arXiv subjects

Xun Yu Zhou

Publications and source records attributed to Xun Yu Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Dynamic mean-variance portfolio selection with no-shorting constraints and unknown investment opportunity sets

We study continuous-time mean-variance portfolio selection with no-shorting constraints and unknown investment opportunity sets from a reinforcement learning (RL) perspective. The problem is a constrained stochastic linear -- quadratic control problem for which the entropy-regularized exploratory formulation of Wang et al. (2020) leads to difficulty in theoretical analysis, because enforcing the constraint on the support of randomized policies nullifies the tractable Gaussian exploration. To tackle this challenge, we introduce an auxiliary exploratory problem without entropy in which exploratory policies are still Gaussian whose samples may violate the no-shorting requirement but their means satisfy it. We then prove that, for a suitable choice of exploration variance, the mean of the optimal Gaussian policy of the auxiliary problem coincides with the optimal policy of the original problem. Motivated by this theoretical result, we develop a model-free RL algorithm that learns the optimal policy of the auxiliary (and hence the original) problem directly from trajectory data without estimating the investment opportunity set. A numerical example demonstrates the performance of the proposed algorithm.

math.OC↗

arXiv2026

ART for Diffusion Sampling: Continuous-Time Control and Actor-Critic Learning

We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on fixed prescriptions and can therefore be suboptimal. To address this limitation, we propose Adaptive Reparameterized Time (ART), a continuous-time control formulation that learns a time change by treating the speed of the sampling clock as the control, so that a uniform grid on the learned clock induces adaptive timesteps in the original diffusion time. Based on a leading-order Euler error surrogate, ART provides a principled objective for allocating timesteps along the sampling trajectory. To solve this deterministic control problem, we introduce ART-RL, an auxiliary randomized formulation with Gaussian policies that turns schedule learning into a continuous-time reinforcement learning problem. We prove that the randomized ART-RL formulation is equivalent to ART at the optimizer level, in the sense that its optimal Gaussian policy recovers the optimal ART time-warping rate through its mean. We further establish policy evaluation and policy improvement characterizations and derive trajectory-based moment identities that yield implementable actor--critic updates for learning the schedule. Across experiments ranging from controlled low-dimensional settings to image generation, ART-RL can be plugged into existing diffusion samplers by changing only the timestep grid, consistently improving sample quality over strong baseline schedules at matched budgets while leaving the rest of the sampling pipeline unchanged. The learned schedules also exhibit broad generalization, transferring without retraining across sampling budgets, datasets, solvers, pipelines, and representation spaces.

cs.LG↗

arXiv2026

A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

cs.LG↗

arXiv2026

Optimal exit strategies of CPT gamblers in unfair gambles

In this paper we study optimal exit strategies of gamblers with cumulative prospect theory (CPT) preferences in games where the expected payoff is strictly negative at each play, and formulate the problem as optimal stopping on asymmetric random walks. Applying a geometric transformation of the underlying cumulative gain/loss process, engaging randomized strategies and changing the decision variable from stopping times to probability distribution of the accumulated gain or loss at exit time, we solve the problem via the Skorokhod embedding. Drastically different from the fair gamble problem studied by He et al. (2019a), we show that the unfair problem in the infinite time horizon has finite values for a wide range of CPT parameter specifications. We then present the analytical solutions in the case of piece-wise power utility and power probability distortion functions. Compared to the strategies used in fair gambling, the CPT gamblers in unfair gambles are less loss-tolerant and choose not to gamble at all when the games are sufficiently unfavorable.

q-fin.MF↗

arXiv2026

Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian

Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM- and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models.

stat.ML↗

arXiv2026

Amortized Guidance for Image Inpainting with Pretrained Diffusion Models

We study image inpainting with generative diffusion models. Existing methods typically either train dedicated task-specific models, or adapt a pretrained diffusion model separately for each masked image at deployment. We introduce a middle-ground model, termed Amortized Inpainting with Diffusion (AID), which keeps a pretrained diffusion backbone fixed, trains a small reusable guidance module offline, and then reuses it across masked images without per-instance optimization. We formulate it as a deterministic guidance problem with a supervised terminal objective. To make this problem learnable in high dimensions, we derive an auxiliary Gaussian formulation and prove that solving this randomized problem recovers the optimal deterministic guidance field. This bridge yields a principled continuous-time actor--critic algorithm for learning the guidance module in a fully data-driven manner. Empirically, on AFHQv2 and FFHQ under the pixel EDM pipeline and on ImageNet under the latent EDM2 pipeline, AID consistently improves the quality--speed trade-off over strong fixed-backbone and amortized inpainting baselines across multiple mask types, while adding less than one percent trainable overhead.

