• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

Astrophysical and high-energy-density laboratory plasmas often have large-amplitude, sub-Larmor-scale electromagnetic fluctuations excited by various kinetic-streaming or anisotropy-driven instabilities. The Weibel (or the filamentation) instability is particularly important because it can rapidly generate strong magnetic fields, even in the absence of seed fields. Particles propagating in collisionless plasmas with such small-scale magnetic fields undergo stochastic deflections similar to Coulomb collisions, with the magnetic pitch-angle diffusion coefficient representing the effective "collision" frequency. We show that this effect of the plasma "quasi-collisionality" can strongly affect the growth rate and evolution of the Weibel instability in the deeply nonlinear regime. This result is especially important for understanding cosmic-ray-driven turbulence in an upstream region of a collisionless shock of a gamma-ray burst or a supernova. We demonstrate that the quasi-collisions caused by the fields generated in the upstream suppress the instability slightly but can never shut it down completely. This confirms the assumptions made in the self-similar model of the collisionless foreshock.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Mathematical models in Biology and Medicine (2017/2018)
• Event: Workshop on "Mathematical Methods in Biology and Medicine" (2017)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

As a fundamental process converting magnetic to plasma energy in high-energy astrophysical plasmas, relativistic magnetic reconnection is a leading explanation for the acceleration of particles to the ultrarelativistic energies necessary to power nonthermal emission (especially X-rays and gamma-rays) in pulsar magnetospheres and pulsar wind nebulae, coronae and jets of accreting black holes, and gamma-ray bursts. An important objective of plasma astrophysics is therefore the characterization of nonthermal particle acceleration (NTPA) effected by reconnection. Reconnection-powered NTPA has been demonstrated over a wide range of physical conditions using large two-dimensional (2D) kinetic simulations. However, its robustness in realistic 3D reconnection -- in particular, whether the 3D relativistic drift-kink instability (RDKI) disrupts NTPA -- has not been systematically investigated, although pioneering 3D simulations have observed NTPA in isolated cases. Here we present the first comprehensive study of NTPA in 3D relativistic reconnection in collisionless electron-positron plasmas, characterizing NTPA as the strength of 3D effects is varied systematically via the length in the third dimension and the strength of the guide magnetic field. We find that, while the RDKI prominently perturbs 3D reconnecting current sheets, it does not suppress particle acceleration, even for zero guide field; fully 3D reconnection robustly and efficiently produces nonthermal power-law particle spectra closely resembling those obtained in 2D. This finding provides strong support for reconnection as the key mechanism powering high-energy flares in various astrophysical systems. We also show that strong guide fields significantly inhibit NTPA, slowing reconnection and limiting the energy available for plasma energization, yielding steeper and shorter power-law spectra.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

We present results from particle-in-cell simulations of driven turbulence in magnetized, collisionless, and relativistic pair plasmas. We find that the fluctuations are consistent with the classical k −5/3 ¡Ñ magnetic energy spectrum at fluid scales and a steeper k −4 ¡Ñ spectrum at sub-Larmor scales, where k¡Ñ is the wave vector perpendicular to the mean field. We demonstrate the development of a nonthermal, power-law particle energy distribution f(E)¡­E−¥á, with an index ¥á that decreases with increasing magnetization and increases with an increasing system size (relative to the characteristic Larmor radius). Our simulations indicate that turbulence can be a viable source of energetic particles in high-energy astrophysical systems, such as pulsar wind nebulae, if scalings asymptotically become insensitive to the system size.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

It is shown that grains streaming through a fluid are generically unstable if their velocity, projected along some direction, matches the phase velocity of a fluid wave. This can occur whenever grains stream faster than a fluid wave. The wave itself can be quite general--sound waves, magnetosonic waves, epicyclic oscillations, and Brunt-V\"ais\"al\"a oscillations each generate instabilities, for example. A simple expression for this "resonant drag instability" (RDI) growth rate is derived. This expression (i) illustrates why such instabilities are so virulent and generic, and (ii) allows for simple analytic computation of RDI growth rates and properties for different fluid systems. As examples, we introduce several new instabilities, which could see application across a variety of astrophysical systems from protoplanetary disks to galactic outflows.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

