We study a stochastic model for the installation of renewable energy capacity under demand uncertainty and jump driven dynamics. The system is governed by a multidimensional Ornstein–Uhlenbeck (OU) process driven by a subordinator, capturing abrupt variations in renewable generation and electricity load. Installation decisions are modeled through control actions that increase capacity in response to environmental and economic conditions. We consider two distinct solution approaches. First, we implement a structured threshold based control rule, where capacity is increased proportionally when the stochastic capacity factor falls below a fixed level. This formulation leads to a nonlinear partial integro-differential equation (PIDE), which we solve by reformulating it as a backward stochastic differential equation with jumps. We extend the DBDP solver in Huré et al. (2020) to the pure jump setting, employing a dual neural network architecture to approximate both the value function and the jump sensitivity. Second, to benchmark the performance of the first approach, we propose a deep control algorithm that directly learns a state and time dependent feedback threshold policy by minimizing the expected cost functional using neural networks. This approach preserves the threshold based structure while allowing the threshold to adapt dynamically to the evolving system state, enabling more flexible and responsive interventions. Numerical experiments on calibrated data highlight the strengths of both methods. While the threshold based BSDE approach offers interpretability and tractability, the deep control strategy achieves improved performance through flexibility in capacity allocation. Together, these tools provide a robust framework for decision support in long term renewable energy expansion under uncertainty.

We study the numerical approximation of backward stochastic Volterra integral equations (BSVIEs) and their reflected extensions, which naturally arise in problems with time inconsistency, path dependent preferences, and recursive utilities with memory. These equations generalize classical BSDEs by involving two dimensional time structures and more intricate dependencies. We begin by developing a well posedness and measurability framework for BSVIEs in product probability spaces. Our approach relies on a representation of the solution as a parametrized family of backward stochastic equations indexed by the initial time, and draws on results of Stricker and Yor to ensure that the two parameter solution is well defined in a joint measurable sense. We then introduce a discrete time learning scheme based on a recursive backward representation of the BSVIE, combining the discretization of Hamaguchi and Taguchi with deep neural networks. A detailed convergence analysis is provided, generalizing the framework of deep BSDE solvers to the two dimensional BSVIE setting. Finally, we extend the solver to reflected BSVIEs, motivated by applications in delayed recursive utility with lower constraints.

We formulate a Contract for Difference (CfD) with early exit options as a two-player zero-sum Dynkin game, capturing the strategic interaction between an electricity producer and a regulatory authority. The payoff structure includes running revenues, early termination penalties, and a terminal settlement, while the underlying electricity prices follow mean-reverting dynamics. The value of the game and the associated feedback optimal stopping rules are characterized through a doubly reflected backward stochastic differential equation (DRBSDE). In addition, we provide a structural characterization of the reflection mechanism, showing how the cumulative penalty processes are generated by the interaction between the value process and the time dependent barriers. To approximate the solution of the DRBSDE, we propose a learning-based numerical method that combines time discretization with neural network approximations of the backward components along simulated price trajectories. The approach avoids explicit state space discretization, accommodates time dependent barriers, and is applicable in moderately high-dimensional settings. A convergence result is established to justify the link between the continuous-time formulation and its numerical approximation. The proposed Deep DRBSDE solver is illustrated on a CfD model driven by 24-dimensional mean-reverting electricity prices representing multiple European market zones. In addition, a symmetric benchmark Dynkin game in dimension 20 and a mean-field extension are considered to assess the validity of the solver in controlled settings. The numerical results demonstrate stable training behavior and a consistent approximation of the contract value and optimal stopping regions across the considered examples.

We study stochastic control problems in a financial market subject to default risk, modeled via a progressive enlargement of filtration. The default event is represented by a random time, and the associated information flow is incorporated through an enlarged filtration framework. We establish a stochastic maximum principle for controlled diffusion processes with a single jump at the default time. The approach relies on the martingale representation under the enlarged filtration, which allows us to derive the adjoint equations and characterize necessary optimality conditions. The presence of default introduces additional terms in both the state dynamics and the adjoint processes, reflecting the impact of jump risk and information flow on optimal strategies. We apply the theoretical results to two explicitly solvable portfolio optimization problems under default risk. In particular, we obtain closed form expressions for the optimal controls in both the logarithmic and exponential utility cases. We further analyze how the default mechanism influences optimal strategies through the filtration structure and model parameters. These results contribute to a deeper understanding of optimal decision making in the presence of random time events.

