L4

We study fully discrete approximation of the 2D Euler equations for ideal homogeneous fluids. We focus on spectral methods and  discuss rates of convergence of velocity and vorticity under different assumptions on the smoothness of the data.

In this seminar we introduce a portfolio optimisation framework, in which the use of rough path signatures (Lyons, 1998) provides a novel method of incorporating path-dependencies in the joint signal-asset dynamics, naturally extending traditional factor models, while keeping the resulting formulas lightweight, tractable and easily interpretable. Specifically, we achieve this by representing a trading strategy as a linear functional applied to the signature of a path (which we refer to as “Signature Trading” or “Sig-Trading”). This allows the modeller to efficiently encode the evolution of past time-series observations into the optimisation problem. In particular, we derive a concise formulation of the dynamic mean-variance criterion alongside an explicit solution in our setting, which naturally incorporates a drawdown control in the optimal strategy over a finite time horizon. Secondly, we draw parallels between classical portfolio stategies and Sig-Trading strategies and explain how the latter leads to a pathwise extension of the classical setting via the “Signature Efficient Frontier”. Finally, we give explicit examples when trading under an exogenous signal as well as examples for momentum and pair-trading strategies, demonstrated both on synthetic and market data. Our framework combines the best of both worlds between classical theory (whose appeal lies in clear and concise formulae) and between modern, flexible data-driven methods (usually represented by ML approaches) that can handle more realistic datasets. The advantage of the added flexibility of the latter is that one can bypass common issues such as the accumulation of heteroskedastic and asymmetric residuals during the optimisation phase. Overall, Sig-Trading combines the flexibility of data-driven methods without compromising on the clarity of the classical theory and our presented results provide a compelling toolbox that yields superior results for a large class of trading strategies.

This is based on works with Blanka Horvath and Magnus Wiese.

We introduce the notion of a tensorially absorbing inclusion of C*-algebras, i.e., when a unital inclusion absorbs a strongly self-absorbing C*-algebra. This is a strong condition that ensures certain properties of both algebras (and their intermediate subalgebras) in a very strong sense. We discuss such inclusions, their non-triviality, and how often these inclusions appear.

We extend stochastic optimal transport to path-dependent settings. The problem is to find a semimartingale measure that satisfies general path-dependent constraints, while minimising a cost function on the drift and diffusion coefficients. Duality is established and expressed via non-linear path-dependent partial differential equations (PPDEs). The technique has applications in volatility calibration, including the calibration of path-dependent derivatives, LSV models, and joint SPX-VIX models. It produces a non-parametric volatility model that localises to the features of the derivatives. Another application is in the robust pricing and hedging of American options in continuous time. This is achieved by establishing duality in a space enlarged by the stopping decisions, and showing that the extremal points of martingale measures on the enlarged space are in fact martingale measures on the original space coupled with stopping times.

We show the super-hedging duality for multi-action options which generalise American options to a larger space of actions (possibly uncountable) than {stop, continue}. We put ourselves in the framework of Bouchard & Nutz model relying on analytic measurable selection theorem. Finally we consider information delay on the action component of the product space. Information delay is expressed as a possibility to look into the future in the dual formulation. This is a joint work with Ivan Guo, Shidan Liu and Zhou Zhou.

We consider the reinforcement learning problem of controlling an unknown dynamical system to maximise the long-term average reward along a single trajectory. Most of the literature considers system interactions that occur in discrete time and discrete state-action spaces. Although this standpoint is suitable for games, it is often inadequate for systems in which interactions occur at a high frequency, if not in continuous time, or those whose state spaces are large if not inherently continuous. Perhaps the only exception is the linear quadratic framework for which results exist both in discrete and continuous time. However, its ability to handle continuous states comes with the drawback of a rigid dynamic and reward structure.

        This work aims to overcome these shortcomings by modelling interaction times with a Poisson clock of frequency $\varepsilon^{-1}$ which captures arbitrary time scales from discrete ($\varepsilon=1$) to continuous time ($\varepsilon\downarrow0$). In addition, we consider a generic reward function and model the state dynamics according to a jump process with an arbitrary transition kernel on $\mathbb{R}^d$. We show that the celebrated optimism protocol applies when the sub-tasks (learning and planning) can be performed effectively. We tackle learning by extending the eluder dimension framework and propose an approximate planning method based on a diffusive limit ($\varepsilon\downarrow0$) approximation of the jump process.

        Overall, our algorithm enjoys a regret of order $\tilde{\mathcal{O}}(\sqrt{T})$ or $\tilde{\mathcal{O}}(\varepsilon^{1/2} T+\sqrt{T})$ with the approximate planning. As the frequency of interactions blows up, the approximation error $\varepsilon^{1/2} T$ vanishes, showing that $\tilde{\mathcal{O}}(\sqrt{T})$ is attainable in near-continuous time.

In this short course, we survey the celebrated weak and strong compactness theorems proved by Karen Uhlenbeck in 1982. These results are fundamental to the gauge theory and have found numerous applications to geometry, topology, and theoretical physics. The proof is based on the ingenious idea of putting connections into ``Uhlenbeck--Coulomb gauge'', which enables the use of standard elliptic and/or nonlinear PDE techniques, as well as involved local-to-global patching arguments. We aim at giving detailed explanation of the proof, and we shall also discuss the relation between Uhlenbeck's compactness and the classical geometric problem of isometric immersions of submanifolds into Euclidean spaces.