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• Benth, Fred Espen; Ortiz-Latorre, Salvador & Tarquini, Leonardo (2026). Signature McKean-Vlasov stochastic differential equations. arXiv. doi: 10.48550/ARXIV.2606.22964. Full text in Research Archive Show summary

  McKean-Vlasov-type stochastic differential equations (SDEs) are characterized by coefficients depending on both the state and the law of the solution. In this work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric p-rough path lift of its solution, with $p\in(2,3)$. After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean-Vlasov SDEs. Finally, we consider the associated particle system and propagation of chaos is established.

• Moda, Massimiliano; Hout, Karel J. in 't; Vanmaele, Michèle & Benth, Fred Espen (2026). Numerical Valuation of European Options Under Two-Asset Infinite-Activity Exponential Lévy Models. Applied Mathematical Finance. ISSN 1350-486X. doi: 10.1080/1350486X.2026.2715128. Full text in Research Archive
• Gyawali, Sanjay; Nakstad, Espen Rostrup; Sletvold, Bjørn; Lurås, Hilde; Wisløff, Torbjørn & Benth, Fred Espen [Show all 7 contributors for this article] (2026). Infection dynamics across regions during the COVID-19 pandemic in Norway. Journal of Infection and Public Health. ISSN 1876-0341. 19(9), p. 1–12. doi: 10.1016/j.jiph.2026.103328. Full text in Research Archive Show summary

  Background: Urban areas often act as early epicentres of pandemics, yet transmission dynamics across regions with different demographic profiles remains poorly understood. We aimed to characterise COVID-19 transmission in Norway and assess how it interacted with infection control measures across urban and rural settings. Methods: We analysed daily COVID-19 cases across three pandemic waves (Alpha, Delta, Omicron). For local trends, 36 sites in the Greater Oslo Region were grouped by infection trajectories using group-based trajectory modelling. For national trends, we examined the cities of Oslo, Bergen, Trondheim, Stavanger, Tromsø, and Akershus county. Infection trends were analysed using linear mixed models and lagged regression analyses and related to demographic characteristics. Results: In the Greater Oslo Region, three distinct site groups-high, moderate, and low infection-were identified. The high infection group had the highest population density, largest proportion of crowded housing and immigrant population and preceded the moderate and low groups by 5-13 days during the Alpha wave. Nationally, Oslo's infection rates peaked 11-24 days before other cities during the Alpha wave, with shorter lags during the Delta and Omicron waves. Despite similar control measures, infection rates differed substantially, with the high infection group and Oslo city consistently bearing the greatest burden. Conclusions: Densely populated areas with crowded housing act as early warning signals, with infection growth preceding that in other areas by days to weeks. Systematic monitoring of such high-burden urban areas could provide valuable lead time for policymakers and support more targeted, timely responses in future pandemics.

• Agram, Nacira; Benth, Fred Espen; Pucci, Giulia & Rems, Jan (2026). A deep learning approach to renewable capacity installation under jump uncertainty. Mathematics and Computers in Simulation. ISSN 0378-4754. 250, p. 152–177. doi: 10.1016/j.matcom.2026.06.024. Full text in Research Archive Show summary

  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.

• Regragui, Mustapha; Hout, Karel J. in ’t; Vanmaele, Michèle & Benth, Fred Espen (2026). Numerical methods for solving PIDEs arising in swing option pricing under a two-factor mean-reverting model with jumps. Applied Numerical Mathematics. ISSN 0168-9274. 227, p. 204–231. doi: 10.48550/arXiv.2511.01587. Full text in Research Archive
• Benth, Fred Espen; Harang, Fabian Andsem & Straum, Fride Josefine Emberland (2026). Universal Approximation on Non-geometric Rough Paths and Applications to Financial Derivatives Pricing. Applied Mathematical Finance. ISSN 1350-486X. doi: 10.1080/1350486X.2026.2659017. Full text in Research Archive Show summary

  We present a novel perspective on the universal approximation theorem for rough path functionals, introducing a polynomial-based approximation class. We extend universal approximation to non-geometric rough paths within the tensor algebra. This development addresses critical needs in finance, where no-arbitrage conditions necessitate Itô integration. Furthermore, our findings motivate a hypothesis for payoff functionals in financial markets, allowing straightforward analysis of signature payoffs proposed in Arribas (Derivatives Pricing Using Signature Payoffs; 2018).

• Heupink, Lieke Fleur; Nakstad, Espen Rostrup; Lurås, Hilde; Wiik, Pål; Lippestad, Kristine & Benth, Fred Espen [Show all 7 contributors for this article] (2026). Short term forecast of new daily pandemic hospitalizations: A time series model for a single hospital. Epidemics. ISSN 1755-4365. 54. doi: 10.1016/j.epidem.2026.100894. Full text in Research Archive
• Benth, Fred Espen & Veraart, Almut E.D. (2026). Research frontiers in ambit stochastics: In memory of Ole E. Barndorff-Nielsen. Bernoulli. ISSN 1350-7265. 32(1), p. 24–48. doi: 10.3150/25-bej1902. Full text in Research Archive Show summary

  This article surveys key aspects of ambit stochastics and remembers Ole E. Barndorff-Nielsen’s important contributions to the foundation and advancement of this new research field over the last two decades. It also highlights some of the emerging trends in ambit stochastics.

• Benth, Fred Espen; Deelstra, Griselda & Kozpınar, Sinem (2025). Pricing energy quanto options: a regime-switching framework with stochastic interest rates. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2025.2541874. Full text in Research Archive Show summary

  Energy quanto options are risk management instruments gaining a lot of attraction from the actors in energy markets due to their nature, which allows to handle volumetric and price risk simultaneously, in contrast to typical contracts that solely take price risk into consideration. The payoff of these options is constructed in such a way that it resembles the product of the payouts from two options, each with an energy-related underlying. If the underlying assets are considered to be energy futures contracts, which are more easily traded assets than temperature and energy spots themselves, the payoff of the energy quanto option can be treated as the product of the payoff of two European vanilla options. This paper investigates the valuation of energy quanto options written on temperature and electricity futures in the presence of regime-switching stochastic interest rates. The electricity futures price is governed by a continuous Markov-modulated model with mean reversion, and the dynamics of temperature futures evolve without regime-switching but retain seasonality and mean reversion. Using a change of measure and fast Fourier transform techniques, we obtain a pricing formula for energy quanto options on futures. Numerical examples are carried out to demonstrate the implementation of the model.

