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Representation of infinite-dimensional forward price models in commodity markets. (English) Zbl 1322.60100
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 D. Filipović [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 Lévy 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 Lévy 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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60G10 Stationary stochastic processes
91G80 Financial applications of other theories
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47G10 Integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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