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Term structure models driven by Wiener processes and Poisson measures: existence and positivity. (English) Zbl 1207.91068
The authors focus on term structure models where the forward rate is driven by a possibly infinite Wiener process and a homogeneous Poisson random measure. As is rather common in this type of research, they switch to the Musiela-parametrization of the forward rate, in order to make the analysis amenable to the application of Hilbert-space theory. This then leads to the question whether a particular stochastic integral equation describing implicitly the movement of this forward curve has a unique solution in a particular sense. Here the drift and the volatilities are allowed to be suitable functions of the prevailing forward curve, whereas these curves take their values in an appropriate Hilbert space. The question is answered affirmatively, and as such this existence result is a considerable extension, and far from trivial at that, of similar results in a diffusion setting or in the setting of Lévy processes. As the authors remark, in practice one is often interested in term-structure models producing positive forward curves. They actually give a characterization in the setting of their Wiener-Poisson-type model of such positivity preserving term-structure models. As one might expect from these experts in the field, their paper is far from trivial, but nevertheless extremely clearly written, and with excellent and appropriate references. In my view this paper will become one of the landmarks in the Hilbert-space theory of term-structure models.

MSC:
91G30 Interest rates, asset pricing, etc. (stochastic models)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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