• Agram, Nacira; Benth, Fred Espen & Pucci, Giulia (2025). Installation of renewable capacities to meet energy demand and emission constraints under uncertainty. IMA Journal of Management Mathematics. ISSN 1471-678X. 37(1), p. 39–60. doi: 10.1093/imaman/dpaf023. Full text in Research Archive Show summary

  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.

• Hvidsten, Tobias Verheugen; Roithner, Maximilian; Benth, Fred Espen & Zeyringer, Marianne (2025). Driving towards net-zero: the impact of electric vehicle flexibility participation on a future Norwegian electricity system. Environmental Research: Infrastructure and Sustainability. 5(2). doi: 10.1088/2634-4505/ade0e6. Full text in Research Archive Show summary

  Electric vehicle (EV) batteries have a proven flexibility potential which could serve as an alternative to conventional electricity storage solutions. EV batteries could support the balancing of supply and demand, and the integration of variable renewable energy into the electricity system. The flexibility potential from EVs, in distinction to conventional battery storage, depends on the vehicle user’s willingness and opportunity to make their vehicle available for flexibility. This rate of participation is often not considered in modelling studies, despite the impact EV flexibility could have on the electricity system. This work presents a modelling study of the Norwegian electricity system using the high spatial and temporal resolution electricity system model (highRES). Two different charging strategies for EVs, flexible charging and vehicle-to-grid (V2G), are optimised in conjunction with the design and dispatch of the electricity system. By varying the rate of participation in the charging strategies we demonstrate their benefits for a future net-zero electricity system. Our findings show EVs’ potential to eliminate the need for stationary battery storage with just 50% participation in V2G. In addition, we find that the EV flexibility contributes to reductions in the total system cost by almost 4% and 15% assuming 100% participation in flexible charging and V2G, respectively.

• Greevenbroek, Koen van; Grochowicz, Aleksander; Zeyringer, Marianne & Benth, Fred Espen (2025). Trading off regional and overall energy system design flexibility in the net-zero transition. Nature Sustainability. doi: 10.1038/s41893-025-01556-2. Full text in Research Archive Show summary

  Thetransitiontonet-zeroemissionsinEuropeisdeterminedbyapatchworkofcountry-levelandEUwide policy, creating coordination challenges in an interconnected system. We use an optimisation model to mapoutnear-optimal energy system designs for 2050, focussing on the planning flexibility of individual regions while maintaining overall system robustness against different weather years, cost assumptions, and land use limitations. Our results reveal extensive flexibility at a regional level, where only few technologies (solar around the Adriatic and wind on the British Isles and in Germany) cannot be substituted. National policymakers can influence renewable energy export and hydrogen strategies significantly, provided they coordinate this with the remaining European system. However, stronger commitment to solar in Southern Europe and Germany unlocks more design options for Europe overall. These results on regional trade-offs facilitate more meaningful policy discussions which are crucial in the transition to a sustainable energy system.

• Benth, Fred Espen & Eyjolfsson, Heidar (2024). Robustness of Hilbert space-valued stochastic volatility models. Finance and Stochastics. ISSN 0949-2984. 28, p. 1117–1146. doi: 10.1007/s00780-024-00542-4. Full text in Research Archive Show summary

  In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

• Grochowicz, Aleksander; Benth, Fred Espen & Zeyringer, Marianne (2024). Spatio-temporal smoothing and dynamics of different electricity flexibility options for highly renewable energy systems—Case study for Norway. Applied Energy. ISSN 0306-2619. 356. doi: 10.1016/j.apenergy.2023.122338. Full text in Research Archive Show summary

  In this article, we investigate the mismatch of renewable electricity production to demand and how flexibility options enabling spatial and temporal smoothing can reduce risks of variability. As a case study we pick a simplified (partial) 2-region representation of the Norwegian electricity system and focus on wind power. We represent regional electricity production and demand through two stochastic processes: the wind capacity factors are modelled as a two-dimensional Ornstein–Uhlenbeck process and electricity demand consists of realistic base load and temperature-induced load coming from a deseasonalised autoregressive process. We validate these processes, that we have trained on historical data, through Monte Carlo simulations allowing us to generate many statistically representative weather years. For the investigated realisations (weather years) we study deviations of production from demand under different wind capacities, and introduce different scenarios where flexibility options like storage and transmission are available. Our analysis shows that simulated loss values are reduced significantly by cooperation between regions and either mode of flexibility. Combining storage and transmission leads to even more synergies and helps to stabilise production levels and thus reduces likelihoods of inadequacy of renewable power systems.

• Vorobeva, Ekaterina; Eggen, Mari Dahl; Midtfjord, Alise Danielle; Benth, Fred Espen; Hupe, Patrick & Brissaud, Quentin [Show all 8 contributors for this article] (2024). Estimating stratospheric polar vortex strength using ambient ocean-generated infrasound and stochastics-based machine learning. Quarterly Journal of the Royal Meteorological Society. ISSN 0035-9009. 150, p. 2712–2726. doi: 10.1002/qj.4731. Full text in Research Archive Show summary

  There are sparse opportunities for direct measurement of upper stratospheric winds, yet improving their representation in subseasonal-to-seasonal prediction models can have significant benefits. There is solid evidence from previous research that global atmospheric infrasound waves are sensitive to stratospheric dynamics. However, there is a lack of results providing a direct mapping between infrasound recordings and polar-cap upper stratospheric winds. The global International Monitoring System (IMS), which monitors compliance with the Comprehensive Nuclear-Test-Ban Treaty, includes ground-based stations that can be used to characterize the infrasound soundscape continuously. In this study, multi-station IMS infrasound data were utilized along with a machine-learning supported stochastic model, Delay-SDE-net, to demonstrate how a near-real-time estimate of the polar-cap averaged zonal wind at 1-hPa pressure level can be found from infrasound data. The infrasound was filtered to a temporal low-frequency regime dominated by microbaroms, which are ambient-noise infrasonic waves continuously radiated into the atmosphere from nonlinear interaction between counter-propagating ocean surface waves. Delay-SDE-net was trained on 5 years (2014–2018) of infrasound data from three stations and the ERA5 reanalysis 1-hPa polar-cap averaged zonal wind. Using infrasound in 2019–2020 for validation, we demonstrate a prediction of the polar-cap averaged zonal wind, with an error standard deviation of around 12 m·s compared with ERA5. These findings highlight the potential of using infrasound data for near-real-time measurements of upper stratospheric dynamics. A long-term goal is to improve high-top atmospheric model accuracy, which can have significant implications for weather and climate prediction.

