# Gaussian approximations with applications to retirement products

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Dr Thomas Bernhardt, Dr Ronnie Loeffen  No more applications being accepted  Funded PhD Project (Students Worldwide)

There is an ongoing retirement crisis in the world.

Governments and insurance companies are searching for retirement products that cope with rising life expectancy and uncertain financial markets.

New retirement products adjust the income of retirees with experienced mortality rates and financial returns.

However, that means that the retiree bears all the risk. Predicting how long the retiree receives a stable income is crucial. Imagine a group of people who pool their funds together. Each member receives an income from the funds in the pool. The income depends on the remaining lifetimes of the people that form a sequence of independent random variables. Now, Donsker's theorem tells us that we can approximate such a pool with a Gaussian process. However, identically distributed lifetimes is a strong assumption. People differ according to their age, mortality rates, savings and more. The interesting mathematical questions that follow are how to incorporate the differences? How to get robust results that can cope with the uncertainty of mortality rates? How to deliver a stable income to retirees?

Expected Outcome

The successful applicant will develop mathematical tools for the analysis and evaluation of new retirement products. That includes but is not limited to generalizations of Donsker's theorem suitable for compound distributions and the mathematical explanation of phenomena observed in numerical studies.

Applicants should have a good background in Mathematics. Ideal candidates have strong knowledge about probability theory, including compound distributions, Donsker's theorem, and martingale theory.

Review Process:

We will invite shortlisted candidates to a Zoom interview, which takes place at the end of January. One to two weeks before the interview, candidates will be emailed an exercise to solve typical for an advanced probability course. The interview questions will focus on the candidate's attempt/solution. We are most interested in the candidate's mathematical ability to prove a statement accurately.

You need to submit:

1) a CV (including contact details of two academic references),

2) copy of MSc degree (or equivalent, or evidence of the expected date of obtaining) in Mathematics (or closely related field),

3) transcripts of grades in English,

4) a personal statement indicating interest in this specific research project; hear you can also point out anything you would like us to know about you which is not covered by your other documents; be brief and do not repeat your other documents,

5) copy of your passport if you will need a visa to study in the UK,

6) evidence of English language ability if English is not your first language (can be obtained at a later stage).

The submission deadline is the 3rd January 2022. You can apply online via the following link:

For further inquiries, contact Doctor Thomas Bernhardt:

## Funding Notes

This is a funded PhD studentship, covering fees and stipend (£15,609 in 2021-22).
Open to all applicants.
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