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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 12.00 - 13:00, 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 Enfale Farooq, the Research Manager, on E.Farooq@lse.ac.uk for further information about any of these seminars.

Upcoming speakers:

Thursday 1 December 2022 - Linus Wunderlich (QMUL)
Hybrid - 12:00 on Zoom, and MAR.2.10

Neural networks for high-dimensional parametric option pricing using PDEs
In this talk we will discuss the deep parametric PDE method for parametric option pricing in high dimensions, underlying theoretical results for neural networks and an application in risk management.
The deep parametric PDE method uses deep neural networks to solve parametric partial differential equations, such as those arising in option pricing. Especially for a large number of risk factors, the efficiency of neural networks for high dimensional problems is beneficial. We investigate this efficiency theoretically by presenting approximation rates for networks with smooth activation functions. 

Previous seminars in the series: 

Thursday 3 November 2022 - Alexandre Pannier (Imperial College London)

On the ergodic behaviour of affine Volterra processes
We show the existence of a stationary measure for a class of multidimensional stochastic Volterra systems of affine type. These processes are in general not Markovian, a shortcoming which hinders their large-time analysis. We circumvent this issue by lifting the system to a measure-valued stochastic PDE introduced by Cuchiero and Teichmann, whence we retrieve the Markov property. Leveraging on the associated generalised Feller property, we extend the Krylov-Bogoliubov theorem to this infinite-dimensional setting and thus establish an approach to the existence of invariant measures. We present concrete examples, including the rough Heston model from Mathematical Finance.

Thursday 20 October 2022 - David Itkin (Imperial College London)

Open Markets in Stochastic Portfolio Theory and Rank Jacobi Processes
Stochastic portfolio theory is a framework to study large equity markets over long time horizons. In such settings investors are often confined to trading in an “open market” setup consisting of only assets with high capitalizations. In this work we relax previously studied notions of open markets and develop a tractable framework for them under mild structural conditions on the market.
Within this framework we also introduce a large parametric class of processes, which we call rank Jacobi processes. They produce a stable capital distribution curve consistent with empirical observations. Moreover, there are explicit expressions for the growth-optimal portfolio, and they are also shown to serve as worst-case models for a robust asymptotic growth problem under model ambiguity.
Lastly, the rank Jacobi models are shown to be stable with respect to the total number of stocks in the market. Time permitting, we will show that, under suitable assumptions on the parameters, the capital distribution curves converge to a limiting quantity as the size of the market tends to infinity. This convergence result provides a theoretical explanation for an important empirically observed phenomenon.

This talk is based on joint work with Martin Larsson.

Thursday 6 October 2022 - Joe Jackson (University of Texas)

Well-posedness for non-Markovian quadratic BSDE systems with special structure
In this talk I will discuss some recent existence and uniqueness results for non-Markovian quadratic BSDE systems. Much of the talk will actually be about linear BSDEs, because it turns out that estimates for an appropriate class of linear BSDEs can be used to obtain existence results for quadratic systems. Indeed, the Malliavin derivative of a BSDE satisfies a linear BSDE, and strong enough estimates on the Malliavin derivative can be used to obtain existence. The difficulty in executing this strategy in the quadratic case is that the relevant linear BSDEs have unbounded coefficients, which a-priori can only be estimated in a space we call bmo. In a series of recent works with Gordan Žitković, I studied linear BSDEs with bmo coefficients systematically, and the following picture has emerged: both existence and uniqueness may fail for such equations, but can be recovered under various structural conditions.

2021/222019/202018/192017/18, 2016/17, 2015/16 

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