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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].

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