In this paper, we introduce a model-based deep-learning approach to solve finite-horizon continuous-time stochastic control problems with jumps. We iteratively train two neural networks: one to represent the optimal policy and the other to approximate the value function. Leveraging a continuous-time version of the dynamic programming principle, we derive two different training objectives based on the Hamilton-Jacobi-Bellman equation, ensuring that the networks capture the underlying stochastic dynamics. Empirical evaluations on different problems illustrate the accuracy and scalability of our approach, demonstrating its effectiveness in solving complex high-dimensional stochastic control tasks. Code is available at https://github.com/jdupret97/

This paper studies continuous-time stochastic control problems whose controlled states are fully non-Markovian and depend on unknown model parameters. Such problems arise naturally in path-dependent stochastic differential equations, rough-volatility hedging, and systems driven by fractional Brownian motion. Building on the discrete skeleton approach developed in earlier work, we propose a Monte Carlo learning methodology for the associated embedded backward dynamic programming equation. Our main contribution is twofold. First, we construct explicit dominating training laws and Radon--Nikodym weights for several representative classes of non-Markovian controlled systems. This yields an off-model training architecture in which a fixed synthetic dataset is generated under a reference law, while the dynamic programming operators associated with a target model are recovered by importance sampling. Second, we use this structure to design an adaptive update mechanism under parametric model uncertainty, so that repeated recalibration can be performed by reweighting the same training sample rather than regenerating new trajectories. For fixed parameters, we establish non-asymptotic error bounds for the approximation of the embedded dynamic programming equation via deep neural networks. For adaptive learning, we derive quantitative estimates that separate Monte Carlo approximation error from model-risk error. Numerical experiments illustrate both the off-model training mechanism and the adaptive importance-sampling update in structured linear-quadratic examples.

Generative AI can be framed as the problem of learning a model that maps simple reference measures into complex data distributions, and it has recently found a strong connection to the classical theory of the Schrödinger bridge problems (SBPs) due partly to their common nature of interpolating between prescribed marginals via entropy-regularized stochastic dynamics. However, the classical SBP enforces hard terminal constraints, which often leads to instability in practical implementations, especially in high-dimensional or data-scarce regimes. To address this challenge, we follow the idea of the so-called soft-constrained Schrödinger bridge problem (SCSBP), in which the terminal constraint is replaced by a general penalty function. This relaxation leads to a more flexible stochastic control formulation of McKean-Vlasov type. We establish the existence of optimal solutions for all penalty levels and prove that, as the penalty grows, both the controls and value functions converge to those of the classical SBP at a linear rate. Our analysis builds on Doob's h-transform representations, the stability results of Schrödinger potentials, Gamma-convergence, and a novel fixed-point argument that couples an optimization problem over the space of measures with an auxiliary entropic optimal transport problem. These results not only provide the first quantitative convergence guarantees for soft-constrained bridges but also shed light on how penalty regularization enables robust generative modeling, fine-tuning, and transfer learning.

We propose a physics-informed neural network policy iteration (PINN-PI) framework for solving stochastic optimal control problems governed by second-order Hamilton--Jacobi--Bellman (HJB) equations. At each iteration, a neural network is trained to approximate the value function by minimizing the residual of a linear PDE induced by a fixed policy. This linear structure enables systematic $L^2$ error control at each policy evaluation step, and allows us to derive explicit Lipschitz-type bounds that quantify how value gradient errors propagate to the policy updates. This interpretability provides a theoretical basis for evaluating policy quality during training. Our method extends recent deterministic PINN-based approaches to stochastic settings, inheriting the global exponential convergence guarantees of classical policy iteration under mild conditions. We demonstrate the effectiveness of our method on several benchmark problems, including stochastic cartpole, pendulum problems and high-dimensional linear quadratic regulation (LQR) problems in up to 10D.

