Multi-Task Dynamic Pricing in Credit Market with Contextual Information

We study the dynamic pricing problem faced by a broker that buys and sells a large number of financial securities in the credit market, such as corporate bonds, government bonds, loans, and other credit-related securities. One challenge in pricing these securities is their infrequent trading, which leads to insufficient data for individual pricing. However, many of these securities share structural features that can be utilized. Building on this, we propose a multi-task dynamic pricing framework that leverages these shared structures across securities, enhancing pricing accuracy through learning. In our framework, a security is fully characterized by a $d$ dimensional contextual/feature vector. The customer will buy (sell) the security from the broker if the broker quotes a price lower (higher) than that of the competitors. We assume a linear contextual model for the competitor's pricing, with unknown parameters a prior. The parameters for pricing different securities may or may not be similar to each other. The firm's objective is to minimize the expected regret, namely, the expected revenue loss against a clairvoyant policy which has the knowledge of the parameters of the competitor's pricing model. We show that the regret of our policy is better than both a policy that treats each security individually and a policy that treats all securities as the same. Moreover, the regret is bounded by $\tilde{O} ( \delta_{\max} \sqrt{T M d} + M d ) $, where $M$ is the number of securities and $\delta_{\max}$ characterizes the overall dissimilarity across securities in the basket.

Exploratory Optimal Stopping: A Singular Control Formulation

This paper explores continuous-time and state-space optimal stopping problems from a reinforcement learning perspective. We begin by formulating the stopping problem using randomized stopping times, where the decision maker's control is represented by the probability of stopping within a given time--specifically, a bounded, non-decreasing, c\`adl\`ag control process. To encourage exploration and facilitate learning, we introduce a regularized version of the problem by penalizing it with the cumulative residual entropy of the randomized stopping time. The regularized problem takes the form of an (n+1)-dimensional degenerate singular stochastic control with finite-fuel. We address this through the dynamic programming principle, which enables us to identify the unique optimal exploratory strategy. For the specific case of a real option problem, we derive a semi-explicit solution to the regularized problem, allowing us to assess the impact of entropy regularization and analyze the vanishing entropy limit. Finally, we propose a reinforcement learning algorithm based on policy iteration. We show both policy improvement and policy convergence results for our proposed algorithm.

Inference of Utilities and Time Preference in Sequential Decision-Making

This paper introduces 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.

Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization

Diffusion models have emerged as a powerful tool rivaling GANs in generating high-quality samples with improved fidelity, flexibility, and robustness. A key component of these models is to learn the score function through score matching. Despite empirical success on various tasks, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. As a first step toward answering this question, this paper establishes a mathematical framework for analyzing score estimation using neural networks trained by gradient descent. Our analysis covers both the optimization and the generalization aspects of the learning procedure. In particular, we propose a parametric form to formulate the denoising score-matching problem as a regression with noisy labels. Compared to the standard supervised learning setup, the score-matching problem introduces distinct challenges, including unbounded input, vector-valued output, and an additional time variable, preventing existing techniques from being applied directly. In this paper, we show that with a properly designed neural network architecture, the score function can be accurately approximated by a reproducing kernel Hilbert space induced by neural tangent kernels. Furthermore, by applying an early-stopping rule for gradient descent and leveraging certain coupling arguments between neural network training and kernel regression, we establish the first generalization error (sample complexity) bounds for learning the score function despite the presence of noise in the observations. Our analysis is grounded in a novel parametric form of the neural network and an innovative connection between score matching and regression analysis, facilitating the application of advanced statistical and optimization techniques.

Fast Policy Learning for Linear Quadratic Control with Entropy Regularization

This paper proposes and analyzes two new policy learning methods: regularized policy gradient (RPG) and iterative policy optimization (IPO), for a class of discounted linear-quadratic control (LQC) problems over an infinite time horizon with entropy regularization. Assuming access to the exact policy evaluation, both proposed approaches are proven to converge linearly in finding optimal policies of the regularized LQC. Moreover, the IPO method can achieve a super-linear convergence rate once it enters a local region around the optimal policy. Finally, when the optimal policy for an RL problem with a known environment is appropriately transferred as the initial policy to an RL problem with an unknown environment, the IPO method is shown to enable a super-linear convergence rate if the two environments are sufficiently close. Performances of these proposed algorithms are supported by numerical examples.