We report the results of a direct comparison between different kinetic models of collisionless plasma turbulence in two spatial dimensions. The models considered include a first principles fully-kinetic (FK) description, two widely used reduced models [gyrokinetic (GK) and hybrid-kinetic (HK) with fluid electrons], and a novel reduced gyrokinetic approach (KREHM). Two different ion beta (â i ) regimes are considered: 0.1 and 0.5. For â i =0.5 , good agreement between the GK and FK models is found at scales ranging from the ion to the electron gyroradius, thus providing firm evidence for a kinetic Alfv'en cascade scenario. In the same range, the HK model produces shallower spectral slopes, presumably due to the lack of electron Landau damping. For â i =0.1 , a detailed analysis of spectral ratios reveals a slight disagreement between the GK and FK descriptions at kinetic scales, even though kinetic Alfv'en fluctuations likely still play a significant role. The discrepancy can be traced back to scales above the ion gyroradius, where the FK and HK results seem to suggest the presence of fast magnetosonic and ion Bernstein modes in both plasma beta regimes, but with a more notable deviation from GK in the low-beta case. The identified practical limits and strengths of reduced-kinetic approximations, compared here against the fully-kinetic model on a case-by-case basis, may provide valuable insight into the main kinetic effects at play in turbulent collisionless plasmas, such as the solar wind.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

The current understanding of magnetohydrodynamic (MHD) turbulence envisions turbulent eddies which are anisotropic in all three directions. In the plane perpendicular to the local mean magnetic field, this implies that such eddies become current-sheetlike structures at small scales. We analyze the role of magnetic reconnection in these structures and conclude that reconnection becomes important at a scale ¥ë¡­LS −4/7L, where SL is the outer-scale (L) Lundquist number and ¥ë is the smallest of the field-perpendicular eddy dimensions. This scale is larger than the scale set by the resistive diffusion of eddies, therefore implying a fundamentally different route to energy dissipation than that predicted by the Kolmogorov-like phenomenology. In particular, our analysis predicts the existence of the subinertial, reconnection interval of MHD turbulence, with the estimated scaling of the Fourier energy spectrum E(k¡Ñ)¡ðk−5/2¡Ñ, where k¡Ñ is the wave number perpendicular to the local mean magnetic field. The same calculation is also performed for high (perpendicular) magnetic Prandtl number plasmas (Pm), where the reconnection scale is found to be ¥ë/L¡­S−4/7LPm−2/7.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

A combined analytic and computational gyrokinetic approach is developed to address the question of the scaling of pedestal turbulent transport with arbitrary levels of E×B shear. Due to strong gradients and shaping in the pedestal, the instabilities of interest are not curvature-driven like the core instabilities. By extensive numerical (gyrokinetic) simulations, it is demonstrated that pedestal modes respond to shear suppression very much like the predictions of a basic analytic decorrelation theory. The quantitative agreement between the two provides us with a new dependable, first principles (physics based) theoretical framework to predict the efficacy of shear suppression in burning plasmas that lie in a low-shear regime not accessed by present experiments.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

The two-dimensional Terry-Horton equation is shown to exhibit the Dimits shift when suitably modified to capture both the nonlinear enhancement of zonal/drift-wave interactions and the existence of residual Rosenbluth-Hinton states. This phenomena persists through numerous simplifications of the equation, including a quasilinear approximation as well as a four-mode truncation. Analytic progress on the truncated system is reported, focused on determining the growth rates of zonal flows and calculating the upper bound of the Dimits shift. The results for the truncated system are then used to estimate the Dimits shift of the fully nonlinear system. A new understanding is thus developed on the fundamental nature of the Dimits shift, both on its operation and its eventual termination.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

Breaking the up-down symmetry of the tokamak poloidal cross-section can significantly increase the spontaneous rotation due to turbulent momentum transport. In this work, we optimize the shape of flux surfaces with both tilted elongation and tilted triangularity in order to maximize this drive of intrinsic rotation. Nonlinear gyrokinetic simulations demonstrate that adding optimally-tilted triangularity can double the momentum transport of a tilted elliptical shape. This work indicates that tilting the elongation and triangularity in an ITER-like device can reduce the energy transport and drive intrinsic rotation with an Alfv\'{e}n Mach number on the order of 1% . This rotation is four times larger than the rotation expected in ITER and is sufficient to stabilize MHD instabilities. It is shown that this optimal shape can be created using the shaping coils of several experiments.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