We mathematically analyze and numerically study an actor-critic machine learning algorithm for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) partial differential equations from stochastic control theory. The architecture of the critic (the estimator for the value function) is structured so that the boundary condition is always perfectly satisfied (rather than being included in the training loss) and utilizes a biased gradient which reduces computational cost. The actor (the estimator for the optimal control) is trained by minimizing the integral of the Hamiltonian over the domain, where the Hamiltonian is estimated using the critic. We show that the training dynamics of the actor and critic neural networks converge in a Sobolev-type space to a certain infinite-dimensional ordinary differential equation (ODE) as the number of hidden units in the actor and critic $\rightarrow \infty$. Further, under a convexity-like assumption on the Hamiltonian, we prove that any fixed point of this limit ODE is a solution of the original stochastic control problem. This provides an important guarantee for the algorithm's performance in light of the fact that finite-width neural networks may only converge to a local minimizers (and not optimal solutions) due to the non-convexity of their loss functions. In our numerical studies, we demonstrate that the algorithm can solve stochastic control problems accurately in up to 200 dimensions. In particular, we construct a series of increasingly complex stochastic control problems with known analytic solutions and study the algorithm's numerical performance on them. These problems range from a linear-quadratic regulator equation to highly challenging equations with non-convex Hamiltonians, allowing us to identify and analyze the strengths and limitations of this neural actor-critic method for solving HJB equations.

We introduce a continuous policy-value iteration algorithm where the approximations of the value function of a stochastic control problem and the optimal control are simultaneously updated through Langevin-type dynamics. This framework applies to both the entropy-regularized relaxed control problems and the classical control problems, with infinite horizon. We establish policy improvement and demonstrate convergence to the optimal control under the monotonicity condition of the Hamiltonian. By utilizing Langevin-type stochastic differential equations for continuous updates along the policy iteration direction, our approach enables the use of distribution sampling and non-convex learning techniques in machine learning to optimize the value function and identify the optimal control simultaneously.

We propose a machine learning algorithm for solving finite-horizon stochastic control problems based on a deep neural network representation of the optimal policy functions. The algorithm has three features: (1) It can solve high-dimensional (e.g., over 100 dimensions) and finite-horizon time-inhomogeneous stochastic control problems. (2) It has a monotonicity of performance improvement in each iteration, leading to good convergence properties. (3) It does not rely on the Bellman equation. To demonstrate the efficiency of the algorithm, it is applied to solve various finite-horizon time-inhomogeneous problems including recursive utility optimization under a stochastic volatility model, a multi-sector stochastic growth, and optimal control under a dynamic stochastic integration of climate and economy model with eight-dimensional state vectors and 600 time periods.

This work introduces a novel paradigm for solving optimal control problems for hybrid dynamical systems under uncertainties. Robotic systems having contact with the environment can be modeled as hybrid systems. Controller design for hybrid systems under disturbances is complicated by the discontinuous jump dynamics, mode changes with inconsistent state dimensions, and variations in jumping timing and states caused by noise. We formulate this problem into a stochastic control problem with hybrid transition constraints and propose the Hybrid Path Integral (H-PI) framework to obtain the optimal controller. Despite random mode changes across stochastic path samples, we show that the ratio between hybrid path distributions with varying drift terms remains analogous to the smooth path distributions. We then show that the optimal controller can be obtained by evaluating a path integral with hybrid constraints. Importance sampling for path distributions with hybrid dynamics constraints is introduced to reduce the variance of the path integral evaluation, where we leverage the recently developed Hybrid iterative-Linear-Quadratic-Regulator (H-iLQR) controller to induce a hybrid path distribution proposal with low variance. The proposed method is validated through numerical experiments on various hybrid systems and extensive ablation studies. All the sampling processes are conducted in parallel on a Graphics Processing Unit (GPU).

Bismut [1] first introduced linear backward stochastic differential equations (BSDEs in short) as the adjoint equation of the classical stochastic optimal control problem. In 1990, Pardoux and Peng firstly proved the existence and uniqueness of nonlinear BSDEs with Lipschitz condition [2]. Since then, the theory of BSDEs has been studied by many researchers and applied in a wide range of areas, such as in stochastic optimal control and mathematical finance [3, 4]. When a BSDE is coupled with a (forward) stochastic differential equation (SDE in short), the system is usually called a forward-backward stochastic differential equation (FBSDE in short). We can refer to the literatures in [5, 6, 7, 8, 9, 10] which studied the existence, uniqueness and the applications of coupled or fully-coupled FBSDEs.

We propose a comprehensive framework for policy gradient methods tailored to continuous time reinforcement learning. This is based on the connection between stochastic control problems and randomised problems, enabling applications across various classes of Markovian continuous time control problems, beyond diffusion models, including e.g. regular, impulse and optimal stopping/switching problems. By utilizing change of measure in the control randomisation technique, we derive a new policy gradient representation for these randomised problems, featuring parametrised intensity policies. We further develop actor-critic algorithms specifically designed to address general Markovian stochastic control issues. Our framework is demonstrated through its application to optimal switching problems, with two numerical case studies in the energy sector focusing on real options.