• Benth, Fred Espen; Eggen, Mari Dahl & Eisenberg, Paul (2024). Ornstein–Uhlenbeck processes in Hilbert space and autoregressive moving-average time series. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 97(1), p. 55–80. doi: 10.1080/17442508.2024.2422122. Full text in Research Archive Show summary

  The relationship between parabolic stochastic partial differential equations and autoregressive moving average (ARMA) time series on the real line is established. This is done in light of semigroup theory, under which the parabolic stochastic partial differential equation admits an Ornstein–Uhlenbeck process in Hilbert space. Hilbert-valued AR(1) (or, ARH(1) for short) processes are shown to naturally appear from sampled Ornstein–Uhlenbeck processes. An error representation of approximating AR(1) time series for evaluated ARH(1) processes is derived for the time dimension. Further, by proper projections of ARH(1) processes into Hilbertian subspaces a spatial error representation is derived for evaluation of such projections. The result shows convergence for approximating ARMA times series with increasing spatial dimensions. A numerical example demonstrates our theoretical results for the stochastic heat equation. The results provide a functional data analysis approach to ARH(1) processes.

• Benth, Fred Espen & Sgarra, Carlo (2024). A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets. Finance and Stochastics. ISSN 0949-2984. 28(4), p. 1035–1076. doi: 10.1007/s00780-024-00546-0. Full text in Research Archive Show summary

  We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

• Benth, Jurate Saltyte; Benth, Fred Espen & Nakstad, Espen Rostrup (2024). Nearly Instantaneous Time-Varying Reproduction Number for Contagious Diseases-a Direct Approach Based on Nonlinear Regression. Journal of Computational Biology. ISSN 1066-5277. 31(8), p. 727–741. doi: 10.1089/cmb.2023.0414. Full text in Research Archive
• Benth, Fred Espen; Schroers, Dennis & Veraart, Almut E.D. (2024). A feasible central limit theorem for realised covariation of SPDEs in the context of functional data. The Annals of Applied Probability. ISSN 1050-5164. 34(2), p. 2208–2242. doi: 10.1214/23-AAP2019. Full text in Research Archive
• Benth, Fred Espen; Lord, Gabriel J.; Nunno, Giulia Di & Petersson, Andreas Erik (2024). The heat modulated infinite dimensional Heston model and its numerical approximation. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2024.2424867. Full text in Research Archive Show summary

  The HEat modulated Infinite DImensional Heston (HEIDIH) is introduced as a concrete case of the general framework of infinite dimensional Heston stochastic volatility models of (F.E. Benth, I.C. Simonsen '18) for the pricing of forward contracts. It is supported by a comprehensive numerical analysis. The model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process. This process is a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process, the solution to a stochastic heat equation on the real half-line. The advection and heat equations are driven by independent space-time Gaussian processes which are white in time and coloured in space, with the latter covariance structure expressed by two different kernels. First, a class of weight-stationary kernels are given, under which regularity results for the HEIDIH model in fractional Sobolev spaces are formulated. In particular, the class includes weighted Matérn kernels. Second, numerical approximation of the model is considered. An error decomposition formula, pointwise in space and time, for a finite-difference scheme is proven. For a special case, essentially sharp convergence rates are obtained when this is combined with a fully discrete finite element approximation of the stochastic heat equation. The analysis takes into account a localization error, a pointwise-in-space finite element discretization error and an error stemming from the noise being sampled pointwise in space. The rates obtained in the analysis are higher than what would be obtained using a standard Sobolev embedding technique. Numerical simulations illustrate the results.

• Benth, Fred Espen; Detering, Nils & Krühner, Paul (2024). Abstract polynomial processes. Electronic Journal of Probability (EJP). 29, p. 1–41. doi: 10.1214/24-EJP1182. Full text in Research Archive Show summary

  We propose a novel approach to polynomial processes, which allows us to analyse such processes on general state spaces. We do not need to specify polynomials explicitly but can work with a general sequence of graded vector spaces of functions on the state space. Elements of these graded vector spaces form the polynomials. By introducing a sequence of vector space complements, we obtain the sets of monomials. The basic tool of our analysis is the polynomial action operator, which is a semigroup of operators mapping conditional expected values of polynomials acting on a polynomial process to polynomials of the same or lower grade. We study abstract polynomial processes under algebraic and topological assumptions on the polynomial actions, and establish an affine drift structure. Moreover, we characterize the covariance structure under similar but slightly stronger conditions. A crucial part in our analysis is the use of the (algebraic or topological) dual of the monomials of grade one, which serves as a linearization of the state space of the polynomial process. We provide several examples of polynomial processes that do not fall into the classical setting but are polynomial processes according to our definition and can be analyzed with the tools we provide here.

• Puica, Mihaela-Alexandra & Benth, Fred Espen (2023). A spatio-temporal model for predicting wind speeds in Southern California. Communications in statistics. Case studies, data analysis and applications.. ISSN 2373-7484. 9(3), p. 321–349. doi: 10.1080/23737484.2023.2217137. Full text in Research Archive
• Grochowicz, Aleksander; Greevenbroek, Koen van; Benth, Fred Espen & Zeyringer, Marianne (2023). Intersecting near-optimal spaces: European power systems with more resilience to weather variability. Energy Economics. ISSN 0140-9883. 118. doi: 10.1016/j.eneco.2022.106496. Full text in Research Archive Show summary

  We suggest a new methodology for designing robust energy systems. For this, we investigate so-called near-optimal solutions to energy system optimisation models; solutions whose objective values deviate only marginally from the optimum. Using a refined method for obtaining explicit geometric descriptions of these near-optimal feasible spaces, we find designs that are as robust as possible to perturbations. This contributes to the ongoing debate on how to define and work with robustness in energy systems modelling. We apply our methods in an investigation using multiple decades of weather data. For the first time, we run a capacity expansion model of the European power system (one node per country) with a three-hourly temporal resolution and 41 years of weather data. While an optimisation with 41 weather years is at the limits of computational feasibility, we use the near-optimal feasible spaces of single years to gain an understanding of the design space over the full time period. Specifically, we intersect all near-optimal feasible spaces for the individual years in order to get designs that are likely to be feasible over the entire time period. We find significant potential for investment flexibility, and verify the feasibility of these designs by simulating the resulting dispatch problem with four decades of weather data. They are characterised by a shift towards more onshore wind and solar power, while emitting more than 50% less CO2 than a cost-optimal solution over that period. Our work builds on recent developments in the field, including techniques such as Modelling to Generate Alternatives (MGA) and Modelling All Alternatives (MAA), and provides new insights into the geometry of near-optimal feasible spaces and the importance of multi-decade weather variability for energy systems design. We also provide an effective way of working with a multi-decade time frame in a highly parallelised manner. Our implementation is open-sourced, adaptable and is based on PyPSA-Eur.