We develop a gyrokinetic treatment for ions in the magnetic presheath, close to the plasma-wall boundary. We focus on magnetic presheaths with a small magnetic field to wall angle, $\alpha \ll 1$ (in radians). Characteristic lengths perpendicular to the wall in such a magnetic presheath scale with the typical ion Larmor orbit size, ${\rho }_{{\rm{i}}}$. The smallest scale length associated with variations parallel to the wall is taken to be across the magnetic field, and ordered $l={\rho }_{{\rm{i}}}/\delta $, where $\delta \ll 1$ is assumed. The scale lengths along the magnetic field line are assumed so long that variations associated with this direction are neglected. These orderings are consistent with what we expect close to the divertor target of a tokamak. We allow for a strong component of the electric field ${\bf{E}}$ in the direction normal to the electron repelling wall, with strong variation in the same direction. The large change of the electric field over an ion Larmor radius distorts the orbit so that it is not circular. We solve for the lowest order orbits by identifying coordinates, which consist of constants of integration, an adiabatic invariant and a gyrophase, associated with periodic ion motion in the system with $\alpha =\delta =0$. By using these new coordinates as variables in the limit $\alpha \sim \delta \ll 1$, we obtain a generalised ion gyrokinetic equation. We find another quantity that is conserved to first order and use this to simplify the gyrokinetic equation, solving it in the case of a collisionless magnetic presheath. Assuming a Boltzmann response for the electrons, a form of the quasineutrality equation that exploits the change of variables is derived. The gyrokinetic and quasineutrality equations give the ion distribution function and electrostatic potential in the magnetic presheath if the entrance boundary condition is specified.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

The use of high-z wall materials attempts to shift the fusion challenge from heat handling to impurity removal. We demonstrate that not only the impurity density in-out asymmetry but also the poloidal flow has a major impact on the radial impurity flux direction. This realization provides the first method of measuring the flux from available diagnostics, without the need of a computationally demanding kinetic calculation of the full bulk ion response. Moreover, it affords insight into optimal tokamak operation to avoid impurity accumulation while allowing free fueling.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

In general, the orbit-averaged radial magnetic drift of trapped particles in stellarators is non-zero due to the three-dimensional nature of the magnetic field. Stellarators in which the orbit-averaged radial magnetic drift vanishes are called omnigeneous, and they exhibit neoclassical transport levels comparable to those of axisymmetric tokamaks. However, the effect of deviations from omnigeneity cannot be neglected in practice, and it is more deleterious at small collisionalities. For sufficiently low collision frequencies (below the values that define the $1/nu $ regime), the components of the drifts tangential to the flux surface become relevant. This article focuses on the study of such collisionality regimes in stellarators close to omnigeneity when the gradient of the non-omnigeneous perturbation is small. First, it is proven that closeness to omnigeneity is required to actually preserve radial locality in the drift-kinetic equation for collisionalities below the $1/nu $ regime. Then, using the derived radially local equation, it is shown that neoclassical transport is determined by two layers located at different regions of phase space. One of the layers corresponds to the so-called $sqrt{nu }$ regime and the other to the so-called superbanana-plateau regime. The importance of the superbanana-plateau layer for the calculation of the tangential electric field is emphasized, as well as the relevance of the latter for neoclassical transport in the collisionality regimes considered in this paper. In particular, the role of the tangential electric field is essential for the emergence of a new subregime of superbanana-plateau transport when the radial electric field is small. A formula for the ion energy flux that includes the $sqrt{nu }$ regime and the superbanana-plateau regime is given. The energy flux scales with the square of the size of the deviation from omnigeneity. Finally, it is explained why below a certain collisionality value the formulation presented in this article ceases to be valid.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

Neoclassical transport in the presence of non-axisymmetric magnetic fields causes a toroidal torque known as neoclassical toroidal viscosity (NTV). The toroidal symmetry of ITER will be broken by the finite number of toroidal field coils and by test blanket modules (TBMs). The addition of ferritic inserts (FIs) will decrease the magnitude of the toroidal field ripple. 3D magnetic equilibria in the presence of toroidal field ripple and ferromagnetic structures are calculated for an ITER steady-state scenario using the Variational Moments Equilibrium Code (VMEC). Neoclassical transport quantities in the presence of these error fields are calculated using the Stellarator Fokker-Planck Iterative Neoclassical Conservative Solver (SFINCS). These calculations fully account for E r , flux surface shaping, multiple species, magnitude of ripple, and collisionality rather than applying approximate analytic NTV formulae. As NTV is a complicated nonlinear function of E r , we study its behavior over a plausible range of E r . We estimate the toroidal flow, and hence E r , using a semi-analytic turbulent intrinsic rotation model and NUBEAM calculations of neutral beam torque. The NTV torque due to TF ripple without ferritic components is found to be comparable in magnitude to the turbulent and NBI torques, though their radial profiles differ. The NTV from the |n|=18 ripple dominates that from lower n perturbations of the TBMs. With the inclusion of FIs, the magnitude of NTV torque is reduced by about 75% near the edge. We present comparisons of several models of tangential magnetic drifts on superbanana-plateau transport at small E r , and we consider the scaling of calculated NTV torque with ripple magnitude.

