Thepresented equations can be efficiently solved using well-known numerical methods.

In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient serving as the hedging strategy, optimized via a loss function enforcing the self-financing portfolio condition. A key challenge arises from the non-smooth nature of option payoffs (e.g., vanilla calls are non-differentiable at-the-money, while digital options are discontinuous), which conflicts with the inherent smoothness of standard neural networks. To address this, we compare unconstrained networks against constrained architectures that explicitly embed the terminal payoff condition, drawing inspiration from PDE-solving techniques. Our framework assumes two tradable assets: the underlying and a liquid call option capturing volatility dynamics. Numerical experiments evaluate the method on simple options with varying non-smoothness, the exotic Equinox option, and scenarios with market jumps for robustness. Results demonstrate superior P&L distributions, highlighting the efficacy of constrained networks in handling realistic payoffs. This work advances machine learning applications in quantitative finance by integrating boundary constraints, offering a practical tool for pricing and hedging in incomplete markets.

This paper introduces \textbf{Measure Learning}, a paradigm for modeling ambiguity via non-linear expectations. We define Neural Expectation Operators as solutions to Backward Stochastic Differential Equations (BSDEs) whose drivers are parameterized by neural networks. The main mathematical contribution is a rigorous well-posedness theorem for BSDEs whose drivers satisfy a local Lipschitz condition in the state variable $y$ and quadratic growth in its martingale component $z$. This result circumvents the classical global Lipschitz assumption, is applicable to common neural network architectures (e.g., with ReLU activations), and holds for exponentially integrable terminal data, which is the sharp condition for this setting. Our primary innovation is to build a constructive bridge between the abstract, and often restrictive, assumptions of the deep theory of quadratic BSDEs and the world of machine learning, demonstrating that these conditions can be met by concrete, verifiable neural network designs. We provide constructive methods for enforcing key axiomatic properties, such as convexity, by architectural design. The theory is extended to the analysis of fully coupled Forward-Backward SDE systems and to the asymptotic analysis of large interacting particle systems, for which we establish both a Law of Large Numbers (propagation of chaos) and a Central Limit Theorem. This work provides the foundational mathematical framework for data-driven modeling under ambiguity.

Stochastic control problems in high dimensions are notoriously difficult to solve due to the curse of dimensionality. An alternative to traditional dynamic programming is Pontryagin's Maximum Principle (PMP), which recasts the problem as a system of Forward-Backward Stochastic Differential Equations (FBSDEs). In this paper, we introduce a formal framework for solving such problems with deep learning by defining a \textbf{Neural Hamiltonian Operator (NHO)}. This operator parameterizes the coupled FBSDE dynamics via neural networks that represent the feedback control and an ansatz for the value function's spatial gradient. We show how the optimal NHO can be found by training the underlying networks to enforce the consistency conditions dictated by the PMP. By adopting this operator-theoretic view, we situate the deep FBSDE method within the rigorous language of statistical inference, framing it as a problem of learning an unknown operator from simulated data. This perspective allows us to prove the universal approximation capabilities of NHOs under general martingale drivers and provides a clear lens for analyzing the significant optimization challenges inherent to this class of models.

Biological and psychological concepts have inspired reinforcement learning algorithms to create new complex behaviors that expand agents' capacity. These behaviors can be seen in the rise of techniques like goal decomposition, curriculum, and intrinsic rewards, which have paved the way for these complex behaviors. One limitation in evaluating these methods is the requirement for engineered extrinsic for realistic environments. A central challenge in engineering the necessary reward function(s) comes from these environments containing states that carry high negative rewards, but provide no feedback to the agent. Death is one such stimuli that fails to provide direct feedback to the agent. In this work, we introduce an intrinsic reward function inspired by early amygdala development and produce this intrinsic reward through a novel memory-augmented neural network (MANN) architecture. We show how this intrinsic motivation serves to deter exploration of terminal states and results in avoidance behavior similar to fear conditioning observed in animals. Furthermore, we demonstrate how modifying a threshold where the fear response is active produces a range of behaviors that are described under the paradigm of general anxiety disorders (GADs). We demonstrate this behavior in the Miniworld Sidewalk environment, which provides a partially observable Markov decision process (POMDP) and a sparse reward with a non-descriptive terminal condition, i.e., death. In effect, this study results in a biologically-inspired neural architecture and framework for fear conditioning paradigms; we empirically demonstrate avoidance behavior in a constructed agent that is able to solve environments with non-descriptive terminal conditions.