• Benth, Fred Espen & Lempa, Jukka (2023). Hedging temperature risk with CDD and HDD temperature futures. Applied Stochastic Models in Business and Industry. ISSN 1524-1904. doi: 10.1002/asmb.2815. Full text in Research Archive Show summary

  Abstract This paper is concerned with managing risk exposure to temperature using weather derivatives. We consider hedging temperature risk using so‐called HDD‐ and CDD‐index futures, which are instruments written on temperatures in specific locations over specific time periods. The temperatures are modelled as continuous‐time autoregressive (CARMA) processes and pricing of the hedging instrument is done under an equivalent pricing measure. We develop hedging strategies for locations, cutoff temperatures, and time periods different to the ones in the traded contracts, allowing for more flexibility in the hedging application. The dynamic hedging strategies are expressed explicitly by the term structure of the volatility. We also provide numerical case studies with temperatures following a CAR(3)‐process to illustrate the temporal behaviour of the hedge under different scenarios.

• Benth, Fred Espen; Detering, Nils & Galimberti, Luca (2023). Pricing options on flow forwards by neural networks in a Hilbert space. Finance and Stochastics. ISSN 0949-2984. 28, p. 81–121. doi: 10.1007/s00780-023-00520-2. Full text in Research Archive
• Benth, Fred Espen & Karbach, Sven (2023). Multivariate continuous-time autoregressive moving-average processes on cones. Stochastic Processes and their Applications. ISSN 0304-4149. 162, p. 299–337. doi: 10.1016/j.spa.2023.05.003. Full text in Research Archive
• Bordin, Chiara; Mishra, Sambeet & Benth, Fred Espen (2023). Pedagogical Perspectives of Interdisciplinary Teaching and Research: An Energy System Modelling Outlook in Relation to Energy Informatics. Energies. 16(15). doi: 10.3390/en16155757. Full text in Research Archive Show summary

  The purpose of this paper is to present and discuss pedagogical frameworks and approaches to developing, delivering, and evaluating a new interdisciplinary course within the domain of energy informatics at both Master’s and PhD levels. This study is needed because many papers on sustainable energy engineering education concentrate on course content but provide very little information on the pedagogical methods employed to deliver that content. The proposed new course is called “smart energy and power systems modelling” and is aimed at discussing how mathematical optimization, in the context of computer science, can contribute to more effectively managing smart energy and power systems. Different pedagogical frameworks are discussed and adapted for the specific domain of energy informatics. An ASSURE model coupled with Bloom’s taxonomy is presented for the design of the course and identification of learning objectives; self-regulated learning strategies are discussed to enhance the learning process; a novel model called GPD (Gaussian Progression of Difficulty) for lecture planning was proposed; a teaching-research nexus is discussed for the course planning and enhancement. Adopting qualitative analyses and an inductive approach, this paper offers a thorough reflection on the strengths and weaknesses of the new course, together with improvement possibilities based on fieldwork and direct experience with the students and colleagues. Opportunities and challenges of interdisciplinary teaching are presented in light of real-world experience, with a particular focus on the interaction between mathematics and computer science to study the specific application of energy and power systems.

• Larsson, Karl; Green, Rikard & Benth, Fred Espen (2023). A stochastic time-series model for solar irradiation. Energy Economics. ISSN 0140-9883. 117. doi: 10.1016/j.eneco.2022.106421. Full text in Research Archive Show summary

  We propose a novel stochastic time series model able to explain the stylized features of daily irradiation level data in 5 cities in Germany. The model is suitable for applications to risk management of photovoltaic power production in renewable energy markets. The suggested dynamics is a low-order autoregressive time series with seasonal level given by an atmospheric clear-sky model. Moreover, we detect a skewness property in the residuals which we explain by a winter–summer regime switch. The stochastic variance is modeled by a seasonally varying GARCH-dynamics. The winter and summer standardized residuals are proposed to be a Gaussian mixture model to capture the bimodal distributions. We estimate the model on the observed data, and perform a validation study. An application to energy markets studying the production at risk for a PV-producer is presented.

• Benth, Fred Espen; Deelstra, Griselda & Kozplnar, And Sinem (2023). Pricing energy quanto options in the framework of Markov-modulated additive processes. IMA Journal of Management Mathematics. ISSN 1471-678X. 34(1), p. 187–220. doi: 10.1093/imaman/dpab032. Full text in Research Archive
• Benth, Fred Espen; Detering, Nils & Galimberti, Luca (2022). Neural networks in Fréchet spaces. Annals of Mathematics and Artificial Intelligence. ISSN 1012-2443. 91, p. 75–103. doi: 10.1007/s10472-022-09824-z. Full text in Research Archive Show summary

  We propose a neural network architecture in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fréchet space X into a Banach space Y. The approximation results are generalising the well known universal approximation theorem for continuous functions from Rn to R, where approximation is done with (multilayer) neural networks Cybenko (1989) Math. Cont. Signals Syst. 2, 303–314 and Hornik et al. (1989) Neural Netw., 2, 359–366 and Funahashi (1989) Neural Netw., 2, 183–192 and Leshno (1993) Neural Netw., 6, 861–867. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Several examples are given of such activation functions. We show furthermore that our neural networks on infinite dimensional spaces can be projected down to finite dimensional subspaces with any desirable accuracy, thus obtaining approximating networks that are easy to implement and allow for fast computation and fitting. The resulting neural network architecture is therefore applicable for prediction tasks based on functional data.

• Schrader, Simon Elias & Benth, Fred Espen (2022). A stochastic study of carbon emission reduction from electrification and interconnecting cable utilization. The Norway and Germany case. Energy Economics. ISSN 0140-9883. 114. doi: 10.1016/j.eneco.2022.106300. Full text in Research Archive Show summary

  We analyse how carbon emissions are reduced utilizing the new NordLink cable between Norway and Germany, taking policy plans of platform electrification in Norway into account. Building a stochastic model based on existing data for Norwegian and German electricity production and demand (Open Power system data, 2020) to account for uncertainty, we found that the cable can be used as an effective way to reduce German emissions by exporting German renewable surplus production on windy days to Norway and importing from Norway other days. We found that such flow of power will lead to a possible reduction of the countries’ total emissions by over 4.1 million tons CO annually. Based on a mean reduction of German CO emissions of about 710 g/kWh by electricity imported from Norway, we found that electrification of the Norwegian oil and gas extraction is inferior to an increased export to Germany in terms of total emission reduction, as electrification only gives a reduction of about 425 g/kWh according to numbers from the Norwegian Oil Directorate (Oljedirektoratet, 2020). A similar calculation assuming a realistic full electrification of the platforms only gave a reduction of 2.4 million tons CO annually. In the long term, we conclude that combining both approaches makes the best use of Norwegian surplus energy production in view of carbon emission reduction.

