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## Studia Mathematica

1999 | 134 | 3 | 251-268
Tytuł artykułu

### Embedding of random vectors into continuous martingales

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let E be a real, separable Banach space and denote by $L^0(Ω,E)$ the space of all E-valued random vectors defined on the probability space Ω. The following result is proved. There exists an extension ${\widetilde Ω}$ of Ω, and a filtration $({\widetilde ℱ}_t)_{t≥0}$ on ${\widetilde Ω}$, such that for every $X ∈ L^0(Ω,E)$ there is an E-valued, continuous $({\widetilde ℱ}_t)$-martingale $(M_t(X))_{t≥0}$ in which X is embedded in the sense that $X = M_τ(X)$ a.s. for an a.s. finite stopping time τ. For E = ℝ this gives a Skorokhod embedding for all $X ∈ L^0(Ω,ℝ)$, and for general E this leads to a representation of random vectors as stochastic integrals relative to a Brownian motion.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
251-268
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-01-29
poprawiono
1998-07-12
Twórcy
autor
• Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Bibliografia
• [1] J. Azéma et M. Yor, Une solution simple au problème de Skorohod, in: Séminaire de Probabilités XIII, Lecture Notes in Math. 721, Springer, 1979, 90-115 and 625-633.
• [2] R. Chacon and J. B. Walsh, One-dimensional potential embedding, in: Séminaire de Probabilités X, Lecture Notes in Math. 511, Springer, 1976, 19-23.
• [3] E. Dettweiler, Banach space valued processes with independent increments and stochastic integration, in: Probability in Banach Spaces IV, Lecture Notes in Math. 990, Springer, 1983, 54-83.
• [4] E. Dettweiler, Stochastic integration of Banach space valued functions, in: L. Arnold and P. Kotelenez (eds.), Stochastic Space-Time Models and Limit Theorems, Reidel, 1985, 53-79.
• [5] L. Dubins, On a theorem of Skorohod, Ann. Math. Statist. 39 (1968), 2094-2097.
• [6] L. Dubins and G. Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913-916.
• [7] R. M. Dudley, Wiener functionals as Itô integrals, Ann. Probab. 5 (1977), 140-141.
• [8] R. M. Dudley, Real Analysis and Probability, Wadsworth & Brooks-Cole, 1989.
• [9] J. Hoffmann - Jοrgensen, Probability in Banach spaces, in: Ecole d'Eté de Probabilités de Saint-Flour VI, Lecture Notes in Math. 598, Springer, 1977, 1-186.
• [10] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, l988.
• [11] A. V. Skorohod [A. V. Skorokhod], Studies in the Theory of Random Processes, Addison-Wesley, 1965.
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Bibliografia
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