cs.CV↗

arXiv2026

Continuous-time q-learning for mean-field control with common noise, part-I: Theoretical foundations

This paper investigates the continuous-time counterpart of the Q-function for entropy-regularized mean-field control (MFC) with controlled common noise, coined as q-function by Jia and Zhou (2023) in the single agent's model. We first show that, under discretely sampled actions, the value function in the exploratory formulation converges to the one in the relaxed control formulation as the time grid refines. Leveraging the relaxed control formulation, we derive the exploratory Hamilton-Jacobi-Bellman (HJB) equation, in which the controlled common noise gives rise to an additional nonlinear functional of policy, rendering the policy iteration intricate. Under certain concavity condition, we establish the existence and uniqueness of the optimal one-step policy iteration via a first-order condition using the partial linear functional derivative with respect to policy. The policy improvement at each iteration is verified by relating to an entropy-regularized optimization problem over the space of policies. In the mean-field setting, we introduce the integrated q-function (Iq-function) defined on the state distribution and the policy, and it is shown that an optimal policy is identified as a two-layer fixed point to the argmax operator of the Iq-function. Finally, we provide the explicit characterization of an optimal policy as a Gaussian distribution in the general linear-quadratic (LQ) setting.

math.OC↗

arXiv2026

Continuous-time q-learning for mean-field control with common noise, part-II: q-learning algorithms

This paper is a continuation work of Ren et al. (2026) aiming to further devise q-learning algorithms for mean-field control (MFC) with controlled common noise. Based on the relaxed control formulation, we first establish the martingale condition of the value function and the Iq-function by evaluating along the conditional state distributions generated by all test policies. As the data in the relaxed control formulation are not observable in practice, we quantify the error incurred when they are replaced by the observable ones in the exploratory formulation under discretely sampled actions. This, together with a two-layer fixed point characterization of an optimal policy in Ren et al. (2026), allows us to propose several algorithms including the Actor-Critic q-learning algorithm, in which the policy is updated in the Actor-step based on the iteration rule induced by the improved Iq-function, and the value function and Iq-function are updated in the Critic-step based on the martingale orthogonality condition using the data from the exploratory formulation. We also establish the convergence of the inner iterations in the Actor-step in an infinite-horizon linear quadratic (LQ) framework. In two examples, within and beyond LQ framework, our q-learning algorithms are implemented with satisfactory performance.

math.OC↗

arXiv2026

Merton's Problem with Recursive Perturbed Utility

The classical Merton investment problem predicts deterministic, state-dependent portfolio rules; however, laboratory and field evidence suggests that individuals often prefer randomized decisions leading to stochastic and noisy choices. Fudenberg et al. (2015) develop the additive perturbed utility theory to explain the preference for randomization in the static setting, which, however, becomes ill-posed or intractable in the dynamic setting. We introduce the recursive perturbed utility (RPU), a special stochastic differential utility that incorporates an entropy-based preference for randomization into a recursive aggregator. RPU endogenizes the intertemporal trade-off between utilities from randomization and bequest via a discounting term dependent on past accumulated randomization, thereby avoiding excessive randomization and yielding a well-posed problem. In a general Markovian incomplete market with CRRA preferences, we prove that the RPU-optimal portfolio policy (in terms of the risk exposure ratio) is Gaussian and can be expressed in closed form, independent of wealth. Its variance is inversely proportional to risk aversion and stock volatility, while its mean is based on the solution to a partial differential equation. Moreover, the mean is the sum of a myopic term and an intertemporal hedging term (against market incompleteness) that intertwines with policy randomization. Finally, we carry out an asymptotic expansion in terms of the perturbed utility weight to show that the optimal mean policy deviates from the classical Merton policy at first order, while the associated relative wealth loss is of a higher order, quantifying the financial cost of the preference for randomization.

q-fin.MF↗

arXiv2025

Generating solution paths of Markovian stochastic differential equations using diffusion models

