arXiv subjects

Yanghui Liu

Publications and source records attributed to Yanghui Liu.

At least 19 recordsLinked to original sources

arXiv2025

Riemann-Skorohod and Stratonovich integrals for Gaussian processes

In this paper we consider Skorohod and Stratonovich-type integrals in a general setting of Gaussian processes. We show that a conversion formula holds when the covariance functions of the Gaussian process are of finite $\rho$-variation for $\rho\geq 1$ and that the diagonals of covariance functions are of finite $\rho'$-variation for $\rho'\geq 1$ such that $\frac{1}{\rho'}+\frac{1}{2\rho}>1$. The difference between the two types of integrals is identified with a Young integral. We also show that the Skorohod integral is the limit of a $[\rho]$-th order Skorohod-Riemann sum.

math.PR↗

arXiv2024

Electrical detection in two-terminal perpendicularly magnetized devices via geometric anomalous Nernst effect

The non-uniform current distribution arisen from either current crowding effect or hot spot effect provides a method to tailor the interaction between thermal gradient and electron transport in magnetically ordered systems. Here we apply the device structural engineering to realize an in-plane inhomogeneous temperature distribution within the conduction channel, and the resulting geometric anomalous Nernst effect (GANE) gives rise to a non-zero 2nd -harmonic resistance whose polarity corresponds to the out-of-plane magnetization of Co/Pt multi-layer thin film, and its amplitude is linearly proportional to the applied current. By optimizing the aspect ratio of convex-shaped device, the effective temperature gradient can reach up to 0.3 K/$\mu$m along the y-direction, leading to a GANE signal of 28.3 $\mu$V. Moreover, we demonstrate electrical write and read operations in the perpendicularly-magnetized Co/Pt-based spin-orbit torque device with a simple two-terminal structure. Our results unveil a new pathway to utilize thermoelectric effects for constructing high-density magnetic memories

cond-mat.mes-hall↗

arXiv2024

Limit theorems for compensated weighted sums and application to numerical approximations

In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its components, a property which allows to transform it into a Skorohod-type Riemann sum. We then establish limit theorem for the compensated sum based on study of the Skorohod-type Riemann sum. Our proof employs techniques from Malliavin calculus and rough path. We apply our limit theorem result to the Euler approximation method for stochastic integrals and additive stochastic differential equations, filling a notable gap in this area of research. We show that the Euler method converges to the solution at the rate $(1/n)^{H+1/2}$, and that this rate is exact in the sense that the asymptotic error distribution solves a linear differential equation.

math.PR↗

arXiv2023

Power variations and limit theorems for stochastic processes controlled by fractional Brownian motions

In this paper we establish limit theorems for power variations of stochastic processes controlled by fractional Brownian motions with Hurst parameter $H\leq 1/2$. We show that the power variations of such processes can be decomposed into the mix of several weighted random sums plus some remainder terms, and the convergences of power variations are dominated by different combinations of those weighted sums depending on whether $H<1/4$, $H=1/4$, or $H>1/4$. We show that when $H\geq 1/4$ the centered power variation converges stably at the rate $n^{-1/2}$, and when $H<1/4$ it converges in probability at the rate $n^{-2H}$. We determine the limit of the mixed weighted sum based on a rough path approach developed in \cite{LT20}.

math.PR↗

arXiv2023

Euler scheme for SDEs driven by fractional Brownian motions: integrability and convergence in law

In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size $n$. Then we use the integrability results to derive the convergence rate in law $n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums.

math.PR↗

arXiv2023

Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability property is now well established. In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size $n$) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter $H>1/3$ and its Malliavin derivatives.

math.PR↗

arXiv2022

Statistical inference for rough volatility: Minimax Theory

Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter $H$. In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of $n^{-1/(4H+2)}$ for estimating $H$ based on n sampled data, extending results known only for the easier case $H>1/2$ so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes.

math.ST↗

arXiv2022

Statistical inference for rough volatility: Central limit theorems

In recent years, there has been a substantive interest in rough volatility models. In this class of models, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter $H < 0.5$. In this paper, we derive a consistent and asymptotically mixed normal estimator of $H$ based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

math.ST↗

arXiv2022

Backward Euler method for stochastic differential equations with non-Lipschitz coefficients

