Schedule for Spring 2024
Seminars are on Thursdays
Time: 4:10pm  5:25pm
Location: Room 903 SSW
Organizers: Steven Campbell, Ioannis Karatzas, Marcel Nutz, Philip Protter
1/18/24

Krzysztof Ciosmak (Toronto)
Title: Localisation for constrained transports Abstract: Martingale optimal transport is a tool that allows for a modelfree pricing of options. It turns out that any martingale transport between given two probabilities is constrained by certain convex sets, dubbed irreducible components. We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures ordered with respect to a lattice cone F of functions on X. Under the assumption that any Ftransport between the two measures is local, we establish the existence of the finest partitioning of X, depending only on the measures and the cone F, into Fconvex sets, called irreducible components, such that any Ftransport between the measures must adhere to this partitioning. Furthermore, we demonstrate that a set, whose sections are contained in the corresponding irreducible components, is a polar set with respect to all Ftransports between the two measures if and only if it is a polar set with respect to all transports. This provides an affirmative answer to a generalisation of a conjecture proposed by Obłój and Siorpaes regarding polar sets in the martingale transport setting. Among our contributions is also a generalisation of the Strassen’s theorem to the setting of generalised convexity. We present applications to the localisation of the Monge—Kantorovich problem, to the martingale transport problem and to the submartingale transport problem in the infinitedimensional setting.

1/25/24 
Moritz Voss (UCLA) Title: Equilibrium in functional stochastic games with meanfield interaction

2/1/24 
Nizar Touzi (NYU) Title: Mean field game of crossholding Abstract: We consider the mean field game of crossholding introduced in Djete and Touzi in the context where the equity value dynamics are affected by a common noise. In this context, the problem exhibits the standard paradigm of meanvariance trade off. Our crucial observation is to search for equilibrium solutions of our mean field game among those models which satisfy an appropriate notion of noarbitrage. Under this condition, it follows that the representative agent optimization step is reduced to a standard portfolio optimization problem with random endowment. 
2/8/2024

NO SEMINAR 
2/15/2024

Graeme Baker (Columbia) Title: Two approaches to meanfield systemic risk models with default cascades

2/22/24

Silvana Pesenti (Toronto) Title: Optimal transport divergences based on scoring functions and their applications. Abstract: We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the MongeKantorovich (MK) optimal transport problem. This gives raise to a rich variety of novel asymmetric MK divergences, which subsume the family of BregmanWasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority the new divergences. Specifically, we derive the optimal coupling of the MK divergences induced by functionals including the mean, generalised quantiles, expectiles, and shortfall measures. Furthermore, we show that while any elicitable lawinvariant convex risk measure gives raise to infinitely many MK divergences, the comonotonic coupling is simultaneously optimal. The novel MK divergences, which can be efficiently calculated, open an array of applications in robust stochastic optimisation. We derive sharp bounds on distortion risk measures under a BregmanWasserstein divergence constraint, and solve for costefficient payoffs under benchmark constraints. 
2/29/24

Harvey Stein (Two Sigma) Title: Functions representable by neural networks
Abstract: It's hard to express the extent to which deep neural networks have transformed machine learning. Networks continue to get larger and more complex and find winder and wider application. On the other hand, some of the classical approximation theorems show that one layer is sufficient to approximate continuous functions on a bounded region. But these theorems are not constructive in nature.
Here, we give an explicit construction to show that all piecewise linear functions can be approximated by ReLU networks with two layers and can be exactly replicated by ReLU networks with three layers without the use of infinite parameters or with two layers if appropriate infinite parameters are available.

3/7/24

Philippe Bergault (Paris Dauphine) Title: A Mean Field Game between Informed Traders and a Broker Abstract: We find closedform solutions to the stochastic game between a broker and a meanfield of informed traders. In the finite player game, the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a Gâteaux derivative approach, we characterise the solution to the game with a system of forwardbackward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory, on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients. (Joint work with Leandro SanchezBetancourt) 
3/14/24

