Skip to main content

Theoretical analysis of numerical schemes for stochastic (partial) differential equations

PGR-P-217

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 Elena Issoglio
Schools
School of Mathematics
Research groups/institutes
Statistics
<h2 class="heading hide-accessible">Summary</h2>

The theory and numerics of stochastic differential equations (SDEs) are well understood in the context of equations with `regular' coefficients. An important effort is currently being made in the field, in the attempt to go beyond the canonical setup. However, the challenge is huge, and results are often very specialised and not easily extendable to different equations. This project aims at developing a solid theoretical analysis of numerical schemes for backward SDEs and stochastic partial differential equations (SPDEs) whose coefficients have very low regularity (typically elements in the class of Schwarz distributions). As an example, the type of SPDEs mentioned above features naturally in physical problems modelled by transport equations in porous media, like water flowing through porous rocks: in this case the velocity of the flow is modified at the level of individual molecules, because the size of the water molecules is comparable to that of rock's pores. This can be mathematically modelled by taking the velocity as a very `rough' function of space (e.g., Schwarz distributions). A starting point for this work would be as follows. We should consider sequences of regularised versions of the stochastic equation under study (i.e., involving a mollification of the `rough' coefficients). This approach is natural but hardly trivial because the convergence (and convergence rate) of the scheme will be strongly depending on a clever choice of the regularising sequence and on the norms that one adopts on the relevant functional spaces. Current theoretical work of Dr Issoglio on these kinds of backward SDEs and SPDEs as well as some early results on numerical methods for SDEs can guide the start of the project. The potential outcomes of such study are likely to be of interest to the wide community of researchers working in stochastic analysis and PDE theory. Keywords: numerical methods, stochastic differential equations, BSDEs, SPDEs, irregular coefficients

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

<p>The earliest start for this project is 1 October 2020.</p>

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

<p>Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/info/130206/applying/91/applying_for_research_degrees">University&rsquo;s website</a>. Please state clearly in the research information section&nbsp;that the research degree you wish to be considered for is &lsquo;Theoretical analysis of numerical schemes for stochastic (partial) differential equations&rsquo; as well as&nbsp;<a href="https://physicalsciences.leeds.ac.uk/staff/41/dr-elena-issoglio">Dr Elena Issoglio</a>&nbsp;as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p><em>We welcome applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.</em></p>

<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>

<p><strong>Self-Funding Students</strong></p> <p><strong>Funding Eligibility</strong></p> <p><strong>UK/EU</strong> &ndash;&nbsp;Leeds Doctoral Scholarship Award paying Academic Fees and Maintenance matching EPSRC rate of &pound;15,009 per year for 3 years, School of Mathematics Scholarship award paying Academic Fees and Maintenance matching EPSRC rate of &pound;15,009 per year for 3 years.&nbsp; Alumni Bursary is available to previous University of Leeds graduates offering 10% discount on Academic Fees.</p> <p><strong>International Students</strong> &ndash;&nbsp;China Scholarship Council-University of Leeds Scholarship Award paying Academic Fees for 3 years,&nbsp;School of Mathematics Scholarship award paying Academic Fees for 3 years, Commonwealth Scholarship and Commonwealth Split Site Scholarships.&nbsp; Alumni Bursary is available to previous University of Leeds graduates offering 10% discount on Academic Fees.</p>

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

<p>For further information regarding your application, please contact Doctoral College Admissions by email:&nbsp;<a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a>, or by telephone: +44 (0)113 343 5057.</p> <p>For further information regarding the project, please contact Dr Elena Issoglio by email:&nbsp;&nbsp;<a href="mailto:E.Issoglio@leeds.ac.uk">E.Issoglio@leeds.ac.uk</a></p>


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