# Abstracts

### Term 2

Date: 15th of March 2023

**Daniel Schwarz **(University of Warwick): * Radner equilibrium and systems of quadratic BSDEs with discontinuous generators *

*/This talk has been cancelled and will be rescheduled in Term 3./*

**Abstract: ** We establish the existence of a Radner Equilibrium in an incomplete, continuous-time financial economy. To prove existence, we formulate a system of quadratic backward stochastic differential equations (BSDEs) which represents the equilibrium problem. Interestingly, the driver of this system of BSDEs features a discontinuity, posing a major challenge for common approaches to establish existence. Exploiting the duality between Markovian BSDEs and PDEs we use unique continuation and backward uniqueness, techniques originally used in the study of PDEs, to show that the set of discontinuity has in fact measure zero. (Joint work with Luis Escauriaza and Hao Xing.)

Date: 8th of March 2023

**Xin Zhi **(University of Warwick): *Optimal Stopping with Trees*

**Abstract: ** In this talk, we will first review a recent method for solving high-dimensional optimal stopping problems using deep Neural Networks. Second, we propose an alternative algorithm replacing Neural Networks by CART-trees which allow for more interpretation of the estimated stopping rules. We apply our algorithm to multiple examples. We in particular compare the performance of the two algorithms with respect to the Bermudan max-call benchmark example. We also show how our algorithm can be used to plot stopping boundaries.

Date: 1st of March 2023

**Olivier Menoukeu Pamen **(University of Liverpool): *Optimal consumption with labour income and borrowing constraints for recursive preferences*

**Abstract: ** In this talk, we present an optimal consumption and investment problem for an investor with liquidity constraints who has isoelastic recursive Epstein-Zin utility preferences and receives a stochastic stream of income. We characterise the optimal consumption strategy as well as the terminal wealth for recursive utility under dynamic liquidity constraints, which prevent the investor to borrow against his stochastic future income. Using duality and backward SDE methods in a possibly non-Markovian diffusion model for the financial market, this gives rise to an interplay of singular control and optimal stopping problems. This talk is based on a joint work with D. Becherer and W. D. Kuissi Kamdem.

Date: 22nd February 2023

**David Bang **(University of Warwick): *Coupling of multidimensional Lévy processes and Wasserstein bounds in the small time stable domain of attraction *

**Abstract: ** We establish upper and lower bounds on the rate of convergence of the Wasserstein distance on the path space for a wide class of Lévy processes attracted to a multidimensional stable law in the small-time regime. In this talk, the main focus will be on the development of two novel couplings between arbitrary pure-jump Lévy processes, used to obtain upper bounds on the Wasserstein distance. We show that the rate of convergence is polynomial for the domain of normal attraction and slower than any polynomial for the domain of non-normal attraction. As an example, we will consider the class of tempered stable processes that are in the small time domain of attraction of a stable process.

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: 1st February 2023

**Paolo Guasoni **(Dublin City University): *General Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents *

**Abstract: **We solve a general equilibrium model in which aggregate consumption has uninsurable growth shocks, rendering the market dynamically incomplete. Several long-lived agents with heterogeneous risk-aversion and time-preference make consumption and investment decisions, trading risky assets and borrowing from and lending to each other. For small growth fluctuations, we obtain closed-form expressions for stock prices, interest rates, and consumption and trading policies. Agents' stochastic discount factors depend on the history of unhedgeable shocks, agents trade assets dynamically, and the dispersion of agents' preferences impacts both the interest rate and asset prices, hence no representative agent exists.

### Term 1

Date: 1st December

**Alex Tse **(University College London): *Periodic portfolio selection with quasi-hyperbolic discounting *

**Abstract: **In this talk, I will introduce a continuous-time portfolio selection problem faced by an agent with S-shaped preference who maximises the discounted utilities derived from the portfolio's periodic performance over an infinite horizon. I will first briefly outline the solution method under a baseline exponential discounting setup. Then I will introduce a time-inconsistent version of the problem featuring quasi-hyperbolic discounting where multiple notions of optimality arise. If the agent is sophisticated who seeks a consistent planning strategy, the problem can then be analysed via a static mean field game where theoretical characterisation of the optimal strategy is provided.

Date: 10th November

**Alex Pannier** (LPSM Paris): *On the ergodic behaviour of affine Volterra processes*

**Abstract:** 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.

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.