This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson proc…
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48 results for “fractional processes”
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…
Deep learning the Hurst parameter of linear fractional processes and assessing its reliabilitystat.ML
Deep learning improves Hurst parameter estimation for fractional processes.
problem Estimating the Hurst parameter in fractional stochastic processes.
method Training Long Short-Term Memory (LSTM) networks on extensive datasets of fBm, fOU, and lfsm processes.
result LSTM outperforms traditional methods in fBm and fOU processes but has limited accuracy on lfsm.
Approximates derivative pricing under fractional stochastic volatility.
problem Derivative pricing under fractional stochastic volatility model.
method Approximate expression derived from deterministic functions and fractional Ornstein-Uhlenbeck process.
result Numerical simulations show the feasibility and effect of long-range dependencies on derivative prices.
Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and practical difficulties in computation and calibration. To address these issues, we…
The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poiss…
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.
Many fractional processes can be represented as an integral over a family of Ornstein-Uhlenbeck processes. This representation naturally lends itself to numerical discretizations, which are shown in this paper to have strong convergence rates of arbitrarily high polynomial order. This explains the potential, but also s…
Volatility roughness studied using fractional noise-driven models.
problem Volatility roughness interpretation.
method Data-reconstructed fractional volatility model with fractional noise.
result Option pricing equation and solution derived using Malliavin calculus.
Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(…
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 . This process has sta…
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.
Defines a new process for financial modeling.
problem Developing a new stochastic process for financial applications.
method Introduces a fractional Cox-Ingersoll-Ross process and proves its properties.
result The process has unique solutions and is strictly positive for certain Hurst parameters.
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.
The sub-fractional CEV modelq-fin.MF
A new model captures option price dynamics using sub-fractional Brownian motion.
problem Capturing the complex price dynamics of financial options.
method Developed a CEV model driven by a mixed sub-fractional Brownian motion.
result Empirical tests show the model effectively captures option price dynamics.
The continuous observation of the financial markets has identified some stylized facts which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationa…
We construct a new process using a fractional Brownian motion and a fractional Ornstein-Uhlenbeck process of the Second Kind as building blocks. We consider the increments of the new process in discrete time and, as a result, we obtain a more parsimonious process with similar autocovariance structure to that of a FARIM…
Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.
problem Volatility uncertainty in fractional Brownian motion.
method Definition and study of multi-dimensional fractional Brownian motion (G-fBm) with Hurst index.
result First results on stochastic calculus for G-fBm with Hurst index > 0.5.
We survey some new progress on the pricing models driven by fractional Brownian motion \cb{or} mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with trans…
In this paper we apply Markovian approximation of the fractional Brownian motion (BM), known as the Dobric-Ojeda (DO) process, to the fractional stochastic volatility model where the instantaneous variance is modelled by a lognormal process with drift and fractional diffusion. Since the DO process is a semi-martingale,…
The paper evaluates integrals for fBm with various Hurst indices.
problem Evaluating integrals for stochastic processes with fractional Brownian motion for different Hurst indices.
method Analytic continuation from complex analysis to extend integral domain.
result Integral formulas for fBm with Hurst indices are derived.
We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black-Scho…
Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processescond-mat.stat-mech
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is…
Score-fPINN tackles high-dimensional FPL equations using fractional score functions.
problem High-dimensional Fokker-Planck-Lévy equations with non-Brownian processes.
method Fractional score function and Physics-informed neural networks (PINN) to solve CoD and numerical overflow.
result Effective solution to high-dimensional FPL equations without fractional Laplacian.
We study fractional stochastic volatility models in which the volatility process is a positive continuous function of a continuous Gaussian process . Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function is globally…
Study provides LDP for non self-similar stochastic volatility models.
problem Analyzing non self-similar stochastic volatility models.
method Short-time large deviation principle (LDP) for models with Volterra process.
result Derives consequences for option prices, implied volatility surfaces, and skew.
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.
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 -th variation for estimating roughness of sample paths.
result Realized volatility exhibits rough behavior with a significantly smaller Hurst exponent than instantaneous volatility.
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.
We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical …
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 or .
result Method shows promising performance in forecasting volatilities, capturing kurtosis and serial dependence.
Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.
problem Fractional isoperimetric inequality and its quantitative aspects.
method Regularization process with a new spirit.
result Stability estimates for fractional Cheeger inequality.
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.
Formula for option pricing in a stochastic volatility model with jumps.
problem Developing a formula for European option pricing in a complex stochastic volatility model.
method Fractional integral of a diffusion process, martingale representation, and Itô calculus for processes with jumps.
result A first-order approximation formula for option prices.
RP-GFRFT unifies fractional order and rotation control for graph signals.
problem Lack of rotation-based spectral control in GFRFT and zero-angle degeneracy in AGFT.
method Rotation-parameterized graph fractional Fourier transform (RP-GFRFT) with degeneracy preserving rotation matrix.
result RP-GFRFT improves spectral filtering performance over existing methods.
Study approximates rough stochastic volatility models using diffusion processes.
problem High computational cost in simulating rough stochastic volatility models.
method Approximates stochastic Volterra equations with an N-dimensional diffusion process.
result Approximations converge strongly with superpolynomial rate in N.
Deep learning predicts path-dependent processes from historical data.
problem Predicting path-dependent processes using historical data.
method Nonparametric regression with deep neural networks.
result Deep learning method converges to theoretical predictions as observation frequency increases.
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.
Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a…
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…
A Gaussian Markov alternative to fractional Brownian motion for pricing financial derivativesq-fin.MF
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 pricing derivatives in markets with long-range dependence and jumps.
problem Deriving pricing formulas for derivatives in markets with long-range dependence and jumps.
method Developed a fractional integro-partial differential equation (PIDE) and used semigroup theory and finite-difference schemes for numerical solutions.
result Closed-form pricing formula for European options and numerical solution for general options.
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 consider a fractional version of the Heston volatility model which is inspired by [16]. Within this model we treat portfolio optimization problems for power utility functions. Using a suitable representation of the fractional part, followed by a reasonable approximation we show that it is possible to cast the proble…
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.
While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessaril…