I am a probabilist working on problems based on stochastic processes, using tools from analysis. Although I am interested mostly in theoretical developments, some of the problems that I study are motivated by applications in physics (for example, the vibration of a string under random perturbations), or finance (such as the evolution of stock prices in markets which exhibit extreme behaviour). My current research program seeks to develop new techniques for analyzing the behaviour of systems modelling complex random phenomena. The program focuses on problems from two distinct areas: (a) stochastic partial differential equations; and (b) heavy-tailed time series.

Stochastic Partial Differential Equations (SPDEs) are mathematical objects used for modeling the behaviour of physical phenomena that evolve simultaneously in space and time, and are subject to random perturbations. Their study requires tools from stochastic analysis, such as Itô calculus or Malliavin calculus. In the classical theory, these equations are perturbed by a space-time Gaussian white noise and have random field solutions only in spatial dimension 1. My program aims to discover and study new properties of the solutions to the stochastic wave and heat equations in higher dimensions, perturbed by more general classes of noise processes (such as fractional noises), as more flexible alternatives to the white noise. These findings will offer new perspectives on the dynamical interplay between the regularity of the noise and the properties exhibited by the random field solution, leading to a deeper understanding of the effect of the noise on the behaviour of solution.

 Heavy tailed time series are encountered frequently in applications in finance, insurance and environmental studies, as models for perturbations that exhibit extreme behaviour. The concept of multivariate regular variation was introduced to describe a similar behaviour in higher dimensions. When we observe processes continuously over a fixed interval of time (or a region in space), we need an infinite-dimensional theory analogous to the theory of multivariate regular variation. The goal of my program is to advance the asymptotic theory for point processes associated with various time series models with values in an infinite-dimensional space of functions, and to apply this theory for deriving new results about the partial sum or partial maximum of the variables in such series. These results will give important new insights into the extreme value theory for time series models which evolve in time and space, and could be used in a variety applications.
[57] Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise (with Panqui Xia and Guangqu Zheng), e-print: arxiv:2310.10784
[56] Gaussian fluctuations for the wave equation under rough perturbations (with Jingyu Huang, Xiong Wang, Panqui Xia and Wangjun Yuan), e-print: arxiv:2307.00103
[55] Continuity in law for solutions of SPDEs with space-time homogeneous Gaussian noise (with Xiao Liang), Stochastics and Dynamics, to appear, e-print: arxiv2305.10330
[54] Hyperbolic Anderson model with time-independent rough noise: Gaussian fluctuations (with Wangjun Yuan), e-print: arxiv2305.05043
[53] Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation (with Guangqu Zheng), e-print: arxiv2302.14178
[52] Parabolic Anderson model with rough noise in space and rough initial conditions (with Le Chen and Yiping Ma), Electronic Communications in Probability 27 (2022), paper no. 65, 1-12, e-print: arxiv2206.11361
[51] Central limit theorems for heat equation with time-independent noise: the regular and rough cases (with Wangjun Yuan), Infinite Dimensional Analysis, Quantum Probability and Related Topics 26 No 2. (2023) Article No. 2250029, e-print: arxiv2205.13105
[50] Spatial integral of the solution to the hyperbolic Anderson model with time-independent noise (with Wangjun Yuan), Stochastic Processes and Their Applications, to appear, e-print: arxiv2201.02319
[49] Stochastic wave equation with Lévy white noise, ALEA, Latin American Journal of Probability and Mathematical Statistics 20 (2023), 463-496, e-print: arxiv2111.14242

以上是Prof. Raluca Balan的研究兴趣和近期出版物，以下是我的研究兴趣：I’m deeply passionate about financial statistics, econometrics, and thoroughly enjoy data analysis and mining. The evolution of the economy, from historical to forward perspectives, is captivating. The landscape is rich with diverse indicators, each holding unique significance. My aim is to unveil the intricate patterns in economic and financial operations within this vast array of indicators.

My love for data analysis and mining is deeply rooted in the pivotal role data plays in understanding finance and the economy. Effective analysis calls for proficient use of statistical software to organize and interpret data. Hence, I strive to enhance my proficiency in tools like R, SPSS, Python, and Amos. Besides, I’m enthusiastic about gaining new software skills specialized for financial and economic analysis.

During my graduate studies, I plan to immerse myself in cutting-edge domains like neural networks, big data machine learning, and deep learning. These areas offer innovative approaches to comprehend and predict economic trends and financial dynamics in our increasingly data-centric world. This exploration aligns with my vision to contribute meaningfully to this dynamic field.

请帮我给Prof. Raluca Balan用英文写一封EMAIL，帮我表达 I am interested in working with her and why I am interested in working with Prof. Anas Abdallah，需要逻辑清晰，论据充分,言辞恳切打动人