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Martingale-valued measures, Ornstein-Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space. (English) Zbl 1133.60020
Émery, Michel (ed.) et al., In memoriam Paul-André Meyer. Séminaire de probabilités XXXIX. Berlin: Springer (ISBN 3-540-30994-2/pbk). Lecture Notes in Mathematics 1874, 171-196 (2006).
The author starts with investigating Hilbert space-valued martingale-valued measures. In particular, he investigates a class of such measures $$M$$ whose covariance structure is determined by a trace class positive operator valued measure (this is precisely the covariance structure found in the martingale part of a Lévy process). Here, it is shown that the covariance of the compensated small jumps is also determined by such operators, which in this case are a continuous superposition of finite rank operators. It is then show how weak and strong stochastic integrals of suitable predictable processes can be developed. In the first of these the integrand $$(F(t,x),\,t\geq 0,\,x\in E)$$ ($$E$$ being a Lusin space) is vector-valued. Generalising an approach due to H. Kunita [Nagoya Math. J. 38, 41–52 (1970; Zbl 0234.60071)], the scalar-valued process $$\int^t_0 \int_E (F(s,x), M(ds, dx))_H$$ is constructed where $$(\cdot,\cdot)_H$$ denotes the inner product in the Hilbert space $$H$$. In the second of these, $$(G(t,x),\,t\geq 0,\,x\in E)$$ is operator-valued. This generalises the stochastic integral of M. Metivier [Semimartingales, a course on stochastic processes (Berlin, 1982; Zbl 0503.60054)] and yields the Hilbert space-valued object $$\int^t_0 \int_E G(s,x)M(ds, dx)$$.
As an application, first the stochastic convolution $$\int^t_0 S(r) dX(r)$$ of a $$C_0$$-semigroup $$(S(r),\, r\geq 0)$$ having infinitesimal generator $$J$$ with a Lévy process $$X= (X(t),\,t\geq 0)$$ is studied. This is then applied to investigate the generalised Langevin equation $$dY(t)= JY(t)+ dX(t)$$ whose unique weak solution is the Ornstein-Uhlenbeck process.
For the entire collection see [Zbl 1092.60003].

##### MSC:
 60G44 Martingales with continuous parameter 60G51 Processes with independent increments; Lévy processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 47D03 Groups and semigroups of linear operators