• Benth, Fred Espen & Galimberti, Luca (2022). Stochastic integrals and Gelfand integration in Fréchet spaces. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 25(2), p. 1–35. doi: 10.1142/S0219025722500072. Full text in Research Archive Show summary

  We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida–Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space.

• Benth, Fred Espen; Schroers, Dennis & Veraart, Almut E. D. (2022). A weak law of large numbers for realised covariation in a Hilbert space setting. Stochastic Processes and their Applications. ISSN 0304-4149. 145, p. 241–268. doi: 10.1016/j.spa.2021.12.011. Full text in Research Archive Show summary

  This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert–Schmidt norm. In addition, we determine convergence rates for common stochastic volatility models in Hilbert spaces.

• Benth, Fred Espen; Nunno, Giulia Di & Schroers, Dennis (2022). A topological proof of Sklar's theorem in arbitrary dimensions. Dependence Modeling. 10(1), p. 22–28. doi: 10.1515/demo-2022-0103. Full text in Research Archive Show summary

  Abstract Copulas are appealing tools in multivariate probability theory and statistics. Nevertheless, the transfer of this concept to infinite dimensions entails some nontrivial topological and functional analytic issues, making a deeper theoretical understanding indispensable toward applications. In this short work, we transfer the well-known property of compactness of the set of copulas in finite dimensions to the infinite-dimensional framework. As an application, we prove Sklar’s theorem in infinite dimensions via a topological argument and the notion of inverse systems.

• Benth, Fred Espen; Detering, Nils & Krühner, Paul (2022). Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. p. 1–23. doi: 10.1080/17442508.2021.2019738. Full text in Research Archive Show summary

  We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

• Harang, Fabian Andsem & Benth, Fred Espen (2021). Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise - Probabilistic Properties and Applications. Electronic Journal of Probability (EJP). 26. doi: 10.1214/21-EJP683. Full text in Research Archive Show summary

  We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from [18] to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al.

• Benth, Fred Espen; Nunno, Giulia Di & Schroers, Dennis (2021). Copula measures and Sklar's theorem in arbitrary dimensions. Scandinavian Journal of Statistics. ISSN 0303-6898. doi: 10.1111/sjos.12559. Full text in Research Archive
• Benth, Fred Espen; Nunno, Giulia Di & Simonsen, Iben Cathrine (2021). Sensitivity analysis in the infinite dimensional Heston model. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 24(2). doi: 10.1142/S0219025721500144. Full text in Research Archive
• Benth, Fred Espen & Lavagnini, Silvia (2021). Correlators of Polynomial Processes. SIAM Journal on Financial Mathematics. 12(4), p. 1374–1415. doi: 10.1137/21M141556X. Full text in Research Archive Show summary

  In the setting of one-dimensional polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for polynomial processes. The developed framework can, for example, be applied in financial pricing, such as for path-dependent options and in a stochastic volatility models context. In applications to options, having closed and compact formulations is attractive for sensitivity analysis and risk management, since Greeks can be derived explicitly.

• Benth, Fred Espen & Simonsen, Iben Cathrine (2021). Metatimes, random measures and cylindrical random variables. Modern Stochastics: Theory and Applications (MSTA). ISSN 2351-6046. 8(3), p. 349–371. doi: 10.15559/21-VMSTA178. Full text in Research Archive Show summary

  Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

• Benth, Fred Espen; Detering, Nils & Lavagnini, Silvia (2021). Accuracy of deep learning in calibrating HJM forward curves. Digital Finance. ISSN 2524-6984. 3, p. 209–248. doi: 10.1007/s42521-021-00030-w. Full text in Research Archive Show summary

  We price European-style options written on forward contracts in a commodity market, which we model with an infinite-dimensional Heath–Jarrow–Morton (HJM) approach. For this purpose, we introduce a new class of state-dependent volatility operators that map the square integrable noise into the Filipović space of forward curves. For calibration, we specify a fully parametrized version of our model and train a neural network to approximate the true option price as a function of the model parameters. This neural network can then be used to calibrate the HJM parameters based on observed option prices. We conduct a numerical case study based on artificially generated option prices in a deterministic volatility setting. In this setting, we derive closed pricing formulas, allowing us to benchmark the neural network based calibration approach. We also study calibration in illiquid markets with a large bid-ask spread. The experiments reveal a high degree of accuracy in recovering the prices after calibration, even if the original meaning of the model parameters is partly lost in the approximation step.

• Benth, Fred Espen; Kutrolli, Gleda & Stefani, Silvana (2021). Dynamic probabilistic forecasting with uncertainty. International Journal of Theoretical and Applied Finance. ISSN 0219-0249. 24(6 & 7). doi: 10.1142/S0219024921500345. Full text in Research Archive Show summary

  In this paper, we introduce a dynamical model for the time evolution of probability density functions incorporating uncertainty in the parameters. The uncertainty follows stochastic processes, thereby defining a new class of stochastic processes with values in the space of probability densities. The purpose is to quantify uncertainty that can be used for probabilistic forecasting. Starting from a set of traded prices of equity indices, we do some empirical studies. We apply our dynamic probabilistic forecasting to option pricing, where our proposed notion of model uncertainty reduces to uncertainty on future volatility. A distribution of option prices follows, reflecting the uncertainty on the distribution of the underlying prices. We associate measures of model uncertainty of prices in the sense of Cont.

• Benth, Fred Espen (2021). Pricing of Commodity and Energy Derivatives for Polynomial Processes. Mathematics. 9(2). doi: 10.3390/math9020124. Full text in Research Archive Show summary

  Operating in energy and commodity markets require a management of risk using derivative products such as forward and futures, as well as options on these. Many of the popular stochastic models for spot dynamics and weather variables developed from empirical studies in commodity and energy markets belong to the class of polynomial jump diffusion processes. We derive a tailor-made framework for efficient polynomial approximation of the main derivatives encountered in commodity and energy markets, encompassing a wide range of arithmetic and geometric models. Our analysis accounts for seasonality effects, delivery periods of forwards and exotic temperature forwards where the underlying “spot” is a nonlinear function of the temperature. We also include in our derivations risk management products such as spread, Asian and quanto options.