This paper introduces a new approach to generating sample paths of unknown Markovian stochastic differential equations (SDEs) using diffusion models, a class of generative AI methods commonly employed in image and video applications. Unlike the traditional Monte Carlo methods for simulating SDEs, which require explicit specifications of the drift and diffusion coefficients, ours takes a model-free, data-driven approach. Given a finite set of sample paths from an SDE, we utilize conditional diffusion models to generate new, synthetic paths of the same SDE. Numerical experiments show that our method consistently outperforms two alternative methods in terms of the Kullback--Leibler (KL) divergence between the distributions of the target SDE paths and the generated ones. Moreover, we present a theoretical error analysis deriving an explicit bound on the said KL divergence. Finally, in simulation and empirical studies, we leverage these synthetically generated sample paths to boost the performance of reinforcement learning algorithms for continuous-time mean--variance portfolio selection, hinting promising applications of our study in financial analysis and decision-making.

cs.LG↗

arXiv2025

Data-Driven Exploration for a Class of Continuous-Time Indefinite Linear--Quadratic Reinforcement Learning Problems

We study reinforcement learning (RL) for the same class of continuous-time stochastic linear--quadratic (LQ) control problems as in \cite{huang2024sublinear}, where volatilities depend on both states and controls while states are scalar-valued and running control rewards are absent. We propose a model-free, data-driven exploration mechanism that adaptively adjusts entropy regularization by the critic and policy variance by the actor. Unlike the constant or deterministic exploration schedules employed in \cite{huang2024sublinear}, which require extensive tuning for implementations and ignore learning progresses during iterations, our adaptive exploratory approach boosts learning efficiency with minimal tuning. Despite its flexibility, our method achieves a sublinear regret bound that matches the best-known model-free results for this class of LQ problems, which were previously derived only with fixed exploration schedules. Numerical experiments demonstrate that adaptive explorations accelerate convergence and improve regret performance compared to the non-adaptive model-free and model-based counterparts.

cs.LG↗

arXiv2024

Mean--Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study

We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes, yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL approach that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black--Scholes markets without factors, we further devise an algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of the Sharpe ratio. We then carry out an extensive empirical study implementing this algorithm to compare its performance and trading characteristics, evaluated under a host of common metrics, with a large number of widely employed portfolio allocation strategies on S\&P 500 constituents. The results demonstrate that the proposed continuous-time RL strategy is consistently among the best, especially in a volatile bear market, and decisively outperforms the model-based continuous-time counterparts by significant margins.

q-fin.PM↗

arXiv2024

Regret of exploratory policy improvement and $q$-learning

We study the convergence of $q$-learning and related algorithms introduced by Jia and Zhou (J. Mach. Learn. Res., 24 (2023), 161) for controlled diffusion processes. For exploratory policy improvement, we establish exponential convergence under growth and regularity assumptions on the model parameters. For q-learning, we derive quantitative error and regret bounds under additional assumptions on the function approximation and the associated stochastic approximation dynamics.

cs.LG↗

arXiv2024

Reward-Directed Score-Based Diffusion Models via q-Learning

We propose a new reinforcement learning (RL) formulation for training continuous-time score-based diffusion models for generative AI to generate samples that maximize reward functions while keeping the generated distributions close to the unknown target data distributions. Different from most existing studies, ours does not involve any pretrained model for the unknown score functions of the noise-perturbed data distributions, nor does it attempt to learn the score functions. Instead, we formulate the problem as entropy-regularized continuous-time RL and show that the optimal stochastic policy has a Gaussian distribution with a known covariance matrix. Based on this result, we parameterize the mean of Gaussian policies and develop an actor--critic type (little) q-learning algorithm to solve the RL problem. A key ingredient in our algorithm design is to obtain noisy observations from the unknown score function via a ratio estimator. Our formulation can also be adapted to solve pure score-matching and fine-tuning pretrained models. Numerically, we show the effectiveness of our approach by comparing its performance with two state-of-the-art RL methods that fine-tune pretrained models on several generative tasks including high-dimensional image generations. Finally, we discuss extensions of our RL formulation to probability flow ODE implementation of diffusion models and to conditional diffusion models.

