# Becoming an expert: Wei Zhu on fractional calculus and its application to insurance

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Wei Zhu has just completed his PhD study in the Institute for Financial and Actuarial Mathematics (IFAM) in the Department of Mathematical Sciences under the supervision of Dr. Corina Constantinescu.

Wei Zhu graduated from Donghua University in Shanghai with a BSc in Mathematics and then an MSc in Financial Mathematics from the University of Liverpool. He has been living in the city of Liverpool for more than 5 years. His original motivation to come to Liverpool is because he is a fan of Liverpool Football Club but now, he loves every aspect of this city, from the fantastic weather to the lovely Scouse accent.

“I was not planning to do a PhD before I came to Liverpool. However, I was lucky enough to meet my supervisor Dr. Corina Constantinescu here. She always makes research so appealing to me, which has motivated my decision to pursue an academic career.” After completiung his Masters Wei stayed on as a PhD student in the Institute for Financial and Actuarial Mathematics (IFAM) and continued working on ruin theory. Under the supervision of Dr. Corina Constantinescu, Wei’s project is dedicated to searching for the application of fractional calculus to insurance and he submitted his PhD thesis “Fractional differential equations in risk theory” in December 2017. His research is aimed at both actuarial science and mathematical journals.

Fractional calculus and its application to insurance

The story of collective risk theory starts from the early 1900s. The traditional approach is to consider the model of the surplus of an insurance company and find the probability that the risk reserve drops below zero, which is known as the ruin probability. The classical risk model describes the surplus of an insurance company over time. In such a model the occurrence of the claims is described by a point process and the amounts of money to be paid by the company at each claim by a sequence of random variables. The company receives a certain amount of premium to cover its liability.

Deriving the ruin probability is a central topic in the risk theory literature. The literature of deriving explicit expressions for the ruin probability of the classical compound Poisson risk model for various claims distributions is abundant in methods and results. Another direction that captured a lot of attention over the last hundred years is to consider altering the assumptions of either independence or memory loss of claim arrivals, thus analysing ruin probabilities in renewal risk models.

On the other hand, fractional calculus is a complex topic having interconnections with various problems of function theory, integral and differential equations, and other branches of analysis. Fractional calculus was born in late 1690s, as old as the traditional calculus, proposed independently by Newton and Leibniz. In the following centuries, it has been continually developed, stimulated by ideas and results in various fields of mathematical analysis. Lots of mathematicians tried to find the “proper” definition for fractional integrals and fractional derivatives.

Meanwhile, it has been shown that fractional-order models are more adequate than previously used integer-order models in various fields. The mathematical modelling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to fractional differential equations.

Considering the application of fractional calculus to insurance, Wei Zhu managed to make very general assumptions on inter-arrival times for the renewal risk model, which includes the classical risk model, Erlang risk model and fractional Poisson risk model. A new family of differential operators are defined in order to construct the fractional integro-differential equations for ruin probabilities in such renewal risk models. Through the characteristic equation approach, specific fractional differential equations for the ruin probabilities can be solved explicitly, allowing for the analysis of the ruin probabilities.

Although his PhD research is mainly in actuarial sciences, Wei Zhu is personally open to many topics, especially those having potential connections with fractional differential equations.

Extracurricular activities

Wei Zhu has attended numerous research conferences, workshops and symposiums during his PhD and presented his research in the UK, China, Australia, France, Japan, Austria and Colombia. In addition Wei has also paid research visits to Nankai University (Japan), Monash University (Australia), Ritsumeikan University (Japan) and the National University of Colombia meeting a host of international actuarial and mathematical science researchers.

As a teaching assistant during his PhD, Wei Zhu has gained a lot of teaching and tutoring experience from being involved in various modules. Since 2016, Wei has been helping his PhD supervisor to introduce a “learning assistants” programme, where senior undergraduate students assist their junior peers in order to improve teaching and student experience in the department.

In addition, since 2014 Wei has participated in the annual summer research internship programme delivering several fundamental mathematical and programming courses to our undergraduate students and helping them to carry out a six-week research project in collaboration with industrial partners.

After graduating, Wei Zhu would like to work in academia and use mathematics to change the world.