• Benth, Fred Espen; Christensen, Troels Sønderby & Rohde, Victor (2021). Multivariate continuous-time modeling of wind indexes and hedging of wind risk. Quantitative finance (Print). ISSN 1469-7688. 21(1), p. 165–183. doi: 10.1080/14697688.2020.1804606. Full text in Research Archive Show summary

  With the introduction of the exchange-traded German wind power futures, opportunities for German wind power producers to hedge their volumetric risk are present. We propose two continuous-time multivariate models for wind power utilization at different wind sites, and discuss the properties and estimation procedures for the models. Applying the models to wind index data for wind sites in Germany and the underlying wind index of exchange-traded wind power futures contracts, the estimation results of both models suggest that they capture key statistical features of the data. We show how these models can be used to find optimal hedging strategies using exchange-traded wind power futures for the owner of a portfolio of so-called tailor-made wind power futures. Both in-sample and out-of-sample hedging scenarios are considered, and, in both cases, significant variance reductions are achieved. Additionally, the risk premium of the German wind power futures is analysed, leading to an indication of the risk premium of tailor-made wind power futures.

• Christensen, Troels Sønderby & Benth, Fred Espen (2020). Modelling the joint behaviour of electricity prices in interconnected markets. Quantitative finance (Print). ISSN 1469-7688. 20(9), p. 1441–1456. doi: 10.1080/14697688.2020.1733059. Full text in Research Archive Show summary

  The liberalization of energy markets worldwide during recent decades has introduced severe implications for the price formation in these markets. Especially within the European day-ahead electricity markets, increased physical connections between different market areas and a joint effort on optimizing the aggregate social welfare have led to highly connected markets. Consequently, observing the exact same hourly day-ahead prices for two or more interconnected electricity markets in Europe happens frequently. This affects the modelling of such prices and in turn the valuation of derivatives written on prices from these market areas. In this paper, we propose a joint model for day-ahead electricity prices in interconnected markets composed of a combination of transformed Ornstein–Uhlenbeck processes. We discuss the properties of the model and propose an estimation procedure based on filtering techniques. Furthermore, the properties of the model reveal that analytical prices are attainable for, e.g., forwards and spread options.

• Benth, Fred Espen; Eikeset, Anne Maria; Levin, Simon A. & Ren, Wanjuan (2020). Analysis of the risk premium in the forward market for salmon. Journal of Commodity Markets. ISSN 2405-8513. p. 1–13. doi: 10.1016/j.jcomm.2019.100122. Full text in Research Archive Show summary

  We analyse forward prices observed at the Fishpool market, and propose a two-factor continuous-time stochastic process for modelling the time dynamics. The data analysis reveals that the two factors can be assumed to be a non-stationary compound Poisson process and a stationary continuous-time autoregressive dynamics, describing the bumps observed in the forward curves. We use the model to analyse the risk premium in the forward markets, and find a negative premium in the long end of the market which is in line with the theory of normal backwardation. However, contracts with short time to maturity have a risk premium with randomly changing sign, pointing towards a hedging pressure also induced by the demand-side of the market.

• Benth, Fred Espen; Detering, Nils & Krühner, Paul (2020). Independent increment processes: a multilinearity preserving property. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2020.1802458. Full text in Research Archive Show summary

  We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space B. It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of B being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.

• Kremer, Marcel; Benth, Fred Espen; Felten, Björn & Kiesel, Rüdiger (2020). Volatility and liquidity on high-frequency electricity futures markets: Empirical analysis and stochastic modeling. International Journal of Theoretical and Applied Finance. ISSN 0219-0249. 23(4). doi: 10.1142/S0219024920500272. Full text in Research Archive
• Benth, Fred Espen; Khedher, Asma & Vanmaele, Michèle (2020). Pricing of commodity derivatives on processes with memory. Risks. 8(1). doi: 10.3390/risks8010008. Full text in Research Archive Show summary

  Spot option prices, forwards and options on forwards relevant for the commodity markets are computed when the underlying process S is modelled as an exponential of a process ξ with memory as, e.g., a Volterra equation driven by a Lévy process. Moreover, the interest rate and a risk premium ρ representing storage costs, illiquidity, convenience yield or insurance costs, are assumed to be stochastic. When the interest rate is deterministic and the risk premium is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity, the process (ξ;ρ) has an affine structure under the pricing measure Q and an explicit expression for the option price is derived in terms of the Fourier transform of the payoff function.

• Krecar, Nikola; Benth, Fred Espen & Gubina, Andrej (2020). Towards definition of the risk premium function. IEEE Transactions on Power Systems. ISSN 0885-8950. 35(2), p. 1085–1098. doi: 10.1109/TPWRS.2019.2938423. Full text in Research Archive Show summary

  Successful trading in electricity markets relies on the market actor's ability to accurately forecast the electricity price. The fundamental electricity price models use market information, provided by various price drivers, including the residual that contains a risk premium. In the past, researchers investigating risk premium focused primarily on daily spot price levels, ignoring the intraday information hindering the accurate risk premium determination. This paper presents a new KGB Method for modelling of risk premium, based on “ex-ante” approach focused on a yearly product. The method involves a novel KGB Model and its linearized formulation, the KGB Linear Model, which enables capturing the influence of renewable energy sources on risk premium. The four key drivers of the KGB Linear Model were used providing an insight into the influence of RES generation on risk premium evolution. The method was tested on historical data from the German electricity market. The results for the 2010-2014 period reveal overall influence of PV production share on risk premium is greater than that of wind production share, both increasing the risk premium due to their variability and uncertainty. Using the KGB Method, market actors can forecast risk premium using information readily available to them.

• Benth, Fred Espen & Rohde, Victor (2019). On non-negative modeling with CARMA processes. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 476(1), p. 196–214. doi: 10.1016/j.jmaa.2018.12.055. Full text in Research Archive Show summary

  Two stationary and non-negative processes that are based on continuous-time autoregressive moving average (CARMA) processes are discussed. First, we consider a generalization of Cox–Ingersoll–Ross (CIR) processes. Next, we consider CARMA processes driven by compound Poisson processes with exponential jumps which are generalizations of Ornstein–Uhlenbeck (OU) processes driven by the same noise. The way in which the two processes generalize CIR and OU processes and the relation between them will be discussed. Furthermore, the stationary distribution, the autocorrelation function, and pricing of zero-coupon bonds are considered.

• Benth, Fred Espen; Piccirilli, Marco & Vargiolu, Tiziano (2019). Mean-reverting additive energy forward curves in a Heath–Jarrow–Morton framework. Mathematics and Financial Economics. ISSN 1862-9679. 13(4), p. 543–577. doi: 10.1007/s11579-019-00237-x. Full text in Research Archive
• Benth, Fred Espen & Pircalabu, Anca (2018). A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures. Applied Mathematical Finance. ISSN 1350-486X. 25(1), p. 36–65. doi: 10.1080/1350486X.2018.1438904. Full text in Research Archive Show summary

  The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.

