Term 1 Year 2023/2024

Date: 13th of December 2023

Anran Hu (University of Oxford): Mean-Field Approximations in Heterogeneous N-Player Games

Abstract: Mean-field games (MFGs) offer a valuable approach to approximating and analyzing the challenging N-player stochastic games. However, existing literature primarily addresses approximation errors in MFGs and N-player games when players are permutation invariant. The rate of convergence remains undetermined for general N-player games. This talk addresses this gap by presenting the first non-asymptotic approximation results for multi-population MFGs (MP-MFGs) compared to heterogeneous N-player games. We initiate our exploration with mean-field type N-player games, featuring K groups of identical and permutation-invariant players. Notably, we establish non-asymptotic approximation error bounds without assuming the uniqueness of Nash equilibrium solutions. The analysis then extends to generic heterogeneous N-player games, encompassing variations in rewards, transition probabilities, and interactions among players that go beyond mean-field type scenarios.

Date: 29th of November 2023

Neofytos Rodosthenous (University College London ): Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities

Abstract: We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Lévy process with positive jumps, we prove the existence as well as explicitly construct a Nash equilibrium. Joint work with Jose Luis Perez and Kazutoshi Yamazaki.

Date: 22nd of November 2023

Albina Danilova (LSE): Order routing and market quality: Who benefits from internalization?

Abstract: Does retail order internalization benefit, via price improvement, or disadvantage, via reduced liquidity, retail traders? To answer this question, we compare two market designs that differ in their mode of liquidity provision: in the setting capturing retail order internalization liquidity is provided by market makers (representing wholesalers) competing for the retail order flow in a Bertrand fashion. Instead, in the open exchange setting price-taking competitive agents act as liquidity providers. We discover that, when liquidity providers are risk averse, routing of marketable orders to wholesalers is preferred by all retail traders: informed, uninformed and noise. Furthermore, most measures of liquidity are unaffected by the market design.

Date: 15th of November 2023

Alvaro Cartea (University of Oxford): Spoofing with Learning Algorithms

Abstract: This paper proposes a dynamic model of the limit order book to derive conditions to test if a trading algorithm will learn to spoof the order book. The testable conditions are simple and easy to implement because they depend only on the parameters of the model. We test the conditions with order book data from Nasdaq and show that market conditions are conducive for an algorithm to learn to spoof the order book. Co-authors: Patrick Chang and Gabriel Garcia-Arenas.

Date: 8th of November 2023

David Itkin (Imperial College London): Ergodic robust maximization of asymptotic growth with stochastic factors

Abstract: We consider an asymptotic robust growth problem under model uncertainty and in the presence of (non-Markovian) stochastic covariance. Building on the previous work of Kardaras & Robertson we fix two inputs representing the instantaneous covariation and invariant density for the asset process X, but additionally allow these quantities to depend on a stochastic factor process Y. Under mild technical assumptions we show that the robust growth optimal strategy is functionally generated and, unexpectedly, does not depend on the factor process Y. Remarkably this remains true even if the joint covariation of X and Y is prescribed as an input. Our result provides a comprehensive answer to a question proposed by Fernholz in 2002. The methods of proof use a combination of techniques from partial differential equations, calculus of variations, and generalized Dirichlet forms. This talk is based on joint work with Benedikt Koch, Martin Larsson and Josef Teichmann.

Date: 25th of October 2023

Ofelia Bonesini (Imperial College London): Correlated equilibria for mean field games with progressive strategies

Abstract: In a discrete space and time framework, we study the mean field game limit for a class of symmetric N-player games based on the notion of correlated equilibrium. We give a definition of correlated solution that allows to construct approximate N-player correlated equilibria that are robust with respect to progressive deviations. We illustrate our definition by way of an example with explicit solutions.

Date: 18th of October 2023

Purba Das (King's College London): Hölder regularity and roughness: Construction and examples

Abstract: We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of Hölder exponent. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.

Date: 4th of October 2023

Moris Strub (WBS): How to choose a model for portfolio selection? A consequentialist approach

Abstract: We propose a consequentialist approach to model selection: Models should be determined not according to statistical criteria, but in view of how they are used. This principle is then studied in detail in the domain of continuous-time portfolio choice. We consider an econometrician with prior beliefs on the likelihood of models to transpire and faced with the task of communicating a single model to an investor. The investor then takes the model communicated by the econometrician and invests according to the strategy maximizing expected utility within this model. The investor receives the consequential performance of trading according to the model communicated by the econometrician in a potentially different model that accurately describes the world. The objective of the econometrician is to choose the model that maximizes the consequential performance of the investor averaged over the likelihood of models to transpire and weighted according to the risk preferences of the econometrician. Our key finding is that it is in the best to communicate a model that is more optimistic than an unbiased estimator would suggest.

Term 3 Year 2022/2023

Date: 14th of July 2023

Samuel Cohen (Oxford): Stability and approximation of projection filters

Abstract: 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)

Date: 21st of June 2023

Tolulope Fadina (Essex): Measures of Risk under Uncertainty

Abstract: A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure and present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Value-at-Risk, a relevant issue for risk management practice.

Date: 24th of May 2023

Umut Cetin (LSE): Speeding up the Euler scheme for killed diffusions

Abstract: Let X be a linear diffusion taking values in (l,r) and consider the standard Euler scheme to compute an approximation to E[g(X(T);T<U]] for a given function g and a deterministic T, where U is the first time that X exits (l.r). It is well-known that the presence of killing introduces a loss of accuracy.. We introduce a drift-implicit Euler method to bring the convergence rate back to the optimal rate that can be obtained in the absence of killing, using the theory of recurrent transformations developed recently. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.