• Thematic program: Models in Plasmas, Earth and Space Science (2017/2018)
• Event: 10th Plasma Kinetics Working Group Meeting (2017)

We investigate the risk premium in cash settled forward contracts on the Baltic Exchange Indices – the so-called Forward Freight Agreements – in the dry bulk shipping markets. We estimate multiple spot price models using Markov Chain Monte Carlo. Using a structure-preserving measure change, we then calibrate the risk premium of traded FFA contracts. Finally we link the risk premium to explanatory variables like e.g., oil prices, demand and supply for shipping and the state of the global economy. Joint work with Jonas Lager and Nikos Nomikos.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We extend the Heston stochastic volatility model to a Hilbert space framework. The stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. We compute the dynam- ics of this process under certain conditions, and project it down to the real line to compare it with the one-dimensional Heston variance process. The stochastic volatility process is defined by a Cholesky decomposition of the variance process. We define another Hilbert-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional of this process. Joint work with Fred Espen Benth.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The newly introduced wind power futures on the European Energy Exchange have brought interesting opportunities for energy market players in Germany. In this paper, we analyze the benefits of wind power futures in the context of both the buyer’s and the seller’s side. From the buyer’s side, we con- sider gas-fired power plants. To increase the competitiveness of such plants, we propose a simple yet powerful spot-based trading strategy taking advantage of wind power futures. The purpose of the trading strategy is two-fold: 1) increase the revenue of running the gas-fired power plant, and 2) minimize the variance of the revenue generated from the strategy using wind power futures. To fa- cilitate optimal hedging decisions, we employ ARMA-GARCH models for the marginal behavior of electricity price, gas price, and wind power production, and a mixed vine copula for the dependency between the variables. We find that significant benefits can be achieved by employing a spot-trading strategy as opposed to a strategy acting in the forward market (conditional on the for- ward spark spread being positive). More importantly, using wind power futures reduces the variance of the spot-trading strategy significantly. From the seller’s side, we have the wind mill owners who are facing a quite volatile revenue due to their exposure to joint price and volumetric risk, which they wish to minimize. By performing a similar analysis as in the case of the gas-fired power plants, we again find that wind power futures are beneficial. Joint work with Anca Pircalabu.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We will give an introduction to short-term electricity markets. We will start with the relation of day-ahead and intraday prices on the EPEX for deliveries in Germany/Austria. In the sequel we will focus on analyzing the intraday market. We will discuss empirical properties of intraday power markets and point out development in recent years. Furthermore, we study the optimal liquidation problem for traders in intraday power markets.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We develop a Monte-Carlo based numerical method for solving discrete- time stochastic optimal control problems with inventory. These are optimal control problems in which the control affects only a deterministically evolving inventory process on a compact state space while the random underlying pro- cess manifests itself through the objective functional. We propose a Regress Later modification of the traditional Regression Monte Carlo which allows to decouple inventory levels in two successive time steps and to include in the basis functions of the regression the dependence on the inventory levels. We develop a backward construction of trajectories for the inventory which enables us to use policy iteration of Longstaff-Schwartz type avoiding nested simulations.Our al- gorithm improves on the grid discretisation procedure largely used in literature and practice, and on the recently proposed control randomisation by Kharroubi et al. (2014). We validate our approach on two numerical examples: one is a benchmark problem of energy arbitrage used to compare different methods available in literature, the other is a high-dimensional problem of the manage- ment of a battery with the purpose of assisting the operations of a wind turbine in providing electricity to a group of buildings in a cost effective way. Joint work with Alessandro Balata.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In this work, we propose a model of electricity demand management through a principal-agent problem, allowing to obtain almost explicit optimal compensations for the consumer. We then illustrate our findings through several numerical experiments, putting the emphasis on the practical implementation of the contracts. (Joint work with Rene Aid and Nizar Touzi).