• Benth, Fred Espen & Suss, Andre (2018). Continuous-Time Autoregressive Moving-Average Processes in Hilbert Space. In Celledoni, Elena; Nunno, Giulia Di; Ebrahimi-Fard, Kurusch & Munthe-Kaas, Hans (Ed.), Computation and Combinatorics in Dynamics, Stochastics and Control. Springer. ISSN 9783030015923. p. 297–320. doi: 10.1007/978-3-030-01593-0_11. Full text in Research Archive Show summary

  The Abel Symposia volume at hand contains a collection of high-quality articles written by the world’s leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory. In recent years we have witnessed a remarkable convergence between individual mathematical disciplines that approach deterministic and stochastic dynamical systems from mathematical analysis, computational mathematics and control theoretical perspectives. Breakthrough developments in these fields now provide a common mathematical framework for attacking many different problems related to differential geometry, analysis and algorithms for stochastic and deterministic dynamics. In the Abel Symposium 2016, which took place from August 16-19 in Rosendal near Bergen, leading researchers in the fields of deterministic and stochastic differential equations, control theory, numerical analysis, algebra and random processes presented and discussed the current state of the art in these diverse fields. The current Abel Symposia volume may serve as a point of departure for exploring these related but diverse fields of research, as well as an indicator of important current and future developments in modern mathematics.

• Benth, Fred Espen & Krühner, Paul (2018). Approximation of forward curve models in commodity markets with arbitrage-free finite-dimensional models. Finance and Stochastics. ISSN 0949-2984. 22(2), p. 327–366. doi: 10.1007/s00780-018-0355-9. Full text in Research Archive Show summary

  In this paper, we show how to approximate Heath–Jarrow–Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have a finite-dimensional state space. Moreover, we recover a closed-form representation of the forward price dynamics in the approximation models and derive the rate of convergence to the true dynamics uniformly over an interval of time to maturity under certain additional smoothness conditions. In the Markovian case, we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.

• Benth, Fred Espen; Ruediger, Barbara & Suess, Andre (2018). Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility. Stochastic Processes and their Applications. ISSN 0304-4149. 128(2), p. 461–486. doi: 10.1016/j.spa.2017.05.005. Full text in Research Archive Show summary

  We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.

• Benth, Fred Espen & Suss, Andre (2018). Cointegration in continuous time for factor models. Mathematics and Financial Economics. ISSN 1862-9679. 13(1), p. 87–114. doi: 10.1007/s11579-018-0221-8. Full text in Research Archive Show summary

  We develop cointegration for multivariate continuous-time stochastic processes, both in finite and infinite dimension. Our definition and analysis are based on factor processes and operators mapping to the space of prices and cointegration. The focus is on commodity markets, where both spot and forward prices are analysed in the context of cointegration. We provide many examples which include the most used continuous-time pricing models, including forward curve models in the Heath-Jarrow-Morton paradigm in Hilbert space.

• Benth, Fred Espen & Simonsen, Iben Cathrine (2018). The Heston stochastic volatility model in Hilbert space. Stochastic Analysis and Applications. ISSN 0736-2994. 36(4), p. 733–750. doi: 10.1080/07362994.2018.1461566. Full text in Research Archive Show summary

  We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.

• Ådland, Roar Os; Benth, Fred Espen & Koekebakker, Steen (2018). Multivariate modeling and analysis of regional ocean freight rates. Transportation Research Part E: Logistics and Transportation Review. ISSN 1366-5545. 113, p. 194–221. doi: 10.1016/j.tre.2017.10.014. Full text in Research Archive Show summary

  In this paper, we propose a new multivariate model for the dynamics of regional ocean freight rates. We show that a cointegrated system of regional spot freight rates can be decomposed into a common non-stationary market factor and stationary regional deviations. The resulting integrated CAR process is new to the literature. By interpreting the common market factor as the global arithmetic average of the regional rates, both the market factor and the regional deviations are observable which simplifies the calibration of the model. Moreover, forward contracts on the market factor can be traded in the Forward Freight Agreement (FFA) market. We calibrate the model to historical spot rate processes and illustrate the term structures of volatility and correlation between the regional prices and the market factor. Our model is an important contribution towards improved modelling and hedging of regional price risk when derivative market liquidity is concentrated in a single global benchmark.

• Benth, Fred Espen; Persio, Luca Di & Lavagnini, Silvia (2018). Stochastic modelling of wind derivatives in energy markets. Risks. 6(2). doi: 10.3390/risks6020056. Full text in Research Archive Show summary

  We model the logarithm of the spot price of electricity with a normal inverse Gaussian (NIG) process and the wind speed and wind power production with two Ornstein–Uhlenbeck processes. In order to reproduce the correlation between the spot price and the wind power production, namely between a pure jump process and a continuous path process, respectively, we replace the small jumps of the NIG process by a Brownian term. We then apply our models to two different problems: first, to study from the stochastic point of view the income from a wind power plant, as the expected value of the product between the electricity spot price and the amount of energy produced; then, to construct and price a European put-type quanto option in the wind energy markets that allows the buyer to hedge against low prices and low wind power production in the plant. Calibration of the proposed models and related price formulas is also provided, according to specific datasets.

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• Benth, Fred Espen & Krühner, Paul (2023). Stochastic Models for Prices Dynamics in Energy and Commodity Markets An Infinite-Dimensional Perspective. Springer. ISBN 9783031403668. 250 p. Full text in Research Archive
• Barndorff-Nielsen, Ole E.; Benth, Fred Espen & Veraart, Almut E. D. (2018). Ambit Stochastics. Springer Nature. ISBN 9783319941288. 402 p. Full text in Research Archive