Date: 10th of May 2023

Max Nendel (Bielefeld University): A Parametric Approach to the Estimation of Convex Risk Functionals based on Wasserstein Distance

Abstract: We explore a static setting for the assessment of risk in the context of mathematical finance and actuarial science that takes into account model uncertainty in the distribution of a possibly infinite-dimensional risk factor. We allow for perturbations around a baseline model, measured via Wasserstein distance, and we investigate to which extent this form of probabilistic imprecision can be parametrized. The aim is to come up with a convex risk functional that incorporates a safety margin with respect to nonparametric uncertainty and still can be approximated through parametrized models.The particular form of the parametrization allows to develop a numerical method, based on neural networks, which gives both the value of the risk functional and the optimal perturbation of the reference measure. Moreover, we study the problem under additional constraints on the perturbations, namely, a mean and a martingale constraint. We show that, in both cases, under suitable conditions on the loss function, it is still possible to estimate the risk functional by passing to a parametric family of perturbed models, which again allows for a numerical approximation via neural networks. The talk is based on joint work with Alessandro Sgarabottolo.

Date: 3rd of May 2023

Eyal Neuman (Imperial College London): Optimality and Statistical Learning of Propagator Models

Abstract: Price impact refers to the empirical fact that execution of a large order affects the risky asset's price in an adverse and persistent manner leading to less favourable prices. Propagator models help us to quantity the price impact. They express price moves in terms of the influence of past trades, convoluted with a price impact kernel function. We consider a class of optimal liquidation problems where the agent's transactions create transient price impact driven by a Volterra-type propagator along with temporary price impact. We formulate these problems as minimization of a revenue-risk functionals, where the agent also exploits available information on a progressively measurable price predicting signal. By using an infinite dimensional stochastic control approach, we derive analytic solutions to these equations which yields an explicit expression for the optimal trading strategy.

We then consider a class of learning problems in which an agent liquidates a risky asset while creating transient price impact driven by an unknown propagator. We characterize the trader's performance as maximization of a revenue-risk functional, where the trader also exploits available information on a price predicting signal. We present a trading algorithm that alternates between exploration and exploitation phases and achieves sublinear regrets with high probability. For the exploration phase we propose a novel approach for non-parametric estimation of the price impact kernel by observing only the visible price process and derive sharp bounds on the convergence rate, which are characterised by the singularity of the propagator.

The talk is based on joint projects with Eduardo Abi-Jaber and Yufei Zhang.

Date: 15th February 2023

Adrien Richou (Université de Bordeaux): BSDEs reflected in a non convex domain: a geometric point of view

Abstract: In a recent paper, we have proved, with J.-F. Chassagneux and S. Nadtochiy, some existence and uniqueness results for BSDEs reflected in a non-convex domain under some restrictive assumptions on the domain and the terminal condition. All these results were obtained by tools and estimates based on the Euclidean structure of $\mathbb{R}^d$. In order to improve these results, at least in dimension $2$, it is also possible to see our domain as a flat manifold with a boundary and to take advantage of geometry tools already developed to tackle martingales in (non flat) manifolds (without boundary). In this talk, I will explain this new approach and the kind of results we are able to obtain. This is a work in progress with M. Arnaudon, J.-F. Chassagneux and S. Nadtochiy.

Date: 3rd November

Leandro Sanchez-Betancourt (King's College London): Internalise or Externalise: Brokers and Informed Trading

Abstract: We study how a broker provides liquidity to an informed trader and to a noise trader. The broker decides how much of the flow she keeps in her books (i.e., internalisation) and how much she unwinds in an exchange (i.e., externalisation). We frame the interactions between the broker and traders as a Stackelberg game. The informed trader knows the stochastic process that drives the drift of the asset price. The order flow of the noise trader is uninformative. We obtain the broker's internalisation and externalisation optimal strategy in closed-form. We show the performance of the broker, the noise trader, and the informed trader for a variety of scenarios. Lastly, we compute the amount of transaction costs that the broker needs to charge to break even.

Date: 6th October

Michael Kupper (Universität Konstanz): Nonlinear semigroups and limit theorems for convex expectations

Abstract: Motivated by model uncertainty, we focus on semigroups of convex monotone operators on spaces of continuous functions. In contrast to the linear theory, the domain of the generator is not invariant. In order to overcome this issue, we consider so-called Lipschitz sets which turn out to be a suitable domain for a weaker notion of the generator. This is defined using Gamma-convergence in an appropriate function space. We show that the Gamma-generator uniquely characterizes the nonlinear semigroup. In particular, we obtain that different approximation schemes lead to the same semigroup. As an application of our results, we show that LLN and CLT type results for convex expectations can be systematically obtained by the so-called Chernoff approximation. The talk is based on joint work with Jonas Blessing, Robert Denk and Max Nendel.

Date: 29th September

Mihail Zervos (LSE): Risk Sharing with Mean-Variance Preferences and Proportional Transaction Costs

Abstract: We consider an economy with two agents. Each of the two agents receives a random endowment flow. We model this cumulative flow as the the stochastic integral of a deterministic function of the economy's state, which we model by means of a general Ito diffusion. Each of the two agents has mean-variance preferences with different risk-aversion coefficients. To hedge against the random fluctuations of their individual endowments, the two agents may enter a risk-sharing agreement to trade a risky asset that is in zero net supply. We determine the agents' optimal equilibrium trading strategies in the presence of proportional transaction costs. In particular, we derive a new free-boundary problem that provides the solution to the agents' optimal equilibrium problem. Furthermore, we derive the explicit solution to this free-boundary problem when the problem data is such that the frictionless optimiser is a strictly increasing or a strictly increasing and then strictly decreasing function of the economy's state.