We study a stochastic model for the installation of renewable energy capacity under demand uncertainty and jump driven dynamics. The system is governed by a multidimensional Ornstein-Uhlenbeck (OU) process driven by a subordinator, capturing abrupt variations in renewable generation and electricity load. Installation decisions are modeled through control actions that increase capacity in response to environmental and economic conditions. We consider two distinct solution approaches. First, we implement a structured threshold based control rule, where capacity is increased proportionally when the stochastic capacity factor falls below a fixed level. This formulation leads to a nonlinear partial integro-differential equation (PIDE), which we solve by reformulating it as a backward stochastic differential equation with jumps. We extend the DBDP solver in [15] to the pure jump setting, employing a dual neural network architecture to approximate both the value function and the jump sensitivity. Second, we propose a fully data driven deep control algorithm that directly learns the optimal feedback policy by minimizing the expected cost functional using neural networks. This approach avoids assumptions on the form of the control rule and enables adaptive interventions based on the evolving system state. Numerical experiments highlight the strengths of both methods. While the threshold based BSDE approach offers interpretability and tractability, the deep control strategy achieves improved performance through flexibility in capacity allocation. Together, these tools provide a robust framework for decision support in long term renewable energy expansion under uncertainty

We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation. This can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations of a constant and noisy observations of the Brownian sheet.

This paper introduces an autonomous vision-based tracking system for a quadrotor unmanned aerial vehicle (UAV) equipped with an onboard camera, designed to track a maneuvering target without external localization sensors or GPS. Accurate capture of dynamic aerial targets is essential to ensure real-time tracking and effective management. The system employs a robust and computationally efficient visual tracking method that combines HSV filter detection with a shape detection algorithm. Target states are estimated using an enhanced extended Kalman filter (EKF), providing precise state predictions. Furthermore, a closed-loop Proportional-Integral-Derivative (PID) controller, based on the estimated states, is implemented to enable the UAV to autonomously follow the moving target. Extensive simulation and experimental results validate the system’s ability to efficiently and reliably track a dynamic target, demonstrating robustness against noise, light reflections, or illumination interference, and ensure stable and rapid tracking using low-cost components.

The current paper focuses on using deep learning methods to optimize the control of conditional McKean–Vlasov jump diffusions. We begin by exploring the dynamics of multi-particle jump-diffusion and presenting the propagation of chaos. The optimal control problem in the context of conditional McKean–Vlasov jump-diffusion is introduced along with the verification theorem (HJB equation). A linear quadratic conditional mean-field (LQ CMF) is discussed to illustrate these theoretical concepts. Then, we introduce a deep-learning algorithm that combines neural networks for optimization with path signatures for conditional expectation estimation. The algorithm is applied to practical examples, including LQ CMF and interbank systemic risk, and we share the resulting numerical outcomes.

We formulate a Contract for Difference (CfD) with early exit options as a two-player zero-sum Dynkin game, reflecting the strategic interaction between an electricity producer and a regulatory entity. The game incorporates penalties for early termination and mean-reverting price dynamics, with the value characterized through a doubly reflected backward stochastic differential equation (DRBSDE). To compute the contract value and optimal stopping strategies, we develop a neural solver that approximates the DRBSDE solution using a sequence of neural networks trained on simulated trajectories. The method avoids discretizing the state space, supports time-dependent barriers, and scales to highdimensional settings. We establish a convergence result and test the method on two scenarios: a benchmark symmetric game in 20 dimensions, and a CfD model with 24-dimensional electricity prices representing multiple European zones. The results demonstrate that the proposed solver accurately captures the contract’s value and optimal stopping regions, with consistent performance across dimensional settings.