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Fred Espen Benth

Publications and source records attributed to Fred Espen Benth.

At least 37 records · Page 2Linked to original sources
arXiv2023

A feasible central limit theorem for realised covariation of SPDEs in the context of functional data

This article establishes an asymptotic theory for volatility estimation in an infinite-dimensional setting. We consider mild solutions of semilinear stochastic partial differential equations and derive a stable central limit theorem for the semigroup adjusted realised covariation (SARCV), which is a consistent estimator of the integrated volatility and a generalisation of the realised quadratic covariation to Hilbert spaces. Moreover, we introduce semigroup adjusted multipower variations (SAMPV) and establish their weak law of large numbers; using SAMPV, we construct a consistent estimator of the asymptotic covariance of the mixed-Gaussian limiting process appearing in the central limit theorem for the SARCV, resulting in a feasible asymptotic theory. Finally, we outline how our results can be applied even if observations are only available on a discrete space-time grid.

math.ST↗
arXiv2023

Intersecting near-optimal spaces: European power systems with more resilience to weather variability

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 3-hourly temporal resolution with 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 up to 50% less $CO_2$ 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 and Modelling All Alternatives, 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.

math.OC↗
arXiv2022

Robustness of Hilbert space-valued stochastic volatility models

In this paper we show that Hilbert space-valued stochastic models are robust with respect to perturbation, 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.

math.PR↗
arXiv2022

Neural Networks in Fréchet spaces

We define a neural network 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 $\mathbb{R}^n$ to $\mathbb{R}$, where approximation is done with (multilayer) neural networks [15, 25, 18, 29]. 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.

math.FA↗
arXiv2022

Pricing options on flow forwards by neural networks in Hilbert space

We propose a new methodology for pricing options on flow forwards by applying infinite-dimensional neural networks. We recast the pricing problem as an optimization problem in a Hilbert space of real-valued function on the positive real line, which is the state space for the term structure dynamics. This optimization problem is solved by facilitating a novel feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural net is built upon the basis of the Hilbert space. We provide an extensive case study that shows excellent numerical efficiency, with superior performance over that of a classical neural net trained on sampling the term structure curves.

q-fin.PR↗
arXiv2021

A Spatio-Temporal Model for Predicting Wind Speeds in Southern California

The share of wind power in fuel mixes worldwide has increased considerably. The main ingredient when deriving wind power predictions are wind speed data; the closer to the wind farms, the better they forecast the power supply. The current paper proposes a hybrid model for predicting wind speeds at convenient locations. It is then applied to Southern California power price area. We build random fields with time series of gridded historical forecasts and actual wind speed observations. We estimate with ordinary kriging the spatial variability of the temporal parameters and derive predictions. The advantages of this work are twofold: (1) an accurate daily wind speed forecast at any location in the area and (2) a general method applicable to other markets.

stat.AP↗
arXiv2021

Accuracy of Deep Learning in Calibrating HJM Forward Curves

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.

q-fin.MF↗
arXiv2021

Correlators of Polynomial Processes

In the setting of 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.

math.PR↗
arXiv2021

Stochastic Integrals and Gelfand Integration in Fréchet Spaces

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.

math.PR↗
arXiv2020

Copula Measures and Sklar's Theorem in Arbitrary Dimensions

Although copulas are used and defined for various infinite-dimensional objects (e.g. Gaussian processes and Markov processes), there is no prevalent notion of a copula that unifies these concepts. We propose a unified approach and define copulas as probability measures on general product spaces. For this we prove Sklar's Theorem in this infinite-dimensional setting. We show how to transfer this result to various function space settings and describe how to model and approximate dependent probability measures in these spaces in the realm of copulas.

math.PR↗
arXiv2020

Sensitivity analysis in the infinite dimensional Heston model

We consider the infinite dimensional Heston stochastic volatility model proposed in \arXiv:1706:03500. The price of a forward contract on a non-storable commodity is modelled by a generalized Ornstein-Uhlenbeck process in the Filipović space with this volatility. We prove different representation formulas for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning of the Greeks. For this we use infinite dimensional Malliavin calculus and a randomization technique.

math.PR↗
arXiv2020

A weak law of large numbers for realised covariation in a Hilbert space setting

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 show that the conditions on the volatility process are valid for most common stochastic volatility models in Hilbert spaces.

math.PR↗
arXiv2020

Abstract polynomial processes

We suggest a novel approach to polynomial processes solely based on a polynomial action operator. With this approach, we can analyse such processes on general state spaces, going far beyond Banach spaces. Moreover, we can be very flexible in the definition of what "polynomial" means. We show that "polynomial process" universally means "affine drift". Simple assumptions on the polynomial action operators lead to stronger characterisations on the polynomial class of processes. In our framework 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 monomials by introducing a sequence of vector space complements. The basic tool of our analysis is the polynomial action operator, which is a semigroup of operators mapping conditional expected values of monomials acting on the polynomial process to monomials of the same or lower grade. Unlike the classical Euclidean case, the polynomial action operator may not form a finite-dimensional subspace after a finite iteration, a property we call locally finite. We study abstract polynomial processes under both 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. Our general framework covers polynomial processes with values in Banach spaces recently studied by Cuchiero and Svaluto-Ferro.

math.PR↗
arXiv2020

Independent increment processes: A multilinearity preserving property

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 generalisation. 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 necessary 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.

math.PR↗
arXiv2020

Stochastic Volterra integral equations and a class of first order stochastic partial differential equations

We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued \Levy 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.

math.PR↗
arXiv2018

Additive energy forward curves in a Heath-Jarrow-Morton framework

One of the peculiarities of power and gas markets is the delivery mechanism of forward contracts. The seller of a futures contract commits to deliver, say, power, over a certain period, while the classical forward is a financial agreement settled on a maturity date. Our purpose is to design a Heath-Jarrow-Morton framework for an additive, mean-reverting, multicommodity market consisting of forward contracts of any delivery period. The main assumption is that forward prices can be represented as affine functions of a universal source of randomness. This allows us to completely characterize the models which prevent arbitrage opportunities: this boils down to finding a density between a risk-neutral measure $\mathbb{Q}$, such that the prices of traded assets like forward contracts are true $\mathbb{Q}$-martingales, and the real world probability measure $\mathbb{P}$, under which forward prices are mean-reverting. The Girsanov kernel for such a transformation turns out to be stochastic and unbounded in the diffusion part, while in the jump part the Girsanov kernel must be deterministic and bounded: thus, in this respect, we prove two results on the martingale property of stochastic exponentials. The first allows to validate measure changes made of two components: an Esscher-type density and a Girsanov transform with stochastic and unbounded kernel. The second uses a different approach and works for the case of continuous density. We apply this framework to two models: a generalized Lucia-Schwartz model and a cross-commodity cointegrated market.

q-fin.MF↗
arXiv2017

Pricing of commodity derivatives on processes with memory

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 Lévy semi-stationary process. Moreover a risk premium \r{ho} representing storage costs, illiquidity, convenience yield or insurance costs is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity. Also the interest rate is assumed to be stochastic. To show the existence of an equivalent pricing measure Q for S we relate the stochastic differential equation for ξ to the generalised Langevin equation. When the interest rate is deterministic the process (ξ; \r{ho}) 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.

q-fin.PR↗