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Filtration des ponts browniens et équations différentielles stochastiques linéaires. (Filtering of Brownian bridges and linear stochastic differential equations). (French) Zbl 0699.60075
Séminaire de probabilités XXIV 1988/89, Lect. Notes Math. 1426, 227-265 (1990).
[For the entire collection see Zbl 0695.00024.]
After some preliminaries about enlargements and reductions of filtrations, the authors study the filtration of the Brownian bridge. If \((X_ t)_{t\geq 0}\) is a one-dimensional Brownian motion, starting from zero, \(\tilde X_ t=X_ t-\int^{t}_{0}s^{-1}X_ sds\) is another Brownian; therefore, the application \(T: X\to \tilde X\) preserves the Wiener measure. With the help of Laguerre polynomials, it is shown that T is strongly mixing.
More generally, the authors consider Gaussian processes defined by: \[ \beta_ t^{\phi}=B_ t-\int^{t}_{0}\phi (s)B_ sds \] where \(\phi\) is a Borel function from \({\mathbb{R}}^+\) to \({\mathbb{R}}\) such that: \[ \forall t>0,\quad \int^{t}_{0}| \phi (s)| \sqrt{s}ds<+\infty; \] properties of the filtration of \(\beta_ t^{\phi}\) are given. Finally, the paper contains a detailed study of the linear stochastic differential equation \[ X_ t=B_ t+\int^{t}_{0}\phi (s)X_ sds, \] where \(B_ t\) is a Brownian motion. Existence and