## Postgraduate research opportunity

# Ruin and related problems for two-dimensional risk process (SUSTech candidates only)

**Value**: Fees and maintenance**Number of awards**: 1**Deadline**: Applications accepted all year round**Key benefits**: This project is available as part of the Leeds-SUSTech Split-Site PhD Programme

### Supervisor(s)

Please contact Dr Lanpeng Ji (Leeds) or Dr Jiawen Gu (SUSTech) to discuss this project further informally.

### Project description

Stochastic analysis of ruin related quantities such as the time of ruin, ruin probabilities, and expected discounted dividend/tax payments is very important in (non-life) insurance applications. For instance, decision makers of an insurance company may decide to take measures in case of high probability of ruin; they may go for reinsurance, increase the premium of certain product or inject capitals. We refer to [1] for detailed discussions on this topic.

Nowadays, insurance companies run different lines of businesses or collaborate with other companies, it is thus of interest to model these businesses using different (probably correlated) risk processes and consider different types of ruin, tax and dividend problems. For instance, suppose an insurance company runs two lines of businesses. Ruin can be considered to occur when one of the businesses is ruined (the surplus becomes negative), both of them are ruined simultaneously, or both of them are ruined (not necessarily simultaneously) within a period of time, say 10 years. In consideration of that, multidimensional risk processes have attracted a lot of attention in recent years; see, e.g., [2-5].

In general, multidimensional problems are much more difficult than the one-dimensional ones. This project will focus on the two-dimensional correlated Brownian motion risk process, with the goal of answering the following questions:

P1: What is the Parisian ruin probability? Is there any relation to the classical ruin probability?

P2: What is the ruin probability for the model with tax? Does the asymptotic tax identity still hold?

P3: What are the optimal dividend strategies for the models with correlation between the two businesses?

The questions in P1 and P2 for the one-dimensional case have been answered in [6] and [7] as special cases. In order to solve P1 or P2, one needs to deal with extremes of Gaussian random fields, which makes the problem very challenging. Investigation on certain convex optimization problems will be needed, and some ideas in the recent contribution [8] could be helpful. Regarding P3, the optimal dividend strategies for a special two-dimensional Brownian motion risk process has been discussed in [5]. Using the theory of stochastic optimal control, the project aims to explore the optimal dividend strategies for models with correlation between the two businesses (e.g., a model similar to [3]).

References:

[1] S. Asmussen & H. Albrecher. Ruin probabilities. Advanced Series on Statistical Science & Applied Probability, 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second ed., 2010.

[2] F. Avram, Z. Palmowski, & M. Pistorius. Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results, Annals of Applied Probability, vol. 19, pp. 2421-2449, 2008.

[3] P. Azcue, N. Muler, & Z. Palmowski. Optimal dividend payments for a two-dimensional insurance risk process, preprint, https://arxiv.org/abs/1603.07019v2, 2016.

[4] L. Ji & S. Robert. Ruin problems of a two-dimensional fractional Brownian motion risk process. Stochastic Models, 2018.

[5] J. Gu, M. Steffensen, & H. Zheng. Optimal dividend strategies of two collaborating businesses in the diffusion approximation model. Mathematics of Operations Research. In press, 2017.

[6] K. Debicki, E. Hashorva, & L. Ji. Parisian ruin of self-similar Gaussian risk processes, Journal of Applied Probability, vol. 52, pp 688-702, 2015.

[7] E. Hashorva, L. Ji, & V. Piterbarg. On the supremum of gamma-reflected processes with fractional Brownian motion as input, Stochastic Process. Appl., vol. 123, pp. 4111-4127, 2013.

### Entry requirements

Applicants should have, or expect to obtain, a minimum equivalent of a UK upper second class honours degree in Mathematics or a related discipline.

If English is not your first language, you must provide evidence that you meet the University’s minimum English Language requirements.

### How to apply

Applications should be submitted via SUSTech in the first instance. Following nomination by SUSTech, formal applications for Split-site research degree study should then be made online through the University of Leeds website. Please state clearly in the research information section that the PhD you wish to be considered for is ‘Ruin and related problems for two-dimensional risk process (SUSTech candidates only)’ as well as Dr Lanpeng Ji as your proposed supervisor.

If you require any further information please contact the Graduate School Office e: maps.pgr.admissions@leeds.ac.uk