Risk-sensitive Markov Decision Process and Learning under General Utility Functions

Reinforcement Learning (RL) has gained substantial attention across diverse application domains and theoretical investigations. Existing literature on RL theory largely focuses on risk-neutral settings where the decision-maker learns to maximize the expected cumulative reward. However, in practical scenarios such as portfolio management and e-commerce recommendations, decision-makers often persist in heterogeneous risk preferences subject to outcome uncertainties, which can not be well-captured by the risk-neural framework. Incorporating these preferences can be approached through utility theory, yet the development of risk-sensitive RL under general utility functions remains an open question for theoretical exploration. In this paper, we consider a scenario where the decision-maker seeks to optimize a general utility function of the cumulative reward in the framework of a Markov decision process (MDP). To facilitate the Dynamic Programming Principle and Bellman equation, we enlarge the state space with an additional dimension that accounts for the cumulative reward. We propose a discretized approximation scheme to the MDP under enlarged state space, which is tractable and key for algorithmic design. We then propose a modified value iteration algorithm that employs an epsilon-covering over the space of cumulative reward. When a simulator is accessible, our algorithm efficiently learns a near-optimal policy with guaranteed sample complexity. In the absence of a simulator, our algorithm, designed with an upper-confidence-bound exploration approach, identifies a near-optimal policy while ensuring a guaranteed regret bound. For both algorithms, we match the theoretical lower bounds for the risk-neutral setting.

Policy Gradient Converges to the Globally Optimal Policy for Nearly Linear-Quadratic Regulators

Nonlinear control systems with partial information to the decision maker are prevalent in a variety of applications. As a step toward studying such nonlinear systems, this work explores reinforcement learning methods for finding the optimal policy in the nearly linear-quadratic regulator systems. In particular, we consider a dynamic system that combines linear and nonlinear components, and is governed by a policy with the same structure. Assuming that the nonlinear component comprises kernels with small Lipschitz coefficients, we characterize the optimization landscape of the cost function. Although the cost function is nonconvex in general, we establish the local strong convexity and smoothness in the vicinity of the global optimizer. Additionally, we propose an initialization mechanism to leverage these properties. Building on the developments, we design a policy gradient algorithm that is guaranteed to converge to the globally optimal policy with a linear rate.

Asymptotic Analysis of Deep Residual Networks

We investigate the asymptotic properties of deep Residual networks (ResNets) as the number of layers increases. We first show the existence of scaling regimes for trained weights markedly different from those implicitly assumed in the neural ODE literature. We study the convergence of the hidden state dynamics in these scaling regimes, showing that one may obtain an ODE, a stochastic differential equation (SDE) or neither of these. In particular, our findings point to the existence of a diffusive regime in which the deep network limit is described by a class of stochastic differential equations (SDEs). Finally, we derive the corresponding scaling limits for the backpropagation dynamics.

Risk-Aware Linear Bandits: Theory and Applications in Smart Order Routing

Motivated by practical considerations in machine learning for financial decision-making, such as risk-aversion and large action space, we initiate the study of risk-aware linear bandits. Specifically, we consider regret minimization under the mean-variance measure when facing a set of actions whose rewards can be expressed as linear functions of (initially) unknown parameters. Driven by the variance-minimizing G-optimal design, we propose the Risk-Aware Explore-then-Commit (RISE) algorithm and the Risk-Aware Successive Elimination (RISE++) algorithm. Then, we rigorously analyze their regret upper bounds to show that, by leveraging the linear structure, the algorithms can dramatically reduce the regret when compared to existing methods. Finally, we demonstrate the performance of the algorithms by conducting extensive numerical experiments in a synthetic smart order routing setup. Our results show that both RISE and RISE++ can outperform the competing methods, especially in complex decision-making scenarios.

Recent Advances in Reinforcement Learning in Finance

The rapid changes in the finance industry due to the increasing amount of data have revolutionized the techniques on data processing and data analysis and brought new theoretical and computational challenges. In contrast to classical stochastic control theory and other analytical approaches for solving financial decision-making problems that heavily reply on model assumptions, new developments from reinforcement learning (RL) are able to make full use of the large amount of financial data with fewer model assumptions and to improve decisions in complex financial environments. This survey paper aims to review the recent developments and use of RL approaches in finance. We give an introduction to Markov decision processes, which is the setting for many of the commonly used RL approaches. Various algorithms are then introduced with a focus on value and policy based methods that do not require any model assumptions. Connections are made with neural networks to extend the framework to encompass deep RL algorithms. Our survey concludes by discussing the application of these RL algorithms in a variety of decision-making problems in finance, including optimal execution, portfolio optimization, option pricing and hedging, market making, smart order routing, and robo-advising.