An integral representation for selfdecomposable banach space valued random variables

Jurek, Zbigniew J.; Vervaat, Wim

doi:10.1007/bf00538800

Cited by 140 publications

(152 citation statements)

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“…[35], section 3.15). The definition was extended to Banach space valued random variables by Jurek and Vervaat [22] (with c still a scalar) while Jurek and Mason [21] considered the finite-dimensional case of "operator self-decomposability" where c is replaced by a semigroup (e −tJ , t ≥ 0), with J an invertible matrix. Jurek [19] also investigated the case where J is a bounded operator in a Banach space.…”

Section: E G(s X)m (Ds Dx)mentioning

confidence: 99%

“…The definition was extended to Banach space valued random variables by Jurek and Vervaat [22] (with c still a scalar) while Jurek and Mason [21] considered the finite-dimensional case of "operator self-decomposability" where c is replaced by a semigroup (e −tJ , t ≥ 0), with J an invertible matrix. Jurek [19] also investigated the case where J is a bounded operator in a Banach space. It is a consequence of results found in [39], [21], [22] and [37] that X is (operator) self-decomposable if and only if it can be embedded as X(0) in a stationary Ornstein-Uhlenbeck process.…”

Section: E G(s X)m (Ds Dx)mentioning

confidence: 99%

“…Jurek [19] also investigated the case where J is a bounded operator in a Banach space. It is a consequence of results found in [39], [21], [22] and [37] that X is (operator) self-decomposable if and only if it can be embedded as X(0) in a stationary Ornstein-Uhlenbeck process. Furthermore a necessary and sufficient condition for the required stationarity is that the Lévy measure ν of X has a certain logarithmic moment, more precisely |x|≥1 log(1 + |x|)ν(dx) < ∞, so we see that this is a condition on the large jumps of X (see also [16]).…”

Section: E G(s X)m (Ds Dx)mentioning

confidence: 99%

“…In the following, we define (X(t), t < 0) to be an independent copy ofX. We recall the Ornstein-Uhlenbeck process (19) …”

Section: S(r)dx(r)mentioning

confidence: 99%

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Martingale-Valued Measures, Ornstein-Uhlenbeck Processes with Jumps and Operator Self-Decomposability in Hilbert Space

Applebaum

In Memoriam Paul-André Meyer

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Summary. We investigate a class of Hilbert space valued martingale-valued measures whose covariance structure is determined by a trace class positive operator valued measure. The paradigm example is the martingale part of a Lévy process. We develop both weak and strong stochastic integration with respect to such martingale-valued measures. As an application, we investigate the stochastic convolution of a C 0 -semigroup with a Lévy process and the associated Ornstein-Uhlenbeck process. We give an infinite dimensional generalisation of the concept of operator self-decomposability and find conditions for random variables of this type to be embedded into a stationary Ornstein-Uhlenbeck process.

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Section: E G(s X)m (Ds Dx)mentioning

confidence: 99%

Section: E G(s X)m (Ds Dx)mentioning

confidence: 99%

Section: E G(s X)m (Ds Dx)mentioning

confidence: 99%

“…In the following, we define (X(t), t < 0) to be an independent copy ofX. We recall the Ornstein-Uhlenbeck process (19) …”

Section: S(r)dx(r)mentioning

confidence: 99%

See 2 more Smart Citations

Martingale-Valued Measures, Ornstein-Uhlenbeck Processes with Jumps and Operator Self-Decomposability in Hilbert Space

Applebaum

In Memoriam Paul-André Meyer

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“…Let Rd 0 A direct verification shows that (4.5) is the Levy measure of some selfdecomposable distribution. In fact, it follows from [JV83] or by a reformulation in [SY84] of a theorem due to Urbanik [U69], that every Levy measure of a selfdecomposable distribution can be written in the form (4.5) with p having the logarithmic moment. On the other hand, Sato [S98] showed that vo is …”

Section: Some Invariant Properties Of Type G Distributionsmentioning

confidence: 99%