We study the traditional backward Euler method for $m$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H > 1/2$ whose drift coefficient satisfies the one-sided Lipschitz condition. The backward Euler scheme is proved to be of order $1$ and this rate is optimal by showing the asymptotic error distribution result. Two numerical experiments are performed to validate our claims about the optimality of the rate of convergence.

math.NA↗

arXiv2020

Convergence of trapezoid rule to rough integrals

Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough integrals are usually Riemann-Stieltjes integrals with correction terms that are sometimes seen as unnatural. As opposed to those somewhat artificial correction terms, our endeavor in this note is to produce a trapezoid rule for rough integrals driven by general $d$-dimensional Gaussian processes. Namely we shall approximate a generic rough integral $\int y \, dX$ by Riemann sums avoiding the usual higher order correction terms, making the expression easier to work with and more natural. Our approximations apply to all controlled processes $y$ and to a wide range of Gaussian processes $X$ including fractional Brownian motion with a Hurst parameter $H>1/4$. As a corollary of the trapezoid rule, we also consider the convergence of a midpoint rule for integrals of the form $\int f(X) dX$.

math.PR↗

arXiv2019

On the anticipative nonlinear filtering problem and its stability

In this paper, we consider an anticipative nonlinear filtering problem, in which the observation noise is correlated with the past of the signal. This new signal-observation model has its applications in both finance models with insider trading and in engineering. We derive a new equation for the filter in this context, analyzing both the nonlinear and the linear cases. We also handle the case of a finite filter with Volterra type observation. The performance of our algorithm is presented through numerical experiments.

math.PR↗

arXiv2018

On the necessary and sufficient conditions to solve a heat equation with general additive Gaussian noise

In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

math.PR↗

arXiv2017

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1/2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate $\sqrtτ$ as $τ\rightarrow \infty$, when the solution is observed continuously on the time interval $[0,τ]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyse the particular case of the fractional Ornstein-Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.

math.PR↗

arXiv2017

Crank-Nicolson scheme for stochastic differential equations driven by fractional Brownian motions

We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, \dots, B^{m})$ with Hurst parameter $H \in (\frac 12,1)$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, \dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slower rate than for the one-dimensional SDEs. Precisely, we shall prove that when $m=1$ and when the drift term is zero, the Crank-Nicolson scheme achieves the exact convergence rate $n^{-2H}$, while in the case $m=1$ and the drift term is non-zero, the exact rate turns out to be $n^{-\frac12 -H}$. In the general case when $m>1$, the exact rate equals $n^{\frac12 -2H}$. In all these cases the limiting distribution of the leading error is proved to satisfy some linear SDE driven by Brownian motions independent of the given fractional Brownian motions.

math.PR↗

arXiv2017

Discrete rough paths and limit theorems

In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough path techniques. As a by-product, we provide a natural explanation of the various new asymptotic behaviors in contrast with the classical unweighted random sum case. We apply our principle to derive some weighted type Breuer-Major theorems, which generalize previous results to random sums that do not have to be in a finite sum of chaos. In this context, a Breuer-Major type criterion in notion of Hermite rank is obtained. We also consider some applications to realized power variations and to Ito's formulas in law. In the end, we study the asymptotic behavior of weighted quadratic variations for some multi-dimensional Gaussian processes.

math.PR↗

arXiv2017

First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case

In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac13 \frac12$. The current contribution generalizes the modified Euler scheme to the rough case $\frac13<H<\frac12$. Namely, we show a convergence rate of order $n^{\frac12-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Hölder norm of this new rough path has an estimate which is independent of the step-size of the scheme.

math.PR↗

arXiv2017

Modified Euler approximation scheme for stochastic differential equations driven by fractional Brownian motions

For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $H> \frac12$ it is known that the classical Euler scheme has the rate of convergence $2H-1$. In this paper we introduce a new numerical scheme which is closer to the classical Euler scheme for diffusion processes, in the sense that it has the rate of convergence $2H-\frac12$. In particular, the rate of convergence becomes $\frac 12$ when $H$ is formally set to $\frac 12$ (the rate of Euler scheme for classical Brownian motion). The rate of weak convergence is also deduced for this scheme. The main tools are fractional calculus and Malliavin calculus. We also apply our approach to the classical Euler scheme.

math.PR↗

arXiv2016

Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $γ_n^{-1}$, where $γ_n=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $γ_n=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $γ_n=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $γ_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.

math.PR↗