# Probability, Stochastic Modelling and Financial Mathematics

## Group Members

These web pages describe the Probability, Stochastic Modelling and Financial Mathematics research group at the University of Leeds.

## News

Upcoming
Our seminars take place approximately bi-weekly, normally on Thursday afternoons. A list of upcoming and recent talks can be found on the departemental seminar page.
14 April 2016 — seminar talk
Gabriele Stabile (La Sapienza University, Rome)
Optimal timing of an annuity purchase: a free boundary analysis.

Life annuities provide a long-life income stream that helps individuals to manage the longevity risk. Locating the optimal age (time) at which to purchase an irreversible life annuity is a problem that has received considerable attention in the literature over the past decade (see [1],[2],[3]). Intuitively, one might think that individuals should annuitize their wealth in case their financial investments performs poorly, due to the fear that the performance might become even worst. An alternative reasoning suggest that, in order to have an acceptable annuity payment, individuals should switch to annuities if the financial performance are high enough. This paper aims at investigating the annuitization problem with a rigorous mathematical analysis. From a mathematical point of view, such problem is formulated as a continuous time optimal stopping problem. The optimal time of the annuity purchases depends both on the individual's wealth and life expectancy. It is characterized as the first time the individual's wealth cross an unknown boundary, that divides the time-wealth plane into the so called continuation and stopping regions (where it is optimal respectively to postpone or immediately purchase an annuity). We study the regularity of this boundary, and we prove that it solves an integral equation of Volterra type.

References

[1] Gerrard, D., HĂ¸jgaard, B., and Vigna, E. (2012), Choosing the optimal annuitization time post-retirement Quantitative Finance 12, 1143-1159.

[2] Hainaut, D., and Deelstra, G. (2014), Optimal timing for annuitization, based on jump diffusion fund and stochastic mortality, Journal of Economic Dynamics & Control 44, 124-146.

[3] Milevsky, M.A., and Young, V.R.(2007), Annuitization and asset allocation, Journal of Economic Dynamics & Control 31, 3138-3177.

17 March 2016 — seminar talk
Aretha Teckentrup (University of Warwick)
Gaussian Process Emulators in Bayesian Inverse Problems.

A major challenge in the application of sampling methods to large scale inverse problems, is the high computational cost associated with solving the forward model for a given set of input parameters. To overcome this difficulty, we consider using a surrogate model that approximates the solution of the forward model at a much lower computational cost. We focus in particular on Gaussian process emulators, and analyse the error in the posterior distribution resulting from this approximation.

3 March 2016 — seminar talk
Alexander Veretennikov (University of Leeds)
Multivariate Bernstein polynomials.

About 100 years ago Sergey Bernstein - a big name primarily in PDE and probability but also in some other areas such as constructive function theory - published a two page note in French in a local Kharkov mathematical journal about a new probabilistic way to establish Polynomial Weierstrass' approximation theorem, the proof being based on the LLN for Bernoulli trials. Since then, this topic was developed enormously and became a self-contained branch of approximation theory with a flavour of probability. Bernstein's note was translated into English (and other languages) and presented verbatim and with comments in other journals, in some cases under the names of the translators, although, of course, without a pretence that the result is their. At least in two places (known to me) in Canada and Romania mathematical schools specialising in this area arose and probably the speaker is simply not aware of other such places; new advances related directly to this celebrated note are publishing practically every year. In particular, multivariate versions of Bernstein polynomials were introduced as late as in the 60s. Derivatives and rates of convergence for these polynomials under various assumptions were also investigated both theoretically and via modelling, new versions for approximations on infinite intervals were proposed, etc. In stochastic calculus, as many of the readers of this abstract may know, there is a popular way of proving Ito's formula based on a version of Weierstrass' theorem with derivatives''. In fact, this is a well-known but (in dimension >1) a bit non-standard stuff, which most of standard textbooks usually avoid (while in 1D approximations along with several derivatives is a trivial consequence of the standard Bernstein polynomial technique). So a natural curiosity arose whether or not this may be provided by Bernstein (multivariate) polynomials, and the speaker was unable to find a good reference (although a lot of more or less close materials were found). After several attempts of googling and yandexing, he decided to write and publish a new paper on the topic (2015). Exactly this paper will be presented as a tribute to S.N.Bernstein. So far, the original text is only in Russian at (Siberian) Mathematical Trudy, yet with an abridged English version at arXiv 1507.05235. The work is joint with Evguenia Veretennikova. There is an educated guess that this presentation should be available to a much wider than usual audience and literally everybody is welcome.

25 February 2016 — seminar talk
Mohammud Foondun (Loughborough)
On some qualitative behaviour for a class of fractional stochastic heat equations.

In this talk, we will look at a class of stochastic heat equation with the fractional Laplacian. Under various conditions, we will show how one can approximate these equations by a class of interacting stochastic differential equations. If time permits, we will also describe some recent results about equations involving time-fractional derivatives.

11 February 2016 — seminar talk
Anthony Lee (University of Warwick)
Variance estimation and allocation in the particle filter.

We introduce estimators of the variance and weakly consistent estimators of the asymptotic variance of particle filter approximations. These estimators are defined using only a single realization of a particle filter, and can therefore be used to report estimates of Monte Carlo error together with such approximations. We also provide weakly consistent estimators of individual terms appearing in the asymptotic variance, again using a single realization of a particle filter. When the number of particles in the particle filter is allowed to vary over time, the latter permits approximation of their asymptotically optimal allocation. Some of the estimators are unbiased, and hence generalize the i.i.d. sample variance to the non-i.i.d. particle filter setting. [arXiv:1509.00394]

10 December 2015 — seminar talk