## Key facts

- Type of research degree
- PhD
- Application deadline
- Ongoing deadline
- Country eligibility
- International (open to all nationalities, including the UK)
- Funding
- Competition funded
- Supervisors
- Dr Leonid Bogachev and Dr Vladimir V. Kisil
- Schools
- School of Mathematics
- Research groups/institutes
- Probability and Financial Mathematics, Statistics

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.<br />

<p>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 <em>y</em>'(<em>x</em>) + <em>y</em>(<em>x</em>) = <em>p</em> <em>y</em>(2<em>x</em>) + (1-<em>p</em>) <em>y</em>(<em>x</em>/2) (0<<em>p</em><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 <em>p</em>) or loses a half (with probability 1-<em>p</em>). Clearly, <em>y</em>(<em>x</em>) = 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 <em>p</em> < 0.5; this analytic result can be obtained using martingale techniques of probability theory [2].</p> <p>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.</p> <p>A rich source of functional and functional-differential equations with rescaling is the "archetypal equation" <em>y</em>(<em>x</em>) = E[<em>y</em>(<em>α</em>(<em>x</em>-<em>β</em>))], where <em>α</em>, <em>β</em> 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 <em>x</em> → <em>α</em>(<em>x</em>-<em>β</em>).</p> <p>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]).</p> <p><strong>References</strong></p> <ol> <li>Aczél, J. and Dhombres, J. <em>Functional Equations in Several Variables, with Applications to Mathematics, Information Theory and to the Natural and Social Sciences</em>. 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: <em>Topics in Stochastic Analysis and Nonparametric Estimation</em> (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. <em>Proc. Roy. Soc. A</em>, <strong>471</strong> (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. <em>SIAM Reviews</em>, <strong>41</strong> (1999), 45–76. (<a href="https://doi.org/10.1137/S0036144598338446">doi:10.1137/S0036144598338446</a>)</li> <li>Kolesnik, A.D. and Ratanov, N. <em>Telegraph Processes and Option Pricing</em>. 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. <em>Proc. Roy. Soc. London A</em>, <strong>322</strong> (1971), 447–468. (<a href="https://doi.org/10.1098/rspa.1971.0078">doi:10.1098/rspa.1971.0078</a>)</li> <li>Revuz, D. <em>Markov Chains</em>, 2nd edn. North-Holland, Amsterdam, 1984.</li> </ol>

<p>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 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.</p> <p>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.</p> <p>You will be based within a strong research group in Probability and Financial Mathematics (<a href="https://eps.leeds.ac.uk/maths-statistics/doc/probability-financial-mathematics">https://eps.leeds.ac.uk/maths-statistics/doc/probability-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.</p> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p> </p>

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.

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.

<p><strong>Self-Funding and externally sponsored students are welcome to apply.</strong></p> <p><strong>UK </strong>– The <a href="https://phd.leeds.ac.uk/funding/209-leeds-doctoral-scholarships-2022">Leeds Doctoral Scholarships</a>, <a href="https://phd.leeds.ac.uk/funding/198-akroyd-and-brown-scholarship-2022">Akroyd & Brown</a>, <a href="https://phd.leeds.ac.uk/funding/199-frank-parkinson-scholarship-2022">Frank Parkinson</a> and <a href="https://phd.leeds.ac.uk/funding/204-boothman-reynolds-and-smithells-scholarship-2022">Boothman, Reynolds & Smithells</a> Scholarships are available to UK applicants. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK </strong>– 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> 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.</p>

<p>For further information regarding your application, please contact Doctoral College Admissions by email: <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>, or by telephone: +44 (0)113 343 5057</p> <p>For further informaiton regarding the project, please contact Dr Leonid Bogachev by email: <a href="mailto:L.V.Bogachev@leeds.ac.uk">L.V.Bogachev@leeds.ac.uk</a></p>

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