an:06336049
Zbl 1322.60100
Benth, Fred Espen; Kr??hner, Paul
Representation of infinite-dimensional forward price models in commodity markets
EN
Commun. Math. Stat. 2, No. 1, 47-106 (2014).
00335429
2014
j
60H15 60G51 60G10 91G80 47B10 47G10 46E35
forward price models; commodity markets; stochastic partial differential equation; L??vy process; infinite-dimensional stochastic processes; Ornstein-Uhlenbeck processes; stationary processes; Heath-Jarrow-Morton approach; Hilbert space; Hilbert-Schmidt operators; integral operators
Summary: We study the forward price dynamics in commodity markets realised as a process with values in a Hilbert space of absolutely continuous functions defined by \textit{D. Filipovi??} [Lect. Notes. Math. 1760, 134 p. (2001; Zbl 1008.91038)]. The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite-dimensional L??vy process. It is shown that the associated spot price dynamics can be expressed as a sum of Ornstein-Uhlenbeck processes, or more generally, as a sum of certain stationary processes. These results link the possibly infinite-dimensional forward dynamics to classical commodity spot models. We continue with a detailed analysis of multiplication and integral operators on the Hilbert spaces and show that Hilbert-Schmidt operators are essentially integral operators. The covariance operator of the L??vy process driving the forward dynamics and the diffusion term can both be specified in terms of such operators, and we analyse in several examples the consequences on model dynamics and their probabilistic properties. Also, we represent the forward price for contracts delivering over a period in terms of an integral operator, a case being relevant for power and gas markets. In several examples, we reduce our general model to existing commodity spot and forward dynamics.
Zbl 1008.91038