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We give new insights into the theory of the dynamic behavior of com- modity prices with an infinite horizon rational expectations equilibrium model for spot and futures commodity prices. Numerical simulations of the model emphasize the heterogeneity that exists in the behavior of commodity prices by showing the link between the physical characteristics of a market and some stylized facts of commodity futures prices. They show the impact of storage costs on both the variability of the basis and on the Samuelson effect. Finally, the simulations of the model show that an increase in the speculative activity on commodity futures markets has an overall positive effect on risk premia. However, not all of the agents benefit from it.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In this paper we find an approximation to a non-semimartingale Volterra-type process by semimartingales, and furthermore, in the setting of gen- eralized Lebesgue-Stieltjes integration, we find an approximation to the pathwise stochastic integral with this non-semimartingale process as noise. A link to the Itˆo integral and an algorithm for numerical simulation are presented. Joint work with Giulia Di Nunno.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The recent price coupling of many European electricity markets has triggered a fundamental change in the interaction of day-ahead prices, challeng- ing additionally the modeling of the joint behavior of prices in interconnected markets. We propose a regime-switching AR-GARCH copula to model pairs of day-ahead electricity prices in coupled European markets. While capturing key stylized facts empirically substantiated in the literature, this model easily allows us to 1) deviate from the assumption of normal margins and 2) include a more detailed description of the dependence between prices. We base our empirical study on four pairs of prices, namely Germany-France, Germany- Netherlands, Netherlands-Belgium and Germany-Western Denmark. We find that the marginal dynamics are better described by the flexible skew t distribu- tion than the benchmark normal distribution. Also, we find significant evidence of tail dependence in all pairs of interconnected areas we consider. As appli- cations of the proposed empirical model, we consider the pricing of financial transmission rights and the forecasting of tail quantiles. In both applications, we highlight the effects of the distributional assumptions for the margins and the tail dependence. Joint work with Fred Espen Benth.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The growing penetration of non-programmable renewable sources, like solar and wind, introduced in the latest years market uncertainties in the quan- tity of electricity produced, which can possibly originate price spikes. Capacity markets have exactly the purpose of providing new potential capacity when that present in the market is already allocated and there is a sudden drop in supply (due for example to unexpected adverse weather events). In this talk we will present the different capacity remuneration mechanisms, and analyze in more detail the so-called reliability option, which is a call option sold by producers to transmit system operators. This option has the important advantage of shaving possible price peaks, but its correct pricing require non-trivial techniques.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In this paper we develop a statistical arbitrage trading strategy with two key elements in high frequency trading: stop-loss and leverage. We con- sider, as in Bertram (2009), a mean-reverting process for the security price with proportional transaction costs; we show how to introduce stop-loss and lever- age in an optimal trading strategy. We focus on repeated strategies using a self-financing portfolio. For every given stop-loss level we derive analytically the optimal investment strategy consisting of optimal leverage and market en- try/exit levels. First we show that the optimal strategy a la Bertram depends on the probabilities to reach entry/exit levels, on average First-Passage-Times and on average First-Exit-Times from an interval. Then, when the underlying log- price follows an Ornstein-Uhlenbeck process, we deduce analytical expressions for average First-Exit-Times and we write the long-run return of the strategy as an elementary function of the stop-loss. Finally we describe how to apply the strategy to a generic continuous mean-reverting process. Following industry practice of pairs trading we consider two examples of pairs in the energy futures’ market. We report in detail the analysis for two spreads on Heating-Oil and Gas-Oil futures in a year and a half sample of half-hour market prices. Joint work with Tommaso Santagostino Baldi.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In recent years, renewable energy has gained importance in producing power in many markets. The aim of this article is to model photovoltaic (PV) production for three transmission operators in Germany. PV power can only be generated during sun hours and the cloud cover will determine its overall production. Therefore, we propose a model that takes into account the sun intensity as a seasonal function. We model the deseasonalized data by an au- toregressive process to capture the stochastic dynamics in the data. We present two applications based on our suggested model. First, we build a relationship between electricity spot prices and PV production where the higher the volume of PV production, the lower the power prices. As a further application, we discuss virtual power plant derivatives and energy quanto options. Joint work with Fred Espen Benth.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

