Title. Martingale model risk
Abstract. We consider the general framework of distributionally robust optimization under a martingale restriction. We provide explicit expressions for model risk sensitivities in this context by considering deviations in the Wasserstein distance and the corresponding adapted one. As a by-product we obtain an explicit expression for first order model risk hedging.
Title. Linear-Quadratic Mean Field Games in Hilbert Spaces
Abstract. Originally developed in finite-dimensional spaces, mean field games (MFGs) have become pivotal in addressing large-scale problems involving numerous interacting agents, and have found extensive applications in economics and finance. However, there are scenarios where Euclidean spaces do not adequately capture the essence of a problem such as non-Markovian systems. A clear and intuitive example is systems involving time delays.
This talk presents a comprehensive study of linear-quadratic (LQ) MFGs in Hilbert spaces, generalizing the classic LQ MFG theory to scenarios involving $N$ agents with dynamics governed by infinite-dimensional stochastic equations. In this framework, both state and control processes of each agent take values in separable Hilbert spaces. All agents are coupled through the average state of the population which appears in their linear dynamics and quadratic cost functional. Specifically, the dynamics of each agent incorporates an infinite-dimensional noise, namely a $Q$-Wiener process, and an unbounded operator. The diffusion coefficient of each agent is stochastic involving the state, control, and average state processes. We
first study the well-posedness of a system of $N$ coupled semilinear infinite-dimensional stochastic evolution equations establishing the foundation of MFGs in Hilbert spaces. We then specialize to $N$-player LQ games described above and study the asymptotic behavior as the number of agents, $N$, approaches infinity. We develop an infinite-dimensional variant of the Nash Certainty Equivalence principle and characterize a unique Nash equilibrium for the limiting MFG. Finally, we study the connections between the $N$-player game and the limiting MFG, demonstrating that the empirical average state converges to the mean field and that the resulting limiting best-response strategies form an $\epsilon$-Nash equilibrium for the $N$-player game in Hilbert spaces.
Title. Exponential Expression Rates for Neural Operator Approximations to the Solution Operator of Certain FBSDEs
Abstract. The numerical solution to forward-backwards stochastic differential equations (FBSDEs) plays a central role in optimal control and its applications to game theory, economics, finance, and insurance. Most classical numerical and modern deep-learning schemes, however, have the disadvantage that they must be re-run every time the user specifies a new set of parameters and/or terminal conditions for an FBSDE, meaning that these methods cannot feasibly solve large families of FBSDEs. One possible solution is to consider a neural operator (NO) which ``learns to solve FBSDEs''; the NO outputs the solution to an FBSDE given inputs: a terminal condition and a generator of the backward process. Though the existence of such NOs is not surprising, it is unclear if they can be implemented using a few parameters. We establish exponential rates for NO approximations of the solution operator to a broad class of fully coupled FBSDEs with random terminal time. Our result is based on new exponential approximation rates for a class of convolutional NOs, which can efficiently encode Green's function to the Elliptic boundary problems associated with our FBSDEs. Joint work with: Takashi Furuya
Title. Optimal consumption under loss-averse multiplicative habit-formation preferences
Abstract. We consider a loss-averse version of the multiplicative habit formation preference and the corresponding optimal investment and consumption strategies over an infinite horizon. The agent's consumption preference is depicted by a general S-shaped utility function of her consumption-to-habit ratio. By considering the concave envelope of the S-shaped utility and the associated dual value function, we provide a thorough analysis of the HJB equation for the concavified problem via studying a related nonlinear free boundary problem. Based on established properties of the solution to this free boundary problem, we obtain the optimal consumption and investment policies in feedback form. Some new and technical verification arguments are developed to cope with generality of the utility function. The equivalence between the original problem and the concavified problem readily follows from the structure of the feedback policies. We also discuss some quantitative properties of the optimal policies under several commonly used S-shaped utilities, complemented by illustrative numerical examples and their financial implications.
Title. A Resource Sharing Model with Local Time Interactions
Abstract. We consider a resource sharing model for $N$ financial firms which takes the form of a semimartingale reflected Brownian motion in the positive orthant. When the asset value of any firm hits zero, the other firms contribute to a local time reflection term to keep the distressed firm's assets non-negative. Depending on a critical parameter $\alpha$, which reflects either friction or subsidy, the interval of existence for this scheme is shown to be either infinite, or almost surely finite. In the mean-field limit, the behaviour of the Fokker--Planck equation also depends on $\alpha$: either solutions exists for all time or the equation exhibits finite-time blowup. The passage from finite to mean-field model is investigated. We also analyze connections between our model and systemic risk models involving hitting times, the up-the-river problem of Aldous, various free boundary PDEs, and Atlas models from Stochastic Portfolio Theory.
Title. From rank-based models with common noise to pathwise entropy solutions of SPDEs
Abstract. We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers by Gess, Souganidis, Lions, and Perthame).
Title. Optimal Loss Reporting in Continuous Time with Full Insurance
Abstract. Experience-rating systems, such as bonus-malus systems, offer discounts on future premiums to insureds who maintain a claim-free record, thereby generating incentives for insureds to not report certain losses. This paper formulates an optimal loss-reporting problem in a continuous-time framework for an insured with full insurance. The insured follows a barrier strategy for reporting losses and aims to maximize the expected exponential utility of her terminal wealth over a random horizon t. In the special case when t has a constant hazard rate, we obtained the optimal barrier strategy in closed form, which is a strictly positive constant. In the general case of a time-varying hazard rate for t, we obtain the optimal barrier strategy in semi-closed form, subject to solving a system of ordinary differential equations. Additionally, we uncover several noteworthy qualitative insights into both the optimal barrier strategy and the associated value functions.
