Three highlighted speakers will give broad overview talks based on their research. The idea is to provide early stage researchers a glimpse into the mathematical and statistical approaches to various types of research in mathematical finance.
Optimal Transport applications in Mathematical Finance
In this talk I will present some applications of Optimal Transport (OT) in Mathematical Finance. Notably, two specific classes of transports have been extensively exploited by the Mathematical Finance community in the past years, the so-called Martingale OT and Causal OT. I will introduce these kinds of transports and discuss the rationale for their success.
Connecting GANs and MFGs
Generative Adversarial Networks (GANs), introduced in 2014 by Goodfellow, have celebrated great empirical success, especially in image generation and processing. Meanwhile, Mean-Field Games (MFGs), as analytically feasible approximations for N-player games, have experienced rapid growth in theoretical studies. In this paper, we establish theoretical connections between GANs and MFGs. Interpreting MFGs as GANs, on one hand, allows us to devise GANs-based algorithm to solve MFGs. Interpreting GANs as MFGs, on the other hand, provides a new and probabilistic foundation for GANs. Moreover, this interpretation helps establish an analytical connection between GANs and Optimal Transport (OT) problems.
Based on joint work with C. Cao of UC Berkeley and M. Lauriere of Princeton U.
Mean field games of optimal stopping and application to entry-exit games in energy markets. We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. We also present a numerical method for computing the equilibrium in the case of potential games and show its convergence. Finally, we briefly present an application to entry and exit games in electricity markets. We shall discuss the modeling of long-term dynamics of the electricity industry, where the renewable producers look for the optimal moment to enter the market, the conventional producers look for the optimal moment to exit, and the interaction between the two types of producers takes place through the market price determined by an exogeneous demand curve and an endogeneous merit order supply curve. (Based on joint works with Peter Tankov, René Aid and Géraldine Bouveret).
Deep Fictitious Play for Stochastic Differential Games
Stochastic differential games can be used to model competitions in Fintech industries, i.e. in P2P lending platforms, insurance markets. Computing Nash equilibria is one of the core objectives in differential games, with a major bottleneck coming from the notorious intractability of N-player games, also known by the curse of dimensionality. To overcome this difficulty, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms, for which we refer as deep fictitious play. The resulted deep learning algorithm is scalable, parallelizable and model-free. We illustrate the performance of proposed algorithms by comparing them to the closed-form solution of the linear-quadratic game. We also prove the convergence of the fictitious play under appropriate assumptions and verify that the convergent limit forms an open-loop Nash equilibrium. Based on the formulation by backward stochastic differential equations, we extend the strategy of deep fictitious play to compute closed-loop Markovian Nash equilibrium for both homogeneous and heterogeneous large N-player games.
Approximating the optimal investment strategy in incomplete markets
We look at the problem of portfolio optimization in a short time horizon in an incomplete market. Closed-form approximating formulas for the optimal trading strategy are obtained using asymptotic techniques. The approximate value function is obtained by constructing classical sub- and super-solutions to the associated Hamilton-Jacobi-Bellman (HJB) equation using a formal expansion in powers of horizon time. Martingale inequalities are used to prove a comparison principle for sub- and super-solutions to the HJB equation.
Neural Networks for Optimal Execution
Stochastic control problems are commonly solved using PDE approaches. However, for high-dimensional problems, this type of approach stumbles upon the curse of dimensionality. In this work, we use neural networks to obtain a non-linear approximation to the solution of a specific kind of stochastic control problem: the optimal execution problem. In this problem, the investor needs to either buy or sell a certain amount during the trading day, and must choose a trading speed to operate. We generate Monte-Carlo samples using parameters extracted from Toronto Stock Exchange data. We use this data to train the model, and compare its solution to the classic PDE solution to optimal execution problems. The neural network model thus outputs the optimal trading speed at any given time of the day and agent’s inventory. In comparing our neural network non-linear approximation to the regular linear one, we develop a new approach towards benchmarking neural networks’ solutions through the use of linear regression. Finally, we use transfer learning to continue learning and improve our model by allowing data auto-correlation to play a part in the training.
Conditional Optimal Stopping
Inspired by recent work of P.-L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For instance, an agent may care only about states where she is still alive at the time of stopping, or a company may condition on not being bankrupt. We observe that conditional optimization is time-inconsistent due to the dynamic change of the conditioning probability and develop an equilibrium approach in the spirit of R. H. Strotz’ work for sophisticated agents in discrete time. Equilibria are found to be essentially unique in the case of a finite time horizon whereas an infinite horizon gives rise to non-uniqueness and other interesting phenomena. We also introduce a theory which generalizes the classical Snell envelope approach for optimal stopping by considering a pair of processes with Snell-type properties. (Joint work with Marcel Nutz)