Energy markets, and in particular, electricity markets, exhibit very peculiar features. The historical series of both futures and spot prices include seasonality, mean-reversion, spikes and small fluctuations. After the pioneer- ing paper by Schwartz, where an Ornstein-Uhlenbeck dynamics is assumed to describe the spot price behavior, several different approaches have been inves- tigated in order to describe the price evolution. A comprehensive presentation of the literature until 2008 is offered in the book by Benth, Saltyte-Benth and Koekebakker [8]. High frequency trading, on the other hand, introduced some new features in commodity prices dynamics: in the paper by Filimonov, Bic- chetti, Maystre and Sornette [11] evidence is shown of endogeneity and struc- tural regime shift, and in order to quantify this level the branching ratio is adopted as a measure of this endogenous impact and a Hawkes processes dy- namics is assumed as a reasonable modeling framework taking into account the self-exciting properties [1]. The purpose of the present paper is to pro- pose a new modeling framework including all the above mentioned features, still keeping a high level of tractability. The model considered allows to obtain the most common derivatives prices in closed or semi-closed form. Here with semi-closed we mean that the Laplace transform of the derivative price admits an explicit expression. The models we are going to introduce can describe the prices dynamics in two different forms, that can be proved to be equivalent: the first is a representation based on random fields, the second is based on Continuous Branching Processes with Immigration (CBI in the following). The idea of adopting a random fields framework for power prices description is not new: O.E. Barndorff-Nielsen, F.E. Benth and A. Veraart introduced the Ambit Fields to this end, showing how this approach can provide a very flexible and still tractable setting for derivatives pricing [2], [3]. A model based on CBI has been proposed recently by Y. Jiao, C. Ma and S. Scotti in view of short interest rate modeling, and in that paper it was shown that, with a suitable choice of the L´evy process driving the CBI dynamics, the model can offer a significant extension of the popular CIR model [12]. The model we propose extends in different ways some relevant models al- ready available in the literature. It belongs to the class of arithmetic models (following the classification proposed by F.E. Benth, J. Salthythe-Benth and S. Koekebakker), and the driving processes are L´evy processes with positive jumps, i.e. subordinators, so it extends the model proposed by F.E. Benth, J. Kallsen and T. Meyer-Brandis [6] by formulating the dynamics via a random field ap- proach, which allows to include some self-exciting features. On the other hand, the random field approach highlights some similarities with the Ambit Field- based models introduced by O.E. Barnorff-Nielsen, F.E. Benth and A. Veraart [3]; the main difference between the model proposed in this paper and the Ambit Field-based models consists in the character of the extra dimension appearing in the random field adopted: while in the Ambit Field setting the parameter of this dimension is a time parameter, in the present setting this will be a pa- rameter of space type. This main difference will be reflected moreover in the integration domain of the integrals defining the dynamics. The features of our modeling approach just outlined, allow to introduce the so- called self-exciting properties in a simple and natural way and, although the pricing formulas for basic contracts like forward will exhibit very small changes with respect to those obtained for the previous models, the present model will exhibit a substantially different risk premium term structure. The presentation will be organized as follows: in Section 2 we’ll introduce the market model we are going to consider, while in Section 3 we shall discuss the relations between our model and the CBI processes. In Section 4 we’ll present some closed formulas for Futures and Option prices when the underlying dynamics is assumed to be given by the model introduced. Section 5 includes a theoretical analysis of the jumps behavior and the self-exciting property. In Section 6 we’ll provide some suggestions about estimation methods for the same model. In this last section, in particular, we are going to highlight the main issues and to propose a theoretical statistical approach. In particular, we are going to derive the maximum likelihood estimator for the parameters of the intensity process. By following the ideas presented in [7] and in [13], the first step to perform will be to de-seasonalise the data. The second step, definitely less trivial, is to split the components Y1 and Y2 emerging from the data. This issue is well analyzed in [7] and [13] and their approach is directly applicable to our framework. Then, we first focus on the process Y1, sometimes called the base signal. Following [7], we look for the ergodic distribution of Y1 fitting the data. By recalling that the ergodic distribution of a CIR diffusion is of Gamma type [10], our model is in agreement with the previous literature (see subsection 5.4.2 in[7]) and we obtain the estimated parameters values for the driving processes. Joint work with Ying Jiao, Chunhua Ma and Simone Scotti. References [1] Bacry, E., Mastromatteo, J. and Muzy, J.-F. Hawkes Processes in Finance, PREPRINT (2015). [2] Barndorff-Nielsen, O.E., Benth, F.E. and Veraart, A. (2013): Modelling en- ergy spot prices by volatility modulated L´evy driven Volterra processes, Bernoulli, 19, 803-845. [3] Barndorff-Nielsen, O.E., Benth, F.E. and Veraart, A. (2014): Modelling Electricity Futures by Ambit Fields, Advances in Applied Probability, 46 (3), 719-745. [4] Barndorff-Nielsen, O.E. and Shephard, N. (2000): Modelling by L´evy Pro- cesses for Financial Econometrics, in L´evy Processes Theory and Applications, eds. Barndorff- Nielsen, Mikosch and Resnick, Boston, Birkhauser. [5] Benth F. E., Cartea A. and Kiesel R. (2008): Pricing forward contracts in power markets by the certainty equivalence principle: explaining the sign of the market risk premium, Journal of Banking and Finance, 32, 2006-2021. [6] Benth, F. E., Kallsen J. and Meyer-Brandis T. (2007): A Non-Gaussian Ornstein- Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing, Appl. Math. Finance, 14(2), 153-169. [7] Benth, F. E., Kiesel, R. and Nazarova A. (2012): A critical empirical study of three electricity price models, Energy Economics, 34, 1589-1616. [8] Benth, F. E., Salthyte-Benth J. and Koekebakker S. (2008): Stochastic Mod- elling of Electricity and Related Markets , World Scientific, Singapore. [9] Benth, F. E. and Sgarra C. (2012): The Risk Premium and the Esscher Transform in Power Markets, Stoch. Anal. Appl., 30(1), 20-43. [10] Cox, J., Ingersoll, J. and Ross, S. (1985): A theory of the term structure of interest rate. Econometrica 53, 385-408. [11] Filimonov, V., Bicchetti, D., Maystre, N., Sornette, D. (2015):Quantifica- tion of the High Level of Endogeneity and Structural Regime Shifts in Com- modity Markets, preprint. [12] Jiao, Y., Ma, C., Scotti, S. (2016): Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling, preprint, hal-01275397v2. [13] Meyer-Brandis, T. and Tankov, P. (2008): Multi-factor jump-diffusion mod- els of electricity prices. International Journal of Theoretical and Applied Fi- nance, 11(5), 503-528.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The use of energy storage to balance electric grids is increasing and, with it, the importance of operational optimisation from the twin viewpoints of cost and system stability. In this paper we assess the real option value of balancing reserve provided by an energy-limited storage unit. The contractual arrangement is a series of American-style call options in an energy imbalance market (EIM), physically covered and delivered by the store, and purchased by the power system operator. We take the EIM price as a general regular one- dimensional diffusion and impose natural economic conditions on the option parameters. In this framework we derive the operational strategy of the storage operator by solving two timing problems: when to purchase energy to load the store (to provide physical cover for the option) and when to sell the option to the system operator. We give necessary and sufficient conditions for the finiteness and positivity of the value function – the total discounted cash flows generated by operation of the storage unit. We also provide a straightforward procedure for the numerical evaluation of the optimal operational strategy (EIM prices at which power should be purchased) and the value function. This is illustrated with an operational and economic analysis using data from the German Amprion EIM. (Joint work with Jan Palczewsk (University of Leeds)).