Title. Parametric Continuity in Problems of Optimal Stopping with Applications to American Options and Stopping Games
Abstract. We use the dual perspective for optimal stopping, introduced by Davis and Karatzas, to derive pathwise conditions for commonly sought properties of optimal stopping problems. In particular, we show that the continuity of the value function, the continuity of the optimal stopping time, the continuity of the spatial derivative of the value function, and the existence of equilibria in a large class of stopping games can be derived from the path properties of the reward process and its Snell envelope. We provide several illustrative examples and applications.
Title. Inference of Utilities and Time Preference in Sequential Decision-Making
Abstract. In this talk, we introduce a novel stochastic control framework to enhance the capabilities of automated investment managers, or robo-advisors, by accurately inferring clients' investment preferences from past activities. Our approach leverages a continuous-time model that incorporates utility functions and a generic discounting scheme of a time-varying rate, tailored to each client's risk tolerance, valuation of daily consumption, and significant life goals. We address the resulting time inconsistency issue through state augmentation and the establishment of the dynamic programming principle and the verification theorem. Additionally, we provide sufficient conditions for the identifiability of client investment preferences. To complement our theoretical developments, we propose a learning algorithm based on maximum likelihood estimation within a discrete-time Markov Decision Process framework, augmented with entropy regularization. We prove that the log-likelihood function is locally concave, facilitating the fast convergence of our proposed algorithm. Practical effectiveness and efficiency are showcased through two numerical examples, including Merton's problem and an investment problem with unhedgeable risks. Our proposed framework not only advances financial technology by improving personalized investment advice but also contributes broadly to other fields such as healthcare, economics, and artificial intelligence, where understanding individual preferences is crucial.
Title. Utility maximization under endogenous pricing
Abstract. We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. We show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which corresponds to a non-linear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case where the driver function of the representative market maker grows at least quadratically or the utility function of the large investor falls faster than quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete or the utility function is exponential.
Title. A mean field optimal stopping problem with common noise
Abstract. This paper focuses on a mean-field optimal stopping problem with common noise, inspired by Talbi, Touzi, and Zhang 2023. The goal is to establish the dynamic programming equation along with its comparison principle and convergence rate of the value function of the large population optimal stopping problem towards those of the mean field limit. The proof of comparison principle relies on the Ishii's Lemma proved in Bayraktar, Ekren, and Zhang 2023.
Title. Spanning Multi-Asset Payoffs with ReLUs
Abstract. We propose a distributional formulation of the spanning problem of a multi-asset payoff by vanilla basket options. This problem is shown to have a unique solution if and only if the payoff function is even and absolutely homogeneous, and we establish a Fourier-based formula to calculate the solution. Financial payoffs are typically piecewise linear, resulting in a solution that may be derived explicitly, yet may also be hard to exploit numerically. One-hidden-layer feedforward neural networks instead provide a natural and efficient numerical alternative for discrete spanning. We test this approach for a selection of archetypal payoffs and obtain better hedging results with vanilla basket options compared to industry-favored approaches based on single-asset vanilla hedges. Reference paper: https://arxiv.org/abs/2403.14231 (joint work with Stéphane Crépey and Hoang-Dung Nguyen, Université Paris-Cité)
Title. Neural Operators Can Play Dynamic Stackelberg Games
Abstract. Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{follower's best-response operator} can be approximately implemented, uniformly on compact sets of controls of the leader, by an \textit{attention-based neural operator}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.
Title. Risk bounds for the marginal expected shortfall under dependence uncertainty
Abstract. Measuring the contribution of a bank or an insurance company to the overall systemic risk of the market is an important issue, especially in the aftermath of the 2007-2009 Financial Crisis and the financial downturn of 2020. Specifically, the Office of the Superintendent of Financial Institutions (OSFI) in Canada recommends a multifaceted approach to address this problem, with two major components being the size of the company and its interconnectedness within the market.
One appropriate quantitative measure of systemic risk contribution is the marginal expected shortfall (MES), introduced by Acharya et al. (2017) and known to actuarial scholars as the risk capital allocation rule based on the expected shortfall risk measure since at least Panjer and Jing (2001). The MES risk measure has attracted considerable attention and has been evaluated under various assumptions on the joint probability distribution of companies’ risks in the market of interest. However, in reality, the univariate probability distributions of distinct companies in the market can be quite distinct, and the dependence structure between the companies is rarely known.
In this paper, we derive the upper and lower bounds of MES under the assumption of full information on the marginal distributions of the companies’ risks and an unknown dependence structure. Furthermore, we also derive improved bounds for the MES risk measure when partial information on companies’ risk exposures–and hence their dependence–is available. We employ two well-known factor models, the multiplicative and additive background risk models, to describe such partial information.
Reference
Acharya, V. V., L. H. Pedersen, T. Philippon, and M. Richardson (2017). Measuring systemic risk. The review of financial studies 30 (1), 2–47.
Panjer, H. H. and J. Jing (2001). Solvency and capital allocation. University of Waterloo, Institute of Insurance and Pension Research.
Title. Optimal Execution for N Traders with Transient Price Impact
Abstract. We study an N-player game of optimal execution in the Obizhaeva—Wang model of transient price impact. A unique equilibrium exists when the problem is regularized by an additional instantaneous cost, but existence fails without regularization. We show how a modification of the model restores existence. The newly found equilibrium is consistent with the limit for small regularization, and has a simpler form. (Joint work with Steven Campbell.)