Numerical approximations of irregular differential equations

Recent advances in the study of regularization by noise phenomena have pushed forward our understanding of numerical approximations for stochastic differential equations with irregular coefficients which are discontinuous or even distributions. However, the current results only treat equations whose coefficients are uniformly bounded which leave out interesting cases that one encounters in practice. This Phd project aims to develop further these advances in numerical approximations of differential equations with unbounded coefficients, and therefore will address practical implementations.

<p>Consider, for instance, a multidimensional stochastic differential equation driven by Brownian motion with a drift and its corresponding Euler-Maruyama scheme. We are interest in the optimal convergence rate of the scheme in situations when the drift is a measurable function of time and space. Note that we do not assume any continuity property on the drift.</p> <p>When the drift is bounded, the recent article “<a data-clk="hl=en&sa=T&ei=ckdzZd2OBYzCmgGp_5zQBg" href="https://projecteuclid.org/journals/annals-of-applied-probability/volume-33/issue-3/Quantifying-a-convergence-theorem-of-Gy%C3%B6ngy-and-Krylov/10.1214/22-AAP1867.short">Quantifying a convergence theorem of Gyöngy and Krylov</a>” by Konstantinos Dareiotis, Máté Gerencsér, Khoa Lê obtains a strong convergence rate of order ½, which is the same as in the classical case when the drift is Lipschitz continuous. An extensions of this result to the case of integrable drift is reported in “<a data-clk="hl=en&sa=T&ei=M0hzZcq-LozCmgGp_5zQBg" href="https://arxiv.org/abs/2110.01343">Taming singular stochastic differential equations: A numerical method”</a> by Khoa Lê and Chengcheng Ling.</p> <p>The  current challenging problem would be extending these results to equations with growing drifts (for example, unbounded measurable drifts).</p> <p>Similar problems for stochastic partial differential equations could be also considered.</p>

<p class="MsoNoSpacing">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 <em><strong>PhD Statistics FT</strong></em> and in the research information section that the research degree you wish to be considered for is <em><strong>Numerical approximations of irregular differential equations</strong></em> as well as <a href="https://eps.leeds.ac.uk/faculty-engineering-physical-sciences/staff/12207/khoa-le">Dr Khoa Le</a> as your proposed supervisor. Please state in the Finance section the funding source you are applying for, if you are self-funding or are externally sponsored (please include the name of your sponsor).</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>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.</p> <p>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.</p> <p class="MsoNoSpacing"><strong>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.</strong></p> <p><strong>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:</strong></p> <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>

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 style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK</strong> – The <a href="http://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> <strong>(closing date: Monday 3 February 2025)</strong>, <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: Tuesday 1 April 2025)</strong> and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> are available to UK applicants.</p> <p><strong>Non-UK</strong> – <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> 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> <strong>(closing date: Monday 6 January 2025) </strong>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>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.</p> <p><strong>Important:</strong> Please note that that the award does <em><strong>not</strong></em> 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. </p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p class="MsoNoSpacing">For questions about this research project, please contact Dr Khoa Le by email to <a href="mailto:K.Le@leeds.ac.uk">K.Le@leeds.ac.uk</a>.</p> <p>For general enquiries about your application, please contact PGR Admissions by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a>. </p>