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• Benth, Fred Espen (2026). Managing weather risk across time and space. Full text in Research Archive
• Benth, Fred Espen (2026). Finance-informed learning and pricing of energy derivatives. Full text in Research Archive
• Benth, Fred Espen (2026). Stochastic volatility and infinite-dimensional Ornstein-Uhlenbeck processes. Full text in Research Archive
• Benth, Fred Espen (2026). Recent advances on term-structure modeling and pricing in commodity markets. Full text in Research Archive
• Benth, Fred Espen (2026). Finance-informed learning and pricing of energy derivatives. Full text in Research Archive
• Benth, Jurate Saltyte; Benth, Fred Espen & Nakstad, Espen Rostrup (2025). Rebuttal to Flaws in the Paper ‘Nearly Instantaneous Time-Varying Reproduction Number for Contagious Diseases—a Direct Approach Based on Nonlinear Regression’. Journal of Computational Biology. ISSN 1066-5277. 32(8), p. 823–823. doi: 10.1089/cmb.2025.0024. Full text in Research Archive
• Benth, Fred Espen (2025). Structure-informed learning and interpretability -- the case of energy markets. Full text in Research Archive
• Benth, Fred Espen (2025). Hva kan en matematiker bidra med i risikohåndtering? Full text in Research Archive
• Benth, Fred Espen (2025). Weather derivatives. Full text in Research Archive
• Benth, Fred Espen (2025). Hedging weather risk in energy markets. Full text in Research Archive
• Benth, Fred Espen (2025). Using informed neural networks to price power options. Full text in Research Archive
• Benth, Fred Espen (2025). Using informed neural networks to price options on power forward -- and some more on forwards. Full text in Research Archive
• Skogvoll, Vidar; Lindstrøm, Tom Louis; Benth, Fred Espen & Haraldsrud, Andreas (2024). Real læring - Realistfabrikken: Hvordan lage matematiker. [Radio]. Podcast. Full text in Research Archive
• Grochowicz, Aleksander; Greevenbroek, Koen van; Benth, Fred Espen & Zeyringer, Marianne (2023). Intersecting Near-Optimal Spaces for Policy Information. Full text in Research Archive
• Eggen, Mari Dahl; Midtfjord, Alise Danielle; Vorobeva, Ekaterina; Benth, Fred Espen; Hupe, Patrick & Brissaud, Quentin [Show all 10 contributors for this article] (2023). Using a machine learning and stochastics-founded model to provide near real-time stratospheric polar vortex diagnostics based on high-latitude infrasound data. Full text in Research Archive Show summary

  Acoustic waves below the frequency limit of human hearing - infrasound - can travel for thousands of kilometres in the atmosphere. The global propagation signature of infrasound is highly sensitive to the wind structure of the stratosphere. This work exploits processed continuous data from three high-latitude infrasound stations to characterize an aspect of the stratospheric polar vortex. Concretely, a mapping is developed which takes the infrasound data from these three stations as input and outputs an estimate of the polar cap zonal mean wind averaged over 60-90 degrees in latitude at the 1 hPa pressure level. This stratospheric diagnostic information is relevant to, for example, sudden stratospheric warming assessment and sub-seasonal prediction. The considered acoustic data is within a low-frequency regime globally dominated by so-called microbarom infrasound, which is continuously radiated into the atmosphere due to nonlinear interaction between counter-propagating ocean surface waves. We trained a stochastics-based machine learning model (delay-SDE-net) to map between a time series of five years (2014-2018) of processed infrasound data and the ERA5 (reanalysis-based) daily average polar cap wind at 1 hPa for the same period. The ERA5 data was hence treated as ground-truth. In the prediction, the delay-SDE-net utilizes time-lagged inputs and their dependencies, as well as the day of the year to account for seasonal differences. In the validation phase, the input was the 2019 and 2020 infrasound time series, and the model inference results in an estimate of the daily average polar cap wind time-series. This result was then compared to the ERA5 representation of the stratospheric diagnostic time-series for the same period. The applied machine learning model is based on stochastics and allows for an interpretable approach to estimate the aleatoric and epistemic prediction uncertainties. It is found that the mapping, which is only informed of the trained model, the day of year, and the infrasound data from three stations, generates a 1 hPa polar cap average wind estimate with a prediction error standard deviation of around 10 m/s compared to ERA5. Focus should be put on the winter months because this is when the coupling between the stratosphere and the troposphere can mostly influence the surface conditions and provide additional prediction skill, in particular during strong and weak stratospheric polar vortex regimes. The infrasound data is available in real-time, and we discuss how the developed approach can be extended to provide near real-time stratospheric polar vortex diagnostics.

• Benth, Fred Espen (2022). Pricing options on flow forwards by neural networks in Hilbert space. Full text in Research Archive
• Benth, Fred Espen (2022). Modellering av risiko i energisystemer. Full text in Research Archive
• Grochowicz, Aleksander; Greevenbroek, Koen van; Benth, Fred Espen & Zeyringer, Marianne (2022). Intersecting near-optimal spaces for robust energy systems. Full text in Research Archive
• Zeyringer, Marianne; Benth, Fred Espen; Roithner, Maximilian; Grochowicz, Aleksander & Sirotko-Sibirskaya, Natalia (2022). Climate-resilient net-zero energy system design. Full text in Research Archive
• Benth, Fred Espen (2022). Klima og vær - data og risiko. Full text in Research Archive
• Benth, Fred Espen (2022). Pricing Options on Flow Forwards by Neural Networks in Hilbert Space. Full text in Research Archive
• Nunno, Giulia Di; Benth, Fred Espen & Simonsen, Iben Cathrine (2021). Infinite dimensional Heston model and sensitivity analysis. Full text in Research Archive
• Benth, Fred Espen & Zeyringer, Marianne (2021). Uncertain energy systems. Full text in Research Archive
• Benth, Fred Espen (2021). Pathwise Gaussian Volterra processes in Hilbert space. Full text in Research Archive
• Benth, Fred Espen & Zeyringer, Marianne (2021). Uncertain Energy Systems. Full text in Research Archive
• Benth, Fred Espen (2021). Hedging volumetric risk in renewable energy markets. Full text in Research Archive
• Benth, Fred Espen & Schrader, Simon Elias (2021). Send krafta til Tyskland. Klassekampen. ISSN 0805-3839. Full text in Research Archive
• Benth, Fred Espen & Zeyringer, Marianne (2021). Er de rekordhøye strømprisene verdt det? Full text in Research Archive
• Benth, Fred Espen (2021). Pathwise Gaussian Volterra processes in Hilbert space. Full text in Research Archive
• Benth, Fred Espen (2019). Stochastic volatility in energy markets. Full text in Research Archive
• Benth, Fred Espen (2019). Stochastic volatility in energy and commodity markets. Full text in Research Archive
• Benth, Fred Espen (2019). Stochastic volatility in commodity markets. Full text in Research Archive
• Benth, Fred Espen (2018). Polynomial processes in Banach space. Full text in Research Archive
• Benth, Fred Espen (2018). Cointegration in continuous time. Full text in Research Archive
• Benth, Fred Espen (2018). Stochastic integration for BSS processes. Full text in Research Archive
• Benth, Fred Espen (2018). Ambit fields and stochastic integration. Full text in Research Archive
• Benth, Fred Espen (2018). Cointegration in continuous time in commodity markets. Full text in Research Archive
• Benth, Fred Espen (2018). Stochastic volatility modeling in power markets. Full text in Research Archive

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