Research
My main area is probability theory and statistics, especially stochastic dynamical processes. I am interested in applying stochastic, statistics and their applications to problems in the real world such as population dynamics, data sciences, social network science, public health, finance, chemistry and numerical analysis. Currently, I am supported by University of Louisville Internal Grant EVPRI "Spatial Population Dynamics with Disease" Funding and American Mathematical Society (AMS) 's Mathematics Research Communities (MRC) Collaboration Funding "Survival Dynamics for Contact Process with Quarantine".
I do both theoretical work and applied problems. Since it is very hard to explain and visualize some my theoretical works here, I will show you some examples of my work in the real-life problems. I hope my work can inspire your interests in probability and statistics.
There are growing signs that the COVID-19 virus has started to spread to rural areas and can impact the rural health care system that is already stretched and lacks resources. To aid in the legislative decision process and proper channelizing of resources, we estimated and compared the county-level change in prevalence rates of COVID-19 by rural-urban status over 3 weeks.
Additionally, we identified hotspots based on estimated covid-19 prevalence rates.We used crowdsourced data on COVID-19 and linked them to county-level demographics, smoking rates, and chronic diseases. We fitted a Bayesian hierarchical spatiotemporal model using the Markov Chain Monte Carlo algorithm in R-studio. We mapped the estimated prevalence rates using ArcGIS 10.8, and identified hotspots using Gettis-Ord local statistics. By checking counties that have a significant increase or decrease of the percentage change in prevalence rates over 14 days, we can determine whether it is appropriate to reopen in the corresponding location.
Bayesian Hierarchical Time Series to capture the seasonal pattern of influenza daily rates. Red dots represent the flu confirmed cases in CDC (Centers for Disease Control and Prevention ). The purple represents respiratory rates. Yellow represents constitutional rates. The black represents downscaled ili rates and orange line is the prediction of ili daily rates.
Time = 2
Time = 3
Time = 5
Epidemic Spreading Process based on the contact network. There are many types of network structure. Each node represents one individual. And red dots stand for infected population, yellow dots stand for recovered population. Nowadays, the existence of the large data make it possible for us to track epidemic process according to the real network in the real life. Similar problems happen in Social Network and Finance cash flow/trading network etc. Exploring the stochastic process on network is a very interesting topic to me.
Regime-switching model and risk-neutral option pricing: The left graph shows the evaluation of the price of a European call option by Fast Fourier Transform algorithm, and compare it with the real value of the option in 2016, May 31st.
Asian option is an option type where the payoff depends on the average price of the underlying asset over a certain period of time as opposed to standard options (American and European) where the payoff depends on the price of the underlying asset at a specific point in maturity time and it is hard to estimate its price. Brownian motion can be applied to affine group to estimate the price of Asian Option.The key is to study probability properties of the functional of Brownian motion on the affine group.