NO SEMINAR 
3/21/24

NO SEMINAR 
3/28/24 
Roger Lee (U Chicago) Title: All AMMs are CFMMs. All DeFi markets have invariants. A DeFi market is arbitragefree if and only if it has an increasing invariant Abstract: In a universal framework that expresses any market system in terms of state transition rules, we prove that every DeFi market system has an invariant function and is thus by definition a CFMM; indeed, all automated market makers (AMMs) are CFMMs. Invariants connect directly to arbitrage and to completeness, according to two fundamental equivalences. First, a DeFi market system is, we prove, arbitragefree if and only if it has a strictly increasing invariant, where "arbitragefree" means that no state can be transformed into a dominated state by any sequence of transactions. Second, the invariant is, we prove, unique if and only if the market system is complete, meaning that it allows transitions between all pairs of states in the state space, in at least one direction. Thus a necessary and sufficient condition for noarbitrage (respectively, for completeness) is the existence of the increasing (respectively, the uniqueness of the) invariant, which, therefore, fulfills in nonlinear DeFi theory a foundational role parallel to the existence (respectively, uniqueness) of the pricing measure in the Fundamental Theorem of Asset Pricing for linear markets. Moreover, a market system is recoverable by its invariant if and only if it is complete. Our examples illustrate (non)existence of various specific types of arbitrage in the context of various specific types of market systems  with or without fees, with or without liquidity operations, and with or without coordination among multiple pools  but the fundamental theorems have full generality, applicable to any DeFi market system and to any notion of arbitrage expressible as a strict partial order. 
4/4/24 
Luhao Zhang (Columbia) Title: Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions Abstract: Decisionmaking in modern stochastic systems, including ecommerce platforms, financial markets, and healthcare systems, has evolved into a multifaceted process that involves information acquisition and adaptive information sources. This paper initiates a study on this integrated process, where these elements are not only fundamental but also interact in a complex and dynamically intertwined manner. We introduce a relatively simple model, which, however, captures the novel elements we consider. Specifically, a decision maker (DM) can choose between an established product A with a known value and a new product B with an unknown value. The DM can observe signals about the unknown value of product B and can also opt to exchange it for product A if B is initially chosen. Mathematically, the model gives rise to a sequential optimal stopping problem with two different informational regimes (before and after buying product B), differentiated by the initial, coarser signal and the subsequent, finer one. We analyze the underlying problems using predominantly viscosity solution techniques, differing from the existing literature on information acquisition which is based on traditional optimal stopping techniques. Additionally, our modeling approach offers a novel framework for developing more complex interactions among decisions, information sources, and information costs through a sequence of nested obstacles. 
4/11/24

Michel Groppe (Goettingen) Title: Lower Complexity Adaptation for Empirical Entropic Optimal

4/18/24 
Johannes Ruf (LSE) Title: The numeraire evariable and reverse information projection 
4/25/24 
Alvaro Cartea (Oxford) Title: Spoofing and Manipulating Order Books with Learning Algorithms Abstract: We propose a dynamic model of the limit order book to derive conditions to test if a trading algorithm will learn to manipulate the order book. Our results show that as a market maker becomes more tolerant to bearing inventory risk, the learning algorithm will find optimal strategies that manipulate the book more frequently. Manipulation occurs to induce mean reversion in inventory to an optimal level and to execute roundtrip trades with limit orders at a higher probability than was otherwise likely to occur; spoofing is a special case when the market maker prefers that manipulative limit orders are not filled. The conditions are tested with order book data from Nasdaq and we show that market conditions are conducive for an algorithm to learn to manipulate the order book. Finally, when two market makers use learning algorithms to trade, their algorithms can learn to coordinate their manipulation. Preprint: https://urldefense.proofpoint.com/v2/url?u=https3A__papers.ssrn.com_sol3_papers.cfm3Fabstract5Fid3D4639959&d=DwIFaQ&c=009klHSCxuh5AI1vNQzSO0KGjl4nbi2Q0M1QLJX9BeE&r=7pQWqsgCHQLqn2i_dni1TYLVnkt_lsEmJnAk4yTN4&m=6WwkfJ_akqZXp08nyNzzW_lM6P8f1QXNIEbmMJ2XiKYXofCpHNG6m_U2tX9apYq&s=Hu7UmhKuGo8eZW5179eI_VSTf9MSvfW0F3286F9itJ4&e=

4:00 p.m. Wednesday, May 1st, 2024

ColumbiaNYU Financial Engineering Colloquium