Neural network-based methods for solving Mean-Field Games (MFGs) equilibria have garnered significant attention for their effectiveness in high-dimensional problems. However, many algorithms struggle with ensuring that the evolution of the density distribution adheres to the required mathematical constraints. This paper investigates a neural network approach to solving MFGs equilibria through a stochastic process perspective. It integrates process-regularized Normalizing Flow (NF) frameworks with state-policy-connected time-series neural networks to address McKean-Vlasov-type Forward-Backward Stochastic Differential Equation (MKV FBSDE) fixed-point problems, equivalent to MFGs equilibria.

Motivated by dynamic risk measures and conditional $g$-expectations, in this work we propose a numerical method to approximate the solution operator given by a Backward Stochastic Differential Equation (BSDE). The main ingredients for this are the Wiener chaos decomposition and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it using neural networks, and we present several numerical examples where we can check the accuracy of the method.

The interplay between stochastic processes and optimal control has been extensively explored in the literature. With the recent surge in the use of diffusion models, stochastic processes have increasingly been applied to sample generation. This paper builds on the log transform, known as the Cole-Hopf transform in Brownian motion contexts, and extends it within a more abstract framework that includes a linear operator. Within this framework, we found that the well-known relationship between the Cole-Hopf transform and optimal transport is a particular instance where the linear operator acts as the infinitesimal generator of a stochastic process. We also introduce a novel scenario where the linear operator is the adjoint of the generator, linking to Bayesian inference under specific initial and terminal conditions. Leveraging this theoretical foundation, we develop a new algorithm, named the HJ-sampler, for Bayesian inference for the inverse problem of a stochastic differential equation with given terminal observations. The HJ-sampler involves two stages: (1) solving the viscous Hamilton-Jacobi partial differential equations, and (2) sampling from the associated stochastic optimal control problem. Our proposed algorithm naturally allows for flexibility in selecting the numerical solver for viscous HJ PDEs. We introduce two variants of the solver: the Riccati-HJ-sampler, based on the Riccati method, and the SGM-HJ-sampler, which utilizes diffusion models. We demonstrate the effectiveness and flexibility of the proposed methods by applying them to solve Bayesian inverse problems involving various stochastic processes and prior distributions, including applications that address model misspecifications and quantifying model uncertainty.

We formulate well-posed continuous-time generative flows for learning distributions that are supported on low-dimensional manifolds through Wasserstein proximal regularizations of $f$-divergences. Wasserstein-1 proximal operators regularize $f$-divergences so that singular distributions can be compared. Meanwhile, Wasserstein-2 proximal operators regularize the paths of the generative flows by adding an optimal transport cost, i.e., a kinetic energy penalization. Via mean-field game theory, we show that the combination of the two proximals is critical for formulating well-posed generative flows. Generative flows can be analyzed through optimality conditions of a mean-field game (MFG), a system of a backward Hamilton-Jacobi (HJ) and a forward continuity partial differential equations (PDEs) whose solution characterizes the optimal generative flow. For learning distributions that are supported on low-dimensional manifolds, the MFG theory shows that the Wasserstein-1 proximal, which addresses the HJ terminal condition, and the Wasserstein-2 proximal, which addresses the HJ dynamics, are both necessary for the corresponding backward-forward PDE system to be well-defined and have a unique solution with provably linear flow trajectories. This implies that the corresponding generative flow is also unique and can therefore be learned in a robust manner even for learning high-dimensional distributions supported on low-dimensional manifolds. The generative flows are learned through adversarial training of continuous-time flows, which bypasses the need for reverse simulation. We demonstrate the efficacy of our approach for generating high-dimensional images without the need to resort to autoencoders or specialized architectures.