42 research outputs found
Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications
We investigate the probabilistic and analytic properties of Volterra
processes constructed as pathwise integrals of deterministic kernels with
respect to the H\"older continuous trajectories of Hilbert-valued Gaussian
processes. To this end, we extend the Volterra sewing lemma from
\cite{HarangTindel} 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. in
\cite{ElEuchRosenbaum}.Comment: 38 page
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
Analytical Approximation for the Price Dynamics of Spark Spread Options
This paper presents an analytic approximation for the pricing dynamics of spark spread options in terms of Fourier transforms. We propose to model the spark spread, that is, the price difference of electricity and gas, directly using a mean-reverting model with diffusion and jumps. The model is analyzed empirically, and shown to fit observed data in the UK reasonably well. The main advantage with the model is that the spark spread of electricity and gas forwards, being forwards with delivery over periods, can be priced analytically. The price dynamics for different spark spread options with electricity and gas forwards as underlying, is analytically derived through Fourier transforms. These pricing expressions allow for efficient numerical valuations via the fast Fourier transform technique.
On stochastic integration for volatility modulated Brownian driven Volterra processes via white noise analysis
This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space of Potthoff–Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.
Preprint of an article published in Infinite Dimensional Analysis Quantum Probability and Related Topics. 2014, 17 (2), DOI: http://dx.doi.org/10.1142/S0219025714500118 © World Scientific Publishing Company http://www.worldscientific.com/worldscinet/idaq
APPROXIMATING LÉVY SEMISTATIONARY PROCESSES VIA FOURIER METHODS IN THE CONTEXT OF POWER MARKETS
The present paper discusses Levy semistationary processes in the context of power markets. A Fourier simulation scheme for obtaining trajectories of these processes is discussed and its rate of convergence is analysed. Finally we put our simulation scheme to work for simulating the price of path dependent options