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.

This paper focuses on minimizing the costs of installing renewable energy capacity while meeting emission constraints under uncertainty in both energy demand and renewable production. We consider a setting where decision-makers must determine when and how much renewable capacity to install, balancing investment costs with future emissions. Our optimization problem combines cost minimization with a probabilistic constraint on total accumulated emissions, reflecting regulatory limits that may be exceeded only with small probability. We examine different investment strategies, allowing for one or multiple installation times, and provide explicit solutions in simplified cases. Our main insight is that, under reasonable assumptions on costs and uncertainty, a single, well-timed investment is optimal and may be delayed to reduce costs when uncertainty and discounting are accounted for. These results challenge common stepwise installation strategies and suggest that committing to a single large investment, possibly postponed, may be more cost-effective and efficient in reaching emission targets. Our findings offer practical guidance for policymakers and energy planners on how to balance costs, timing and environmental goals when expanding renewable energy capacity under uncertainty.

In this paper, we study the optimal control of systems where the state dynamics are governed by a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion, also referred to as a Brownian sheet. These equations can, for example, model the growth of an ecosystem under uncertainty. We first explore some fundamental properties of such linear SPDEs. Next, utilizing time-space white noise calculus, we derive both Pontryagin-type necessary and sufficient conditions for the optimality of the control. Finally, we illustrate our results by solving a linear-quadratic control problem and examining an optimal harvesting problem in the plane. Potential applications to machine learning and to managing random environmental influences are also discussed.

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

We consider the problem of finding the optimal initial investment strategy for a system modelled by a linear McKean–Vlasov (mean-field) stochastic differential equation with delay, driven by Brownian motion and a pure jump Poisson random measure. The goal is to determine the optimal initial values for the system in the period [−𝛿,0], where 𝛿>0 is a delay constant, before the system starts at t = 0. Due to the delay in the dynamics, the system will, after startup, be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By utilizing this equivalence, we can find implicit expressions for the optimal investment. Moreover, we propose a deep neural network-based algorithm to solve the stochastic control problem with delay. Specifically, we employ a multi-layer feed-forward neural network for control modelling in the interval [−𝛿,0], and use back-propagation to train the feed-forward neural network. The gradient of the loss function is computed using stochastic gradient descent (SGD) with respect to the weights of the network.

In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ-algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.