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DR.SC.-02 Evaluation of the dissertation topic

PhD-02 EVALUATION OF THE DISSERTATION TOPIC[1]
GENERAL INFORMATION AND PERSONAL CONTACT INFORMATION OF THE DOCTORAL CANDIDATE
First and last name, and title of the doctoral candidate: / Petra Žugec, mag.math.
Provider of the study programme: / Faculty of Organization and Informatics, University of Zagreb
Name of the study programme: / Mathematics
Scientist ID of the doctoral candidate: / I-316/09
Title of proposed topic in / language of the dissertation: / English
Croatian / Označeni Poissonovi procesi s klasterima i primjene
English / Marked Poisson cluster processes and applications
Area/field/branch (if the doctoral study is performed in a branch): / Natural Sciences/Mathematics/Probability Theory and Statistics
MENTOR(S)
First and last name, and title: / Institution, country: / E-mail:
First mentor: / Dr. Bojan Basrak, Full Professor / Department of Mathematics, Faculty of Science, University of Zagreb /
Second mentor:
The committee appointed for topic evaluation and mentor appointment proposal / First and last name, and title: / Institution, country: / E-mail:
1. Dr. Zoran Vondraček, Full Professor / Department of Mathematics, Faculty of Science, University of Zagreb /
2. Dr. Nikola Sandrić, Assistant Professor / Department of Mathematics, Faculty of Science, University of Zagreb /
3.Dr. Nenad Šuvak, Associate Professor / J.J. Strossmayer
University of Osijek,
Department of
Mathematics, Croatia /
4.
5.
Session of the competent body at which the Committee was appointed, and number of item on the agenda / Assembly of Department of Mathematics, Faculty of Science,November 8 2017, item 4.10
Session of the Ethics Committee at which consent was given to the research proposal
A. Report on the Public Defence of the Dissertation Topic
The candidate talked about her research and proposed dissertation topic at the Probability Seminar in Zagreb on November 28, 2017. Part of the talk was dedicated to the public defence of the dissertation topic.
On that occasion, Petra Žugec presented her plans for future research and announced some preliminary results as well as the results she considers achievable by the methods discussed below. The candidate Petra Žugec showed the knowledge of current literature in herfield, adequate understanding of her subject and presented possible applications of her results in other fields, nonlife insurance mathematics in particular.
B. Evaluation of the Dissertation Topic
(original scientific contribution and evaluation of viability)
The proposed dissertation topic is based intheory of probability and its applications. In particular, it is related to theory of stochastic processes, extreme value theory, risk theory, and limit theorems in probability. Basic question considered by the candidate is the behavior of total sum of claims in a marked Poisson cluster processes, where each point represents a claim characterized by its arrival time, claim size and claim type (mark). The total sum of claims over a period of time is a random quantity of central interest in non-life insurance mathematics for instance, where standard models exclude the possibility of clustering of claims, see Asmussen and Albrecher (2000) or Mikosch (2009).
In recent years, several special models have been proposed to account for the possibility of clustering of events in various applied areas. This includes in particular Hawkes processes, which have been used in insurance mathematics, see e.g. Stabile and Torrisi (2010), Zhu (2013), or Karabash and Zhu (2015), financial mathematics, see e.g. Bacry et al. (2013), seismology, see e.g. Ogata (1988, 1998), criminology, see e.g. Mohler et al. (2011) and in many other fields. However, Hawkes processesshare certain properties with other Poisson cluster models, as described in two volumes book by Daley and Vere Jones (2008) which remains a standard reference to the theory of Poisson cluster processes. In recent years, two interesting dissertation theses were written on this subject by Zhu: Nonlinear Hawkes processes (2013) and Kirchner: Perspectives on Hawkes Processes (2017).
Some central limit theorems for the number of claims arriving over a period of time were found by Daley (1972) and Karabash and Zhu (2015). The latter article considers marked Hawkes processes which can be considered a special case of the model studied by the candidate. It was shown recently that in the unmarked case,multi-type Hawkes processes satisfy even functional central limit theorem, see Bacry et al. (2013). To generalize these results, the candidate proposes to study asymptotic distribution of the total claim amount in the setting where Cramer - Lundberg risk model is augmented with a marked Poisson cluster structure. Thus, the proposed model for arrival of claims in an insurance portfolio is a marked point process of the form

where k's are nonnegative random variables representing arrival times with some degree of clustering and Ak's represent corresponding marks in a rather general metric space S. In the language of point processes theory, marks are assumed merely unpredictable and not independent of the arrival times (Daley and Vere Jones, 2008). It seems natural to assume that the claim sizes can be calculated using a measurable mapping of marks to nonnegative real numbers, f(Ak) say. Hence, the total claim amount over the time interval [0, t] equals

The main goal of the proposed dissertation is to determine the effect of the clustering on the quantity S(t), as t →∞,even in the case when the distribution of the individual claims does not satisfy assumptions of the classical central limit theorem. To achieve her goal,the candidate plans to apply the limit theory for two-dimensional random walks, seeGut(2009). But in order to extend this result for heavy tailed claims with infinite variance or even infinite mean she would need to find an alternative approach, which also seems to be feasible by the proposed methodology. She also plans to apply her results to several special models.
Besides that, the candidate proposed to clarify the notion of conditional intensityfor marked point processeswhich appears to be used impreciselyin more applied literature. The methods suggested by the candidate could be also potentially used to study long-term behaviour of the extreme claims in Poisson cluster model.
The suggested goals seem achievable by the methods proposed by the candidate. This is also clear from the preliminary results she presented within the Probability Seminar on November 28, 2017.
Opinion and recommendation:
Petra Žugec satisfies the eligibility criteria for the acceptance of the dissertation topic. Her proposed thesis topic
Marked Poisson cluster processes and applications
is original, interesting and relevant. It also follows recent developments in the field of her research. We recommend to Mathematics Department to accept the proposal as the dissertation topic. As a mentor we recommend prof.dr.sc. Bojan Basrak.
Recommendation on change or revision of dissertation title:
Proposal for change of mentor and/or appointment of another mentor(specify title, first and last name, institution):
Dissertation defence planned for(specify year and semester):
Fall semester, academic year 2018/2019.
Separate opinion(only if any of the members of the Committee for evaluation of dissertation topic and mentor appointment proposal has a separate opinion)
Signature
(First name and last name of committee member)
ADDITIONAL COMMENTS(as needed):
The committee appointed for topic evaluation and mentor appointment proposal / First and last name, title, institution, country: / Signature:
1.(Committee chair)Dr.Zoran Vondraček, Full Professor, University of Zagreb, Faculty of Science, Croatia
2. Dr. Nikola Sandrić, Assistant Professor, University of Zagreb, Faculty of Science, Croatia
3. Dr. Nenad Šuvak, Associate Professor, J.J. StrossmayerUniversity of Osijek,
Department ofMathematics, Croatia
4.
5.
Zagreb,November 29, 2017Official stamp here

Form DR.SC.-02 Evaluation of the dissertation topic – to be filled out by the Chair of the Committee for topic evaluation and mentor appointment proposal

[1]Please name file as: DR.SC.-02 – Last name and first name of doctoral candidate.doc

Please send the filled-out form DR.SC.-02, in electronic and written format, and signed, to the appropriate Registrar's Office.