Unless otherwise specified, the Stochastic Finance seminar takes place on Wednesdays starting at 11:00 am. While the seminars will run in person, there is also the possibility to join via MS Teams. If you wish to be added to the respective Team, please contact the seminar organiser Miryana Grigorova.Link opens in a new window

All are welcome.

The first session in Term 1 in the academic year 2023/2024 is on the 4th of October.

How to choose a model for portfolio selection? A consequentialist approach 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.

Hölder regularity and roughness: Construction and examples 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.

Correlated equilibria for mean field games with progressive strategies 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.

Ergodic robust maximization of asymptotic growth with stochastic factors 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.

Spoofing with Learning Algorithms 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.

Order routing and market quality: Who benefits from internalization? 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.

Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities 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.

Anran Hu (University of Oxford) Mean-Field Approximations in Heterogeneous N-Player Games 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./td>

The first session in Term 2 is on the 31st of January 2024.

Superdiffusive limits for Bessel-driven stochastic kinetics We explore the scaling of anomalous diffusion in a one-dimensional stochastic kinetic dynamics model. Our model features stochastic drift influenced by external Bessel noise and incorporates internal volatility, which has an arbitrary relationship with this external noise. We identify the superdiffusive scaling exponent for the model, and prove a weak convergence result on the corresponding scale. We show how our result extends to admit, as exogenous noise processes, not only Bessel processes but more general processes satisfying certain asymptotic conditions. We conclude by exploring the connections with stochastic interest rate models. This talk is based on joint work with Conrado Da Costa, Aleks Mijatović and Andrew Wade.

Radner equilibrium and systems of quadratic BSDEs with discontinuous generators 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.)

Trapping Probability in Low-Income Household Capital Dynamics: Insurance and Subsidies Impact

We are examining a risk process characterized by deterministic growth and multiplicative jumps to represent the capital dynamics of low-income households. To account for the higher risk inherent in such environments, we assume that capital losses are proportional to the level of accumulated capital at the time of a jump. Our objective is to calculate the probability of a household falling below the poverty line, known as the trapping probability. "Trapping" arises when a household's capital falls below the poverty threshold, creating a situation from which escape without external help is not possible. We approach the remaining capital distribution as specific instances of the beta distribution and derive closed-form expressions for the trapping probability by analyzing the Laplace transform of the process' infinitesimal generator. Additionally, we investigate the impact of insurance on this probability, particularly when the insurance product provides proportional coverage. Our findings suggest that in certain scenarios, trapping is inevitable without external aid, typically provided in the form of subsidies.

Rough Stochastic Analysis with Jumps Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa Lê, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.

Global solutions for stochastically controlled models Abstract: For a class of evolution equations that possibly have only local solutions, we introduce a stochastic component that ensures that the solutions of the corresponding stochastically perturbed equations are global. The class of partial differential equations amenable for this type of treatment includes the 3D Navier-Stokes equation, the rotating shallow water equation (viscous and inviscid), 3D Euler equation (in vorticity form), 2D Burgers' equation and many other fluid dynamics models. This is based on joint work with Oana Lang (Babes-Bolyai University). https://arxiv.org/abs/2403.05923

Optimal stopping with nonlinear expectation: geometric and algorithmic solutions We use the geometry of functions associated with martingales under nonlinear expectations to solve risk-sensitive Markovian optimal stopping problems. Generalising the linear case due to Dynkin and Yushkievich (1969), the value function is the majorant or pointwise infimum of those functions which dominate the gain function. An emphasis is placed on the geometry of the majorant and pathwise arguments, rather than exploiting convexity, positive homogeneity or related analytical properties. An algorithm is provided to construct the value function at the computational cost of a two-dimensional search. The talk is based on the preprint https://arxiv.org/abs/2306.17623 (with Tomasz Kosmala).

Functional Itô formula and Taylor expansion for non-anticipative maps of rough paths: We rely on the approximation properties of the signature of càdlàg rough paths to derive a functional Itô formula for non-anticipative maps of rough paths. This leads to a functional extension of the Itô formula for càdlàg rough paths (by Friz and Zhang (2018)) which coincides with the change of variable formula formulated by Dupire (2009) as well as by Cont and Fournie (2010), whenever the notions of the regularity of the functionals and the integration coincide. As a byproduct, we show that sufficiently regular non-anticipative path functionals admit a functional Taylor expansion, leading to a far reaching generalization of the recently established results by Dupire and Tissot-Daguette (2022). The talk is based on ongoing joint work with Xin Guo and Francesca Primavera.

Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes. We are interested in the law of the first passage time of an Ornstein-Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first one is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities. This is joint work with my supervisors Dr. Larbi Alili and Dr.Massimiliano Tamborrino. Link to the pre-print: https://arxiv.org/abs/2210.01658

29th September

B 3.02 (Zeeman)

Mihail Zervos (London School of Economics)

Risk Sharing with Mean-Variance Preferences and Proportional Transaction Costs 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.

6th October

Michael Kupper (Universität Konstanz)

Nonlinear semigroups and limit theorems for convex expectations 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.

3rd November

Leandro Sanchez-Betancourt (King's College London)

Internalise or Externalise: Brokers and Informed Trading 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.

10th November

Alexandre Pannier (LPSM Paris)

On the ergodic behaviour of affine Volterra processes 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.

1st December

Alex Tse (University College London)

Periodic portfolio selection with quasi-hyperbolic discounting 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.

General Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents 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.

BSDEs reflected in a non convex domain: a geometric point of view 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.

Coupling of multidimensional Lévy processes and Wasserstein bounds in the small time stable domain of attraction 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.

Optimal consumption with labour income and borrowing constraints for recursive preferences 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.

Optimal Stopping with Trees 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.

Radner equilibrium and systems of quadratic BSDEs with discontinuous generators 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.)

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

Optimality and Statistical Learning of Propagator Models 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.

A Parametric Approach to the Estimation of Convex Risk Functionals based on Wasserstein Distance 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.

Measures of Risk under Uncertainty 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.

Stability and approximation of projection filters 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)