential, and fractional order operators. Here, Gaussian process priors are modifie…
Research on long-range memory in financial and social systems using various models.
problem Understanding the nature of long-range memory in socioeconomic systems.
method Various Markov processes including point processes, stochastic differential equations, and agent-based models.
result New estimators of self-similarity and long-range memory for non-Gaussian systems are needed.
The paper studies the expected number of nodal components for fractional Gaussian fields on manifolds.
problem Estimating the expected number of nodal components for cut-off fractional Gaussian fields.
method Analyzes the behavior of the number of connected components of the zero set of a specific type of Gaussian fields on manifolds.
result The expected number of nodal components is shown to behave differently depending on the value of the parameter s. We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an ε-fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known…
A novel approach calibrates Gaussian process for fast large-scale classification.
problem Deriving fast and accurate classification algorithms with uncertainty quantification.
method Applying Gaussian process regression to classification labels and calibrating predictions.
result The proposed approach provides similar accuracy and uncertainty quantification as Gaussian process classification but with significantly reduced computational resources.
Traditionally, there are three species of classification: unsupervised, supervised, and semi-supervised. Supervised and semi-supervised classification differ by whether or not weight is given to unlabelled observations in the classification procedure. In unsupervised classification, or clustering, all observations are …