cs.LG↗

arXiv2024

Learning to Optimally Stop Diffusion Processes, with Financial Applications

We study optimal stopping for diffusion processes with unknown model primitives within the continuous-time reinforcement learning (RL) framework developed by Wang et al. (2020), and present applications to option pricing and portfolio choice. By penalizing the corresponding variational inequality formulation, we transform the stopping problem into a stochastic optimal control problem with two actions. We then randomize controls into Bernoulli distributions and add an entropy regularizer to encourage exploration. We derive a semi-analytical optimal Bernoulli distribution, based on which we devise RL algorithms using the martingale approach established in Jia and Zhou (2022a). We establish a policy improvement theorem and prove the fast convergence of the resulting policy iterations. We demonstrate the effectiveness of the algorithms in pricing finite-horizon American put options, solving Merton's problem with transaction costs, and scaling to high-dimensional optimal stopping problems. In particular, we show that both the offline and online algorithms achieve high accuracy in learning the value functions and characterizing the associated free boundaries.

math.OC↗

arXiv2024

Sublinear Regret for a Class of Continuous-Time Linear-Quadratic Reinforcement Learning Problems

We study reinforcement learning (RL) for a class of continuous-time linear-quadratic (LQ) control problems for diffusions, where states are scalar-valued and running control rewards are absent but volatilities of the state processes depend on both state and control variables. We apply a model-free approach that relies neither on knowledge of model parameters nor on their estimations, and devise an RL algorithm to learn the optimal policy parameter directly. Our main contributions include the introduction of an exploration schedule and a regret analysis of the proposed algorithm. We provide the convergence rate of the policy parameter to the optimal one, and prove that the algorithm achieves a regret bound of $O(N^{\frac{3}{4}})$ up to a logarithmic factor, where $N$ is the number of learning episodes. We conduct a simulation study to validate the theoretical results and demonstrate the effectiveness and reliability of the proposed algorithm. We also perform numerical comparisons between our method and those of the recent model-based stochastic LQ RL studies adapted to the state- and control-dependent volatility setting, demonstrating a better performance of the former in terms of regret bounds.

cs.LG↗

arXiv2024

Reinforcement Learning for Jump-Diffusions, with Financial Applications

We study continuous-time reinforcement learning (RL) for stochastic control in which system dynamics are governed by jump-diffusion processes. We formulate an entropy-regularized exploratory control problem with stochastic policies to capture the exploration--exploitation balance essential for RL. Unlike the pure diffusion case initially studied by Wang et al. (2020), the derivation of the exploratory dynamics under jump-diffusions calls for a careful formulation of the jump part. Through a theoretical analysis, we find that one can simply use the same policy evaluation and $q$-learning algorithms in Jia and Zhou (2022a, 2023), originally developed for controlled diffusions, without needing to check a priori whether the underlying data come from a pure diffusion or a jump-diffusion. However, we show that the presence of jumps ought to affect parameterizations of actors and critics in general. We investigate as an application the mean--variance portfolio selection problem with stock price modelled as a jump-diffusion, and show that both RL algorithms and parameterizations are invariant with respect to jumps. Finally, we present a detailed study on applying the general theory to option hedging.

cs.LG↗

arXiv2023

Data-Driven Merton's Strategies via Policy Randomization

We study Merton's expected utility maximization problem in an incomplete market, characterized by a factor process in addition to the stock price process, where all the model primitives are unknown. The agent under consideration is a price taker who has access only to the stock and factor value processes and the instantaneous volatility. We propose an auxiliary problem in which the agent can invoke policy randomization according to a specific class of Gaussian distributions, and prove that the mean of its optimal Gaussian policy solves the original Merton problem. With randomized policies, we are in the realm of continuous-time reinforcement learning (RL) recently developed in Wang et al. (2020) and Jia and Zhou (2022a, 2022b, 2023), enabling us to solve the auxiliary problem in a data-driven way without having to estimate the model primitives. Specifically, we establish a policy improvement theorem based on which we design both online and offline actor-critic RL algorithms for learning Merton's strategies. A key insight from this study is that RL in general and policy randomization in particular are useful beyond the purpose for exploration -- they can be employed as a technical tool to solve a problem that cannot be otherwise solved by mere deterministic policies. At last, we carry out both simulation and empirical studies in a stochastic volatility environment to demonstrate the decisive outperformance of the devised RL algorithms in comparison to the conventional model-based, plug-in method.

q-fin.PM↗