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.

This thesis brings together theoretical advances in stochastic optimal control and modern deep learning techniques, with particular emphasis on applications in environmental and energy systems. The first group of contributions investigates optimal control from a theoretical perspective, developing new results and illustrating their relevance through real world applications. The second part explores deep learning methods for solving stochastic differential equations and control problems that are analytically intractable.

We begin by studying impulse control problems for conditional McKean--Vlasov jump diffusions, extending the classical verification theorem to the setting in which the state dynamics depend on their conditional distribution. We then examine an optimal control problem for pollution growth on a spatial network, formulated in a deterministic framework but capturing how environmental policies propagate across interconnected geographical regions. Finally, we develop a model for investment in renewable energy capacity under uncertainty, characterising how optimal installation strategies change in response to fluctuations in energy demand and production. These contributions show how stochastic control can be used to address pressing challenges in environmental regulation and energy planning.

The second line of research focuses on deep learning methods for backward stochastic differential equations (BSDEs) and related formulations, together with direct machine learning approaches for high-dimensional stochastic control. Specifically, we solve Dynkin games by reformulating them as doubly reflected BSDEs, enabling the computation of optimal stopping strategies in energy market contracts. We further develop a deep learning solver for backward stochastic Volterra integral equations (BSVIEs), extending neural BSDE methods to systems with memory. In addition, we propose a machine learning framework for renewable capacity investment under jump uncertainty, treating the problem both through a direct control learning strategy and through a newly developed solver for pure jump BSDEs.

Overall, this thesis lies at the intersection of rigorous mathematical analysis and machine learning-based approaches to stochastic optimal control. On the one hand, we show how careful modeling and theoretical results enable the formulation and study of complex, realistic control problems; on the other hand, we demonstrate how modern machine learning techniques provide powerful tools for solving these problems efficiently. The applications are motivated by urgent questions in environmental and energy sustainability.

Denna avhandling förenar teoretiska framsteg inom stokastisk optimal styrning med moderna djupinlärningsmetoder, med särskild tonvikt på tillämpningar inom miljö- och energisystem. Den första gruppen av bidrag undersöker optimal styrning ur ett teoretiskt perspektiv, utvecklar nya resultat och visar dess relevans genom praktiskt motiverade exempel. Den andra delen behandlar djupinlärningsmetoder för att lösa stokastiska differentialekvationer och styrproblem som annars är analytiskt oöverskådliga.

Vi börjar med att studera impulskontrollproblem för betingade McKean–Vlasov-hoppdiffusioner och utvidgar den klassiska verifikationssatsen till situationer där systemets dynamik beror på dess betingade fördelning. Därefter analyseras ett optimalt styrproblem för utsläppstillväxt på ett rumsligt nätverk, formulerat deterministiskt men avsett att fånga hur miljöpolitiska beslut sprids över sammanlänkade geografiska regioner. Slutligen utvecklar vi en modell för investering i förnybar energikapacitet under osäkerhet, där vi karakteriserar hur optimala installationsstrategier påverkas av variationer i efterfrågan och produktion. Dessa bidrag visar hur stokastisk styrning kan användas för att hantera centrala frågor inom miljöreglering och energiplanering.

Den andra forskningslinjen fokuserar på djupinlärningsmetoder för bakåtriktade stokastiska differentialekvationer (BSDE:er) och relaterade formuleringar, samt direkta maskininlärningsmetoder för högdimensionella stokastiska styrproblem. Vi löser särskilt Dynkin-spel genom att formulera dem som dubbelt reflekterande BSDE:er, vilket möjliggör beräkning av optimala stoppstrategier i energimarknadskontrakt. Vidare utvecklar vi en djupinlärningsbaserad lösare för bakåtriktade stokastiska Volterra-integralekvationer (BSVIE:er), och utvidgar därmed neurala BSDE-metoder till system med minnesstruktur. Dessutom föreslår vi ett maskininlärningsramverk för investeringar i förnybar kapacitet under hopp-osäkerhet, både genom direkt styrinlärning och genom en ny lösare för rena hopp-BSDE:er.

Sammantaget placerar sig denna avhandling i gränslandet mellan rigorös matematisk analys och maskininlärningsbaserade metoder för stokastisk optimal styrning. Å ena sidan visar vi hur noggrann modellering och teoretiska resultat möjliggör formulering och studie av komplexa, realistiska styrproblem; å andra sidan visar vi hur moderna djupinlärningstekniker ger kraftfulla verktyg för att lösa dessa problem på ett effektivt sätt. Tillämpningarna är motiverade av aktuella och angelägna frågor inom miljömässig och energimässig hållbarhet.

This paper studies a model for the optimal control of pollution diffusion over time and space by a centralized economic agent. The controls are the investments in two types of production: a less polluting (”green”) technology and a more polluting (”brown”) one. The goal is to maximize an intertemporal utility function which takes into account the cost of pollution. The main novelty is the fact that the spatial component has a network structure. Moreover, in such a time-space setting, we analyze the trade-off between the use of green and brown technologies: this is also a novelty in such a setting. Extending methods from previous works, we can explicitly solve the problem in the case of strictly convex or linear pollution costs.

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 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.

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 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.