The Archetypal Equation with Rescaling and Related Topics

PGR-P-191

Key facts

Type of research degree: PhD
Application deadline: Ongoing deadline
Project start date: Wednesday 1 October 2025
Country eligibility: International (open to all nationalities, including the UK)
Funding: Competition funded
Source of funding: University of Leeds
Supervisors: Dr Leonid Bogachev
Schools: School of Mathematics
Research groups/institutes: Probability and Financial Mathematics, Statistics

<h2 class="heading hide-accessible">Summary</h2>

Archetypal equation is a functional equation involving averaging (expectation) over a random affine transformation of the argument. This equation is related to many important topics such as the Choquet Deny theorem, Bernoulli convolutions, fractals, subdivision schemes in approximation theory, etc. Investigation of the archetypal equation will involving asymptotic analysis of the corresponding Markov chains with affine jumps. The project may have applications to financial modelling based on random processes with multiplicative jumps.

<h2 class="heading hide-accessible">Full description</h2>

Theory of functional equations is a growing branch of analysis with many deep results and abundant applications (see [1] for a general introduction). A simple functional-differential equation with rescaling is given by y'(x) + y(x) = p y(2x) + (1-p) y(x/2) (0<p<1), which describes e.g. the ruin probability for a gambler who spends at a constant rate (starting with x pounds) but at random time instants decides to bet on the entire current capital and either doubles it (with probability p) or loses a half (with probability 1-p). Clearly, y(x) = const is a solution, and the question is whether or not there are any other bounded, continuous solutions. It turns out that such solutions exist if and only if p < 0.5; this analytic result can be obtained using martingale techniques of probability theory [2]. The equation above exemplifies the "pantograph equation" introduced by Ockendon & Tayler [6] as a mathematical model of the overhead current collection system on an electric locomotive. In fact, the pantograph equation and its various ramifications have emerged in a striking range of applications including number theory, astrophysics, queues and risk theory, stochastic games, quantum theory, population dynamics, imaging of tumours, etc. A rich source of functional and functional-differential equations with rescaling is the "archetypal equation" y(x) = E[y(α(x-β))], where α, β are random coefficients and E denotes expectation [3]. Despite its simple appearance, this equation is related to many important topics, such as the Choquet–Deny theorem, Bernoulli convolutions, self-similar measures and fractals, subdivision schemes in approximation theory, chaotic structures in amorphous materials, and many more. The random recursion behind the archetypal equation, defining a Markov chain with jumps of the form x → α(x-β). In brief, the main objective of this PhD project is to continue a deep investigation of the archetypal equation and its generalizations. Research will naturally involve asymptotic analysis of the corresponding Markov chains, including characterization of their harmonic functions [7]. The project may also include applications to financial modelling based on random processes with multiplicative jumps (cf. [5]). References <ol> <li>Aczél, J. and Dhombres, J. Functional Equations in Several Variables, with Applications to Mathematics, Information Theory and to the Natural and Social Sciences. Cambridge Univ. Press, Cambridge, 1989.</li> <li>Bogachev, L., Derfel, G., Molchanov, S. and Ockendon, J. On bounded solutions of the balanced generalized pantograph equation. In: Topics in Stochastic Analysis and Nonparametric Estimation (P.-L. Chow et al., eds.), pp. 29–49. Springer, New York, 2008. (<a href="https://doi.org/10.1007/978-0-387-75111-5_3">doi:10.1007/978-0-387-75111-5_3</a>)</li> <li>Bogachev, L.V., Derfel, G. and Molchanov, S.A. On bounded continuous solutions of the archetypal equation with rescaling. Proc. Roy. Soc. A, 471 (2015), 20150351, 1–19. (<a href="https://doi.org/10.1098/rspa.2015.0351">doi:10.1098/rspa.2015.0351</a>)</li> <li>Diaconis, P. and Freedman, D. Iterated random functions. SIAM Reviews, 41 (1999), 45–76. (<a href="https://doi.org/10.1137/S0036144598338446">doi:10.1137/S0036144598338446</a>)</li> <li>Kolesnik, A.D. and Ratanov, N. Telegraph Processes and Option Pricing. Springer, Berlin, 2013. (<a href="https://doi.org/10.1007/978-3-642-40526-6">doi:10.1007/978-3-642-40526-6</a>)</li> <li>Ockendon, J.R. and Tayler, A.B. The dynamics of a current collection system for an electric locomotive. Proc. Roy. Soc. London A, 322 (1971), 447–468. (<a href="https://doi.org/10.1098/rspa.1971.0078">doi:10.1098/rspa.1971.0078</a>)</li> <li>Revuz, D. Markov Chains, 2nd edn. North-Holland, Amsterdam, 1984.</li> </ol>

<h2 class="heading">How to apply</h2>

Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>. Please state clearly in the Planned Course of Study section that you are applying for PhD Statistics FT and in the research information section that the research degree you wish to be considered for is The Archetypal Equation with Rescaling and Related Topics as well as <a href="https://eps.leeds.ac.uk/maths/staff/4008/dr-leonid-bogachev">Dr Leonid Bogachev</a> as your proposed supervisor and in the finance section, please state clearly the funding that you are applying for, if you are self-funding or externally sponsored. Successful candidates should have an excellent degree in mathematics and/or statistics, with a good background and research interests in one or more of the following areas: probability; random processes; analysis; mathematical statistics. You will be based within a strong research group in <a href="http://(https://eps.leeds.ac.uk/maths-statistics/doc/probability-financial-mathematics">Probability and Financial Mathematics</a>. Our research focuses on the study and modelling of systems and processes featured by uncertainty and/or complexity, using advanced theoretical, simulation and numerical methods. It covers a vast variety of modern topics both in probability (including theory of random processes and stochastic analysis) and in a wide range of applications in mathematical and other sciences, spanning from nonlinear dynamical systems and mathematical physics through mathematical biology and complexity theory to mathematical finance and economics. If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below). As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability. Applications will be considered after the closing date. Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed. Please note that you must provide the following documents in support of your application by the closing date of Monday 6 January 2025 if applying for the China Scholarship Council-University of Leeds Scholarship, Monday 3 February 2025 if applying for Leeds Doctoral Scholarship or Tuesday 1 April 2025 for Leeds Opportunity Research Scholarship. If you are applying for the School of Mathematics Scholarship 2025/26, or with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application: <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul>

<h2 class="heading heading--sm">Entry requirements</h2>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

<h2 class="heading heading--sm">English language requirements</h2>

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<h2 class="heading">Funding on offer</h2>

Self-Funded or externally sponsored students are welcome to apply. UK – The <a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> (closing date: Monday 3 February 2025), <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> (closing date: Tuesday 1 April 2025) and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> (open from October 2024) are available to UK applicants. Non-UK – <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> (open from October 2024) are available to all International applicants. The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> (closing date: Monday 6 January 2025) is available to nationals of China. The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds. You will be responsible for paying the overtime fee in full in your writing up/overtime year (£320 in Session 2024/25), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award. Important: Please note that that the award does not cover the costs associated with moving to the UK. All such costs (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source. Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.

<h2 class="heading">Contact details</h2>

For further information about your application, please contact PGR Admissions by email to <a href="mailto:EMAIL@leeds.ac.uk">m</a><a href="mailto:maps.pgr.admissions@leeds.ac.uk">aps.pgr.admissions@leeds.ac.uk</a> For further informaiton about this project, please contact Dr Leonid Bogachev by email to <a href="mailto:L.V.Bogachev@leeds.ac.uk">L.V.Bogachev@leeds.ac.uk</a>

<h3 class="heading heading--sm">Linked funding opportunities</h3>

<h3 class="heading heading--sm">Linked research areas</h3>