arXiv subjects

Xiyue Han

Publications and source records attributed to Xiyue Han.

14 recordsLinked to original sources

arXiv2025

Universal portfolios in continuous time: an approach in pathwise It\^o calculus

We provide a simple and straightforward approach to a continuous-time version of Cover's universal portfolio strategies within the model-free context of F\"ollmer's pathwise It\^o calculus. We establish the existence of the universal portfolio strategy and prove that its portfolio value process is the average of all values of constant rebalanced strategies. This result relies on a systematic comparison between two alternative descriptions of self-financing trading strategies within pathwise It\^o calculus. We moreover provide a comparison result for the performance and the realized volatility and variance of constant rebalanced portfolio strategies.

q-fin.MF↗

arXiv2025

On the rate of convergence of estimating the Hurst parameter of rough stochastic volatility models

In [Han \& Schied, 2023, \textit{arXiv 2307.02582}], an easily computable scale-invariant estimator $\widehat{\mathscr{R}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by proving that $\widehat{\mathscr{R}}^s_n$ also consistently estimates the Hurst parameter when applied to the antiderivative of $g \circ X$ for a general nonlinear function $g$. We also establish an almost sure rate of convergence in this general setting. Our result applies, in particular, to the estimation of the Hurst parameter of a wide class of rough stochastic volatility models from discrete observations of the integrated variance, including the rough fractional stochastic volatility model.

q-fin.ST↗

arXiv2024

Projecting Gaussian Ellipsoids While Avoiding Affine Projection Approximation

Recently, 3D Gaussian Splatting has dominated novel-view synthesis with its real-time rendering speed and state-of-the-art rendering quality. However, during the rendering process, the use of the Jacobian of the affine approximation of the projection transformation leads to inevitable errors, resulting in blurriness, artifacts and a lack of scene consistency in the final rendered images. To address this issue, we introduce an ellipsoid-based projection method to calculate the projection of Gaussian ellipsoid onto the image plane, which is the primitive of 3D Gaussian Splatting. As our proposed ellipsoid-based projection method cannot handle Gaussian ellipsoids with camera origins inside them or parts lying below $z=0$ plane in the camera space, we designed a pre-filtering strategy. Experiments over multiple widely adopted benchmark datasets show that our ellipsoid-based projection method can enhance the rendering quality of 3D Gaussian Splatting and its extensions.

cs.CV↗

arXiv2023

Estimating the roughness exponent of stochastic volatility from discrete observations of the integrated variance

We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. The estimator has a very simple form and can be computed with great efficiency on large data sets. It is not derived from distributional assumptions but from strictly pathwise considerations. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models, such as the rough fractional volatility model and the rough Bergomi model. We also demonstrate that our estimator is robust with respect to proxy errors between the integrated and realized variance, and that it can be applied to estimate the roughness exponent directly from the price trajectory. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

q-fin.ST↗

arXiv2023

A criterion for absolute continuity relative to the law of fractional Brownian motion

Let $X$ be the sum of a fractional Brownian motion with Hurst parameter $H$ and an absolutely continuous and adapted drift process. We establish a simple criterion that guarantees that the law of $X$ is absolutely continuous with respect to the law of the original fractional Brownian motion. For $H<1/2$, the trajectories of the derivative of the drift need to be bounded by an almost surely finite random variable; for $H>1/2$, they need to satisfy a H\"older condition with some exponent larger than $2H-1$. These are almost-sure conditions, and no expectation requirements are imposed. For the case in which $X$ arises as the solution of a nonlinear stochastic integral equation driven by fractional Brownian motion, we provide a simple criterion on the drift coefficient under which the law of $X$ is automatically equivalent to the one of fractional Brownian motion.

math.PR↗

arXiv2022

Robust Faber--Schauder approximation based on discrete observations of an antiderivative

We study the problem of reconstructing the Faber--Schauder coefficients of a continuous function $f$ from discrete observations of its antiderivative $F$. For instance, this question arises in financial mathematics when estimating the roughness of volatility from the integrated volatility of an asset price trajectory. Our approach starts with mathematically formulating the reconstruction problem through piecewise quadratic spline interpolation. We then provide a closed-form solution and an in-depth error analysis. These results lead to some surprising observations, which also throw new light on the classical topic of quadratic spline interpolation itself: They show that the well-known instabilities of this method can be located exclusively within the final generation of estimated Faber--Schauder coefficients, which suffer from non-locality and strong dependence on the initial value. By contrast, all other Faber--Schauder coefficients depend only locally on the data, are independent of the initial value, and admit uniform error bounds. We thus conclude that a robust and well-behaved estimator for our problem can be obtained by simply dropping the final-generation coefficients from the estimated Faber--Schauder coefficients.

math.NA↗

arXiv2022

An ultra-high gain single-photon transistor in the microwave regime

A photonic transistor that can switch or amplify an optical signal with a single gate photon requires strong non-linear interaction at the single-photon level. Circuit quantum electrodynamics provides great flexibility to generate such an interaction, and thus could serve as an effective platform to realize a high-performance single-photon transistor. Here we demonstrate such a photonic transistor in the microwave regime. Our device consists of two microwave cavities dispersively coupled to a superconducting qubit. A single gate photon imprints a phase shift on the qubit state through one cavity, and further shifts the resonance frequency of the other cavity. In this way, we realize a gain of the transistor up to 53.4 dB, with an extinction ratio better than 20 dB. Our device outperforms previous devices in the optical regime by several orders in terms of optical gain, which indicates a great potential for application in the field of microwave quantum photonics and quantum information processing.

