arXiv subjects

Binghao Wu

Publications and source records attributed to Binghao Wu.

3 recordsLinked to original sources

arXiv2025

On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion

We give the correct condition for existence of the $k$-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201]. We also show that the $k$-th derivative of the intersection and self-intersection local times of fractional Brownian motion are exponentially integrable for certain parameter values. In addition, we show convergence in distribution when the existence condition is violated for the $k$-th derivative of self-intersection local time of fractional Brownian motion under scaling.

math.PR↗

arXiv2024

On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes

We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results present in the literature for fractional Brownian motion, and in particular give a counter-example to a result in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201] related to this question.

math.PR↗

arXiv2023

A note on a deterministic property to obtain the long run behavior of the range of a stochastic process

A Brownian motion with drift is simply a process $V^η_t$ of the form $V^η_t=B_{t}+ηt$ where $B_{t}$ is a standard Brownian motion and $η>0$ \footnote{The case $η<0$ is deducible by remarking $V^{-η}(t)=-V^η(t)$.} In \cite{tanre2006range}, the authors considered the drifted Brownian motion and studied the statistics of some related sequences defined by certain stopping times. In particular, they provided the law of the range $R_{t}(V^η)$ of $V^η$ as well as its first range process $θ_{V^η}(a)$. In particular, they investigated the asymptotic comportment of $R_{t}(V^η)$ and $θ_{V^η}(a)$. They proved that if $V_{t}^η$ is a Brownian motion with a positive drift $η$ then its range $R_{t}(V^η)=\sup_{0\leq s\leq t}V_{t}^η-\inf_{0\leq s\leq t}V_{t}^η$ is asymptotically equivalent to $ηt$. In other words \begin{equation} \frac{R_{t}(V^η)}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}η.\label{range} \end{equation} In this paper, we show that (\ref{range}) follows from a striking deterministic property. More precisely, we show that the long run behavior of the range of a deterministic continuous function is obtainable straightaway from that of the function itself. Our result can be deemed as the continuous version of a similar one appeared in \cite{mgrw}.

math.PR↗