arXiv research

Let $S_g$ be a closed orientable surface of genus $g \geq 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A \textit{fractional power} of $t_C$ of \textit{exponent} $\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that $h^n = t_C^{\ell}$. Unlike a root of a $t…

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(…

Study solves inverse problems for equations with fractional nonlinearities.

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…

The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stoc…

Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.

The purpose of this paper is to indicate that the recently proposed Momentum fractional least mean squares (mFLMS) algorithm has some serious flaws in its design and analysis. Our apprehensions are based on the evidence we found in the derivation and analysis in the paper titled: \textquotedblleft \textit{Momentum frac…

Accelerators with power-law memory are proposed in the framework of the discrete time approach. To describe discrete accelerators we use the capital stock adjustment principle, which has been suggested by Matthews.The suggested discrete accelerators with memory describe the economic processes with the power-law memory …

Unified treatment of two extension problems using heat equation in Heisenberg group.

Sharp fractional Sobolev inequalities on closed manifolds identified.

The Black-Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time-to-maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian mot…

In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise d…

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

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…

We solve a family of fractional Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility beha…

FPG uses fractional calculus for efficient reinforcement learning with long-term memory.

The paper models cryptocurrency price and volatility with jumps and fractional volatility.

Develops nonparametric regression for non-smooth functions using fractional Laplacian.

The study examines order flow in financial markets using fractional Lévy stable motion.

Analyzed a generalized voter model with power-law herding intensity, revealing anomalous diffusion and long-range memory.

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A yardsale(YS) model in w…

Estimation of functions of $d$ variables is considered using ridge combinations of the form $\textstyle\sum_{k=1}^m c_{1,k} φ(\textstyle\sum_{j=1}^d c_{0,j,k}x_j-b_k)$ where the activation function $φ$ is a function with bounded value and derivative. These include single-hidden layer neural networks, polynomials, …

L2O-CFGD meta-learns hyperparameters for FGD, improving performance.

The paper provides approximations for pricing Asian options using a mixed fractional Brownian motion with jumps.

Extends rough Heston model solution to general λ.

New rough stochastic volatility models using log-modulated fractional Brownian motion.

Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.

We revisit the optimal investment and consumption problem with proportional transaction costs. We prove that both the value function and the slopes of the lines demarcating the no-trading region are analytic functions of cube root of the transaction cost parameter. Also, we can explicitly calculate the coefficients of …

It is shown phenomenologically that the fractional derivative $ξ=D^αu$ of order $α$ of a multifractal function has a power-law tail $\propto |ξ| ^{-p_\star}$ in its cumulative probability, for a suitable range of $α$'s. The exponent is determined by the condition $ζ_{p_\star} = αp_\star$, where $ζ_p$ is the exponent of…

Rough stochastic volatility models have attracted a lot of attentions recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-M…

We develop a variational framework for SDEs driven by fractional noise.

Study on Bitcoin transaction flows and holding times, revealing multifractal and power-law distributions.

Improves probability distribution compression with KT algorithm.

Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Lap…

We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$ satisfying the condi…

Study models market volatility with persistent and temporary impacts.

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method,…

New neural operators model turbulence with memory and randomness.

New model for pricing volatility derivatives considering rough volatility and jumps.

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using classic detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover…

We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known…

A new distribution family extends the $\alpha$-stable distribution with a degree of freedom parameter.

We analyse the dynamics of the Warsaw Stock Exchange index WIG at a daily time horizon before and after its well defined local maxima of the cusp-like shape decorated with oscillations. The rising and falling paths of the index peaks can be described by the Mittag-Leffler function superposed with various types of oscil…

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…

While increasingly deep networks are still in general desired for achieving state-of-the-art performance, for many specific inputs a simpler network might already suffice. Existing works exploited this observation by learning to skip convolutional layers in an input-dependent manner. However, we argue their binary deci…

Modeling financial markets with a novel order flow model.

Empirical studies indicate the existence of long range dependence in the volatility of the underlying asset. This feature can be captured by modeling its return and volatility using functions of a stationary fractional Ornstein--Uhlenbeck (fOU) process with Hurst index $H \in (\frac{1}{2}, 1)$. In this paper, we analyz…

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…