quant-ph↗

arXiv2022

Experimental preparation of generalized cat states for itinerant microwave photons

Generalized cat states represent arbitrary superpositions of coherent states, which are of great importance in various quantum information processing protocols. Here we demonstrate a versatile approach to creating generalized itinerant cat states in the microwave domain, by reflecting coherent state photons from a microwave cavity containing a superconducting qubit. We show that, with a coherent control of the qubit state, a full control over the coherent state superposition can be realized. The prepared cat states are verified through quantum state tomography of the qubit state dependent reflection photon field. We further quantify quantum coherence in the prepared cat states based on the resource theory, revealing a good experimental control on the coherent state superpositions. The photon number statistic and the squeezing properties are also analyzed. Remarkably, fourth-order squeezing is observed in the experimental states. Those results open up new possibilities of applying generalized cat states for the purpose of quantum information processing.

quant-ph↗

arXiv2022

A flying Schrödinger cat in multipartite entangled states

Schrödinger's cat originates from the famous thought experiment querying the counterintuitive quantum superposition of macroscopic objects. As a natural extension, several "cats" (quasi-classical objects) can be prepared into coherent quantum superposition states, which is known as multipartite cat states demonstrating quantum entanglement among macroscopically distinct objects. Here we present a highly scalable approach to deterministically create flying multipartite Schrödinger cat states, by reflecting coherent state photons from a microwave cavity containing a superconducting qubit. We perform full quantum state tomography on the cat states with up to four photonic modes and confirm the existence of quantum entanglement among them. We also witness the hybrid entanglement between discrete-variable states (the qubit) and continuous-variable states (the flying multipartite cat) through a joint quantum state tomography. Our work demonstrates an important experimental control method in the microwave region and provides an enabling step for implementing a series of quantum metrology and quantum information processing protocols based on cat states.

quant-ph↗

arXiv2021

A limit theorem for Bernoulli convolutions and the $Φ$-variation of functions in the Takagi class

We consider a probabilistic approach to compute the Wiener--Young $Φ$-variation of fractal functions in the Takagi class. Here, the $Φ$-variation is understood as a generalization of the quadratic variation or, more generally, the $p^{\text{th}}$ variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions $Φ$ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function $Φ$, our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear $Φ$-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear $Φ$-variation for each function $Φ$ from our class. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.

math.PR↗

arXiv2021

The roughness exponent and its model-free estimation

Motivated by pathwise stochastic calculus, we say that a continuous real-valued function $x$ admits the roughness exponent $R$ if the $p^{\text{th}}$ variation of $x$ converges to zero if $p>1/R$ and to infinity if $p<1/R$. For the sample paths of many stochastic processes, such as fractional Brownian motion, the roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber--Schauder coefficients of $x$ under which the roughness exponent exists and is given as the limit of the classical Gladyshev estimates $\widehat R_n(x)$. This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because it works strictly trajectory-wise and requires no probabilistic assumptions. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber--Schauder expansion of $x$. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence $(\widehat R_n)_{n\in\mathbb N}$. We also discuss how a dynamic change in the roughness parameter of a time series can be detected. Finally, we extend our results to the case in which the $p^{\text{th}}$ variation of $x$ is defined over a sequence of unequally spaced partitions. Our results are illustrated by means of high-frequency financial time series.

math.ST↗

arXiv2021

A Gladyshev theorem for trifractional Brownian motion and $n$-th order fractional Brownian motion

We prove limit theorems for the weighted quadratic variation of trifractional Brownian motion and $n$-th order fractional Brownian motion. Furthermore, a sufficient condition for the $L^P$-convergence of the weighted quadratic variation for Gaussian processes is obtained as a byproduct. As an application, we give a statistical estimator for the self-similarity index of trifractional Brownian motion. These theorems extend results of Baxter, Gladyshev, and Norvaiša.

math.PR↗

arXiv2020

A probabilistic approach to the $Φ$-variation of classical fractal functions with critical roughness

We consider Weierstraß and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$ variation for all $p>1$ but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener--Young $Φ$-variation along the sequence of $b$-adic partitions, where $Φ(x)=x/\sqrt{-\log x}$. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi--van der Waerden functions, we use the CLT for Markov chains if a certain parameter $b$ is odd, and the standard CLT for $b$ even.

math.PR↗

arXiv2020

Step roots of Littlewood polynomials and the extrema of functions in the Takagi class

We give a new approach to characterizing and computing the set of global maximizers and minimizers of the functions in the Takagi class and, in particular, of the Takagi--Landsberg functions. The latter form a family of fractal functions $f_α:[0,1]\to\mathbb R$ parameterized by $α\in(-2,2)$. We show that $f_α$ has a unique maximizer in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $α$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi--Landsberg functions with $α\in(-2,1/2]\cup(1,2)$. For $(1/2,1]$, we show that the step roots are dense in that interval. If $α\in (1/2,1]$ is a step root, then the set of maximizers of $f_α$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$, where $n$ is the degree of the minimal Littlewood polynomial that has $α$ as its step root. In the same way, we determine explicitly the minima of all Takagi--Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$.

math.CA↗