Nonlinear filtering is a central mathematical tool in understanding how we process information. Sadly, the equations involved are often very high dimensional, which may lead to difficulties in applications. One possible resolution (due to D. Brigo and collaborators) is to replace the filter by a low-dimensional approximation, with hopefully small error. In this talk we will see how, in the case where the underlying process is a finite-state Markov Chain, results on the stability of filters can be strengthened to show that this introduces a well-controlled error, leveraging tools from information geometry. (Based on joint work with Eliana Fausti)

In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter H < 0.5. In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

Machine Learning (ML) provides techniques for universal function approximation. In this talk, we apply such techniques to the acceleration of complex derivatives pricing, focusing on Value-at- Risk computations for Bermudan interest rate options. We introduce different applicable ML methods, and we present results from our client projects. Moreover, we propose ways to address regulatory requirements via the model lifecycle process.

A constant weight asset allocation is a popular investment strategy and is optimal under a suitable continuous model. We study the tracking error for the target continuous rebalancing by a feasible finite-time rebalanc- ing under a general multi-dimensional Brownian semimartingale model of asset prices. In a high-frequency asymptotic framework, we derive an asymptotically efficient strategy among simple predictable processes.

Quantum observables determine for instance the stress tensor and heat flux in fluid dynamics. In the talk I will show which properties are used for approximating such quantum observables by classical molecular dynamics and what is new when mean-field molecular dynamics improves the classical setting, using features of the Gibbs distribution.

We study the optimal investment-consumption problem for an agent whose preferences are governed by Epstein–Zin stochastic differential utility and who invests in a constant-parameter Black–Scholes– Merton market. We assume that purchases and sales of the risky asset are subject to proportional transaction costs. We fully characterise all parameter combinations for which the problem is well posed (which may depend on the level of transaction costs). We also provide a full verification argument that relies on no additional technical assumptions and uses primary methods only. Even in the special case of power utility, our arguments are significantly simpler and more elegant than the results in the extant literature. A novel key idea is to parametrise consumption and the value function in terms of the shadow fraction of wealth. The talk is based on joint work with David Hobson and Alex Tse.

Conference webpage

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.