# Joint Risk & Stochastics and Financial Mathematics Seminar

The following seminars have been jointly organised by the Risk and Stochastics Group and the Department of Mathematics. The Seminar normally takes place bi-weekly on Thursdays from 12.00pm-1.00pm in room CLM.7.02 (Clement House, LSE), unless stated below. The series aims to promote communication and discussion of research in the mathematics of insurance and finance and their interface, to encourage interaction between practice and theory in these areas, and to support academic students in related programmes at postgraduate level. All are welcome to attend. Please contact the seminar administrator on seminar@maths.lse.ac.uk for further information about any of these seminars.

## Upcoming Speakers:

#### Thursday 23 November - 12.00 - CLM.7.02 Clement House, LSECharles-Albert Lehalle (Capital Fund Management) Closing The Loop of Optimal Trading: a Mean Field Game of Controls

This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

#### Thursday 7 December - 12.00 - CLM.7.02 Clement House, LSE

Thomas Kruse (Duisburg-Essen)
Multilevel Picard approximations for high-dimensional nonlinear parabolic partial differential equations

In this talk we present a family of new approximation methods for high-dimensional PDEs and BSDEs.

A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. Thereby we obtain a class of multilevel Picard approximations.
Our error analysis proves that for semi-linear heat equations, the computational
complexity of one of the proposed methods is bounded by $O(d\,\eps^{-(4+\delta)})$ for any $\delta > 0$, where $d$ is the dimensionality of the problem and $\eps\in(0,\infty)$ is the prescribed accuracy.

We illustrate the efficiency of one of the proposed approximation methods by means of numerical simulations presenting approximation accuracy against runtime for
several nonlinear PDEs from physics (such as the Allen-Cahn equation) and financial engineering (such as derivative pricing incorporating default risks) in the case of $d=100$ space dimensions.

The talk is based on joint work with W. E, M. Hutzenthaler, and A. Jentzen.

### Previous seminars in the series:

2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010