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In energy markets forward contracts can be of two types: in our ter- minology, forwards and swaps. Who sells a swap contract commits to deliver over a certain period, for instance, power, while by forward we mean the classi- cal financial agreement settled on a maturity date. Our purpose is to design a Heath-Jarrow-Morton framework for an additive, mean-reverting, multidimen- sional market consisting of forward contracts of any maturity date or delivery period. The main assumption is that forward prices can be represented as affine functions of a universal source of randomness. In a Brownian setting, we are able to completely characterize the models which do not allow for arbitrage opportunities. We study the possibility of introducing more general L´evy com- ponents either driving the dynamics of prices or in the context of a stochastic volatility model. Joint work with Fred Espen Benth and Tiziano Vargiolu.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

This presentation is on the simulation of the hour-based German day-ahead power auction, where I apply vector autoregressive (VAR) models, in order to capture the effects of the market infrastructure of the day-ahead auction. This approach ensures that the correct intraday correlation structure is simulated, which will be important for valuing assets with production timing issues (e.g. pumped storages and batteries), thereby creating a more suitable simulation alternative to classic Brownian motion based stochastic simulation for these flexible assets. In order to handle the large dimensionality of the data created by the VAR approach, lasso and elasticnet shrinkages are applied, as well as their adaptive versions. The assessment of these methods is done by performing a classic forecast quality assessment, combined with an evaluation of the (often asymptotic) simulation relevant properties of each model. After estimating the model parameters, simulation from the fitted model is carried out using a block bootstrap. Sanity checks of the appropriateness of the forecasting approach are presented, highlighting both the advantages of the model and the points where future work is necessary.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

