This workshop focuses on recent developments in Mathematical Finance.
Schedule:
09:20-10:00 Huyen Pham (Ecole Polytechnique): Control of Large-Scale Heterogeneous Systems: An Extended Graphon Mean-Field Approach
10:00-10:40 Eyal Neuman (Imperial): Stochastic Graphon Games with Memory
10:40-11:00 Coffee Break
11:00-11:40 Yufei Zhang (Imperial): $\alpha$-Potential Games: A New Paradigm for N-player Dynamic Games
11:40-12:20 Ofelia Bonesini (LSE): Continuous-time Persuasion by Filtering
12:20-14:00 Lunch Break
14:00-14:40 Xin Guo (University of California, Berkeley): Continuous-time Mean Field Games: A Primal-Dual Characterization
14:40-15:20 Pavel Gapeev (LSE): Perpetual American Options in a Two-Dimensional Black-Merton-Scholes Model
15:20-15:50 Coffee Break
15:50-16:20 Umut Cetin (LSE): Kyle Model with Penalties, BSDEs and Entropy
Abstracts of the talks:
- Ofelia Bonesini (LSE): Continuous-time Persuasion by Filtering
- We frame dynamic persuasion in a partial observation stochastic control Leader-Follower game with an ergodic criterion. The Receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the Receiver through a device designed by the Sender that generates the observation process. The commitment of the Sender is enforced. We develop this approach in the case where all dynamics are linear, and the preferences of the Receiver are linear quadratic. We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the Receiver’s value function. An extension to the case of persuasion of a mean field of interacting Receivers is also provided. We illustrate this approach with an application to the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. We show that even in the absence of information cost, it might be optimal for the regulator to blur information available to firms to prevent them from coordinating on a higher level of carbon footprint to reduce their cost of reaching a below average emission target.
This is a joint work with René Aïd, Giorgia Callegaro and Luciano Campi.
- We frame dynamic persuasion in a partial observation stochastic control Leader-Follower game with an ergodic criterion. The Receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the Receiver through a device designed by the Sender that generates the observation process. The commitment of the Sender is enforced. We develop this approach in the case where all dynamics are linear, and the preferences of the Receiver are linear quadratic. We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the Receiver’s value function. An extension to the case of persuasion of a mean field of interacting Receivers is also provided. We illustrate this approach with an application to the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. We show that even in the absence of information cost, it might be optimal for the regulator to blur information available to firms to prevent them from coordinating on a higher level of carbon footprint to reduce their cost of reaching a below average emission target.
- Umut Cetin (LSE): Kyle Model with Penalties, BSDEs and Entropy
- We consider the Kyle model in continuous time, where the informed traders face additional frictions. These frictions may arise due to difficulties in executing large portfolios, or legal penalties in case the informed trader is trading illegally on inside information. The equilibrium is characterised via the solution of a backward stochastic differential equation (BSDE) whose terminal condition is determined as the fixed point of a non-linear operator in equilibrium. A curious connection between the terminal condition of this BSDE and an entropic optimal transport problem appears. We find that informed traders consistently trade a constant multiple of the difference between the fundamental value and their anticipated market price just before their private information is disclosed to the public, reminiscent of the behaviour of a large trader in an Almgren-Chris model.
- Pavel Gapeev (LSE): Perpetual American Options in a Two-Dimensional Black-Merton-Scholes Model
- We study optimal stopping problems for two-dimensional geometric Brownian motions driven by constantly correlated standard Brownian motions on an infinite time interval. These problems are related to the pricing of perpetual American options such as basket options (with an additive payoff structure) and traffic-light options (with a multiplicative payoff structure) in a two-dimensional Black-Merton-Scholes model. We find closed formulas for the value functions expressed in terms of the optimal stopping boundaries which in turn are shown to be unique solutions to the appropriate nonlinear Fredholm integral equations. A key argument in the existence proof is played by a pointwise maximisation of the expressions obtained by the change-of-measure arguments. This provides tight bounds on the optimal stopping boundaries describing its asymptotic behaviour for marginal coordinate values.
- Xin Guo (University of California, Berkeley): Continuous-Time Mean Field Games: A Primal-Dual Characterization
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This talk presents a primal-dual formulation for continuous-time mean field games (MFGs) and establishes a complete analytical characterization of the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player’s control problem with {\it measurable coefficients} is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation, which plays a fundamental role in characterizing NEs of MFGs. Finally, a complete characterization of \emph{all NEs for MFGs} is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by studying the solvability of the dual problem, and in particular through analyzing the regularity of the associated HJB equation.
Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization require neither the convexity of the associated Hamiltonian nor the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.
This is a joint work with Anran Hu, Jiacheng Zhang, and Yufei Zhang.
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- Eyal Neuman (Imperial): Stochastic Graphon Games with Memory
- We analyze finite-player dynamic stochastic games with heterogeneous interactions and non-Markovian linear-quadratic objective functionals, deriving explicit Nash equilibria by means of operator resolvents. When interactions are represented by a weighted graph, we extend the framework to a continuum-player game with interactions modeled by a graphon. We then derive the explicit graphon game Nash equilibrium, by reducing the first-order conditions to an infinite-dimensional coupled system of stochastic Fredholm equations, and introducing a novel approach utilizing the spectral decomposition of the graphon operator in order to solve it. Additionally, we demonstrate the convergence of Nash equilibria from finite-player games to the one of the graphon game as the number of agents grows, providing explicit convergence rates. Finally, we apply these results to diverse stochastic games, including systemic risk models with delay and stochastic differential network games, showcasing our framework’s broad applicability.
This is a joint work with Sturmius Tuschmann
- We analyze finite-player dynamic stochastic games with heterogeneous interactions and non-Markovian linear-quadratic objective functionals, deriving explicit Nash equilibria by means of operator resolvents. When interactions are represented by a weighted graph, we extend the framework to a continuum-player game with interactions modeled by a graphon. We then derive the explicit graphon game Nash equilibrium, by reducing the first-order conditions to an infinite-dimensional coupled system of stochastic Fredholm equations, and introducing a novel approach utilizing the spectral decomposition of the graphon operator in order to solve it. Additionally, we demonstrate the convergence of Nash equilibria from finite-player games to the one of the graphon game as the number of agents grows, providing explicit convergence rates. Finally, we apply these results to diverse stochastic games, including systemic risk models with delay and stochastic differential network games, showcasing our framework’s broad applicability.
- Huyen Pham (Ecole Polytechnique): Control of Large-Scale Heterogeneous Systems: An Extended Graphon Mean-Field Approach
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Networks play a central role in modeling complex systems such as financial markets, power grids, social interactions, and epidemiology. This talk examines dynamical systems of interacting agents/particles with heterogeneous connections, described by graph structures. By employing graphon theory, we analyze the large-population limit, proving a propagation of chaos result that yields a collection of mean-field stochastic differential equations.We further address the control of these non-exchangeable McKean-Vlasov systems from the perspective of a central planner capable of influencing asymmetric interactions. Leveraging tools tailored for this framework, such as derivatives along flows of measures and the corresponding Itô calculus, we establish that the value function of this control problem satisfies a Bellman dynamic programming equation in a function space over the Wasserstein space.To illustrate the applicability of our approach, we present a linear-quadratic graphon model with analytical solutions and apply it to a systemic risk example involving heterogeneous banks.
Based on joint works with: A. De Crescenzo, F. Coppini, F. De Feo, M. Fuhrman and I. Kharroubi.
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- Yufei Zhang (Imperial): $\alpha$-Potential Games: A New Paradigm for N-player Dynamic Games
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Static potential games, pioneered by Monderer and Shapley (1996), are non-cooperative games in which there exists an auxiliary function called static potential function, so that any player’s change in utility function upon unilaterally deviating from her policy can be evaluated through the change in the value of this potential function. The introduction of the potential function is powerful as it simplifies the otherwise challenging task of finding Nash equilibria for non-cooperative games: maximizers of potential functions lead to the game’s Nash equilibria.
In this talk, we propose an analogous and new framework called $\alpha$-potential game for dynamic N-player games, with the potential function in the static setting replaced by an $\alpha$-potential function. We present an analytical characterization of $\alpha$-potential functions for any dynamic game. For stochastic differential games in which the state dynamic is a controlled diffusion, $\alpha$ is explicitly identified in terms of the number of players, and the intensity of interactions and the level of heterogeneity among players. We further show the $\alpha$-Nash equilibrium of the stochastic game can be constructed through an associated conditional McKean-Vlasov control problem. To illustrate our findings, we examine a linear-quadratic game on a graph, where $\alpha$ captures asymmetric interactions and player heterogeneity beyond the mean-field paradigm.
This is a joint work with Xin Guo and Xinyu Li.
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