We study the implications of model uncertainty in a climate-economics framework with three types of capital: "dirty" capital that produces carbon emissions when used for production, "clean" capital that generates no emissions but is initially less productive than dirty capital, and knowledge capital that increases with R\&D investment and leads to technological innovation in green sector productivity. To solve our high-dimensional, non-linear model framework we implement a neural-network-based global solution method. We show there are first-order impacts of model uncertainty on optimal decisions and social valuations in our integrated climate-economic-innovation framework. Accounting for interconnected uncertainty over climate dynamics, economic damages from climate change, and the arrival of a green technological change leads to substantial adjustments to investment in the different capital types in anticipation of technological change and the revelation of climate damage severity.

We present the development and analysis of a reinforcement learning (RL) algorithm designed to solve continuous-space mean field game (MFG) and mean field control (MFC) problems in a unified manner. The proposed approach pairs the actor-critic (AC) paradigm with a representation of the mean field distribution via a parameterized score function, which can be efficiently updated in an online fashion, and uses Langevin dynamics to obtain samples from the resulting distribution. The AC agent and the score function are updated iteratively to converge, either to the MFG equilibrium or the MFC optimum for a given mean field problem, depending on the choice of learning rates. A straightforward modification of the algorithm allows us to solve mixed mean field control games (MFCGs). The performance of our algorithm is evaluated using linear-quadratic benchmarks in the asymptotic infinite horizon framework.

One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs' model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration's FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [Han, Hu and Long, arXiv:2104.12036]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward process.

The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward-backward stochastic differential equations (FBSDEs), and inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide the rigorous theory for the backward deep BSDE method. Specifically, 1. We derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and; 2. We give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory.

In this paper, we propose a numerical methodology for finding the closed-loop Nash equilibrium of stochastic delay differential games through deep learning. These games are prevalent in finance and economics where multi-agent interaction and delayed effects are often desired features in a model, but are introduced at the expense of increased dimensionality of the problem. This increased dimensionality is especially significant as that arising from the number of players is coupled with the potential infinite dimensionality caused by the delay. Our approach involves parameterizing the controls of each player using distinct recurrent neural networks. These recurrent neural network-based controls are then trained using a modified version of Brown's fictitious play, incorporating deep learning techniques. To evaluate the effectiveness of our methodology, we test it on finance-related problems with known solutions. Furthermore, we also develop new problems and derive their analytical Nash equilibrium solutions, which serve as additional benchmarks for assessing the performance of our proposed deep learning approach.

Real-world data can be multimodal distributed, e.g., data describing the opinion divergence in a community, the interspike interval distribution of neurons, and the oscillators natural frequencies. Generating multimodal distributed real-world data has become a challenge to existing generative adversarial networks (GANs). For example, neural stochastic differential equations (Neural SDEs), treated as infinite-dimensional GANs, have demonstrated successful performance mainly in generating unimodal time series data. In this paper, we propose a novel time series generator, named directed chain GANs (DC-GANs), which inserts a time series dataset (called a neighborhood process of the directed chain or input) into the drift and diffusion coefficients of the directed chain SDEs with distributional constraints. DC-GANs can generate new time series of the same distribution as the neighborhood process, and the neighborhood process will provide the key step in learning and generating multimodal distributed time series. The proposed DC-GANs are examined on four datasets, including two stochastic models from social sciences and computational neuroscience, and two real-world datasets on stock prices and energy consumption. To our best knowledge, DC-GANs are the first work that can generate multimodal time series data and consistently outperforms state-of-the-art benchmarks with respect to measures of distribution, data similarity, and predictive ability.

Stochastic optimal control and games have found a wide range of applications, from finance and economics to social sciences, robotics and energy management. Many real-world applications involve complex models which have driven the development of sophisticated numerical methods. Recently, computational methods based on machine learning have been developed for stochastic control problems and games. We review such methods, with a focus on deep learning algorithms that have unlocked the possibility to solve such problems even when the dimension is high or when the structure is very complex, beyond what is feasible with traditional numerical methods. Here, we consider mostly the continuous time and continuous space setting. Many of the new approaches build on recent neural-network based methods for high-dimensional partial differential equations or backward stochastic differential equations, or on model-free reinforcement learning for Markov decision processes that have led to breakthrough results. In this paper we provide an introduction to these methods and summarize state-of-the-art works on machine learning for stochastic control and games.

This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics ({\it e.g.}, the Wasserstein distance). The proposed metrics originate from the maximum mean discrepancy, which we generalize by proposing specific criteria for selecting test function spaces to guarantee the property of being free of CoD. Therefore, we call this class of metrics the generalized maximum mean discrepancy (GMMD). Examples of the selected test function spaces include the reproducing kernel Hilbert space, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of $n$-particle system to the solution to McKean-Vlasov stochastic differential equation; 3. The construction of an $\varepsilon$-Nash equilibrium for a homogeneous $n$-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by GMMD and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein distance and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD.

In stochastic differential games, a Nash equilibrium refers to strategies by which no player has an incentive to deviate. Finding a Nash equilibrium is one of the core problems in noncooperative game theory, however, due to the notorious intractability of N-player game, the computation of the Nash equilibrium has been shown extremely time-consuming and memory demanding, especially for large N [16]. On the other hand, a rich literature on game theory has been developed to study consequences of strategies on interactions between a large group of rational "agents", e.g., system risk caused by inter-bank borrowing and lending, price impacts imposed by agents' optimal liquidation, and market price from monopolistic competition. This makes it crucial to develop efficient theory and fast algorithms for computing the Nash equilibrium of N-player stochastic differential games. Deep neural networks with many layers have been recently shown to do a great job in artificial intelligence (e.g., [2, 39]). The idea behind is to use compositions of simple functions to approximate complicated ones, and there are approximation theorems showing that a wide class of functions on compact subsets can be approximated by a single hidden layer neural network (e.g., [53]). This brings a possibility of solving a high-dimensional system using deep neural networks, and in fact, these techniques have been successfully applied to solve stochastic control problems [20, 29, 1]. In this paper, we propose to build deep neural networks by using strategies of fictitious play, and develop deep learning algorithms for computing the Nash equilibrium of asymmetric N-player non-zerosum stochastic differential games.

In this paper, we propose deep learning algorithms for ranking response surfaces, with applications to optimal stopping problems in financial mathematics. The problem of ranking response surfaces is motivated by estimating optimal feedback policy maps in stochastic control problems, aiming to efficiently find the index associated to the minimal response across the entire continuous input space $\mathcal{X} \subseteq \mathbb{R}^d$. By considering points in $\mathcal{X}$ as pixels and indices of the minimal surfaces as labels, we recast the problem as an image segmentation problem, which assigns a label to every pixel in an image such that pixels with the same label share certain characteristics. This provides an alternative method for efficiently solving the problem instead of using sequential design in our previous work [R. Hu and M. Ludkovski, SIAM/ASA Journal on Uncertainty Quantification, 5 (2017), 212--239]. Deep learning algorithms are scalable, parallel and model-free, i.e., no parametric assumptions needed on the response surfaces. Considering ranking response surfaces as image segmentation allows one to use a broad class of deep neural networks, e.g., UNet, SegNet, DeconvNet, which have been widely applied and numerically proved to possess high accuracy in the field. We also systematically study the dependence of deep learning algorithms on the input data generated on uniform grids or by sequential design sampling, and observe that the performance of deep learning is {\it not} sensitive to the noise and locations (close to/away from boundaries) of training data. We present a few examples including synthetic ones and the Bermudan option pricing problem to show the efficiency and accuracy of this method.