Renewable energy sources such as wind and solar have a high degree of unpredictability and time variation. As a result, balancing supply and demand in real time is becoming ever more challenging and the power grids need greater flexibility on many levels. The proposed approach addresses this challenge by harnessing the inherent flexibility in demand of many types of loads. We develop a distributed control theory and algorithms for automated demand dispatch, which can be used by grid operators as ancillary service to regulate demand- supply balance. The proposed approach uses local control solutions that a) take into account local measurements, constraints, and preferences, and b) lead to a controllable input-output model for the aggregate dynamics. The local control problem can be defined by a family of Markov decision processes, parameterized by a weighting factor that appears in the one-step reward function. This talk introduces a new methodology for solving an entire family of MDPs. In our application to demand control, the focus will be on a family of average-cost optimal control models in which the one-step reward function is defined by Kullback-Leibler divergence with respect to nominal dynamics. The proposed ODE methodology can be seen as a generalization of the linearly solvable MDP framework of Todorov to the case with exogenous disturbances, such as weather or customer behavior.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

This paper analyses the interaction between centralised carbon emis- sive technologies and distributed intermittent non-emissive technologies. In our model, there is a representative consumer who can satisfy her electricity demand by investing in distributed generation (solar panels) and by buying power to a centralised firm at a price he set up. Distributed generation is intermittent and induces an externality cost to the consumer. The firm provides non-random electricity generation subject to carbon price and to transmission costs. The objective of the consumer is to satisfy her demand while minimising investment costs, payment to the firm and intermittency cost. The objective of the firm is to satisfy consumer’s residual demand while minimising investment costs, de- mand deviation costs and maximising payment from the consumer. Investment decisions are formulated as McKean-Vlasov control problems with stochastic coefficients. We provide explicit, model-free solutions to the optimal decision problems faced by each player, the solution of the Pareto optimum and the Stackelberg equilibrium where the firm is the leader. We find that, from the social planner point of view, carbon price or transmission costs are necessary to justify a positive share of distributed capacity in the long-term, whatever the re- spective investment costs of both technologies are. The Stackelberg equilibrium is far from the Pareto equilibrium, leading to a much larger share of distributed energy and to a much higher price for centralised energy. Joint work with Rene Aid, Imen Ben Tahar and Huyen Pham

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

Pricing models of derivative instruments usually fail to provide reli- able results when risks rise and financial crises occur. More advanced stochastic pricing models try to improve the fitting results adding risk factors and/or pa- rameters to the models, incurring the risk of overfitted results. Drawing on these observations, it is proposed a generalisation of the Akaike information criterion suitable to evaluate forecasting power of alternative stochastic pricing models for any fixed arbitrary forecasting time-horizon. The Predictability Informa- tion Criterion (PIC) differs from the classical criteria for evaluating statistical models as it assumes that the random variable to study can ( or cannot) be par- tially predictable, which makes it particularly suitable for studying stochastic pricing models coherently with the semimartingale definition of the price pro- cess. On the basis of this assumption the criterion measures and compares the uncertainty of the predictions of two different alternative models when prices are (or are not) predictable. We conclude with a focus on Crude Oil market by comparing GBM and OU stochastic processes that are generally used for modeling West Texas Intermediate (WTI) oil spot price returns in derivative pricing models.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

By means of Malliavin Calculus we construct an optimal linear strike convention for exchange options under stochastic volatility models. This convention allows us to minimize the difference between the model and implied correlations between the two underlying assets in the spread. Moreover, we show that this optimal convention does not depend on the specific stochastic volatility model. Numerical examples are given. Joint work with Elisa Alos.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

With the emergence of renewable energies (as wind or solar genera- tion), local generation systems are rapidly multiplying integrating renewables, batteries or more conventional plants (such as gas turbines or hydro plants). The impact of random factors (such as demand, energy prices, wind, luminosity etc.) on the management of such local generation systems are significant. Hence, an important issue is to be able to manage efficiently such microgrids in presence of uncertainties. Mathematically, the related optimization problem can be stated in terms of a stochastic control problem which can be reduced to a nonlinear Partial Differential Equation (PDE), known as Hamilton-Jacobi-Bellman (HJB) equation. The presentation focuses on recent forward numerical schemes based on generalized Fokker-Planck representations for nonlinear PDEs in high space dimension. In the specific case of mass conservative PDEs, it is well known that the solution can be probabilistically represented as the marginal densities of a Markov diffusion nonlinear in the sense of Mckean. Then one can design forward interacting particle schemes to approximate numerically the PDEs solu- tion. We present some extensions of this kind of representation and interacting particle scheme associated to a large class of PDEs including the case when they are non-conservative, non integrable with various kind of nonlinearities. (Joint work with Anthony Le Cavil, (HSBC, Paris) and Francesco Russo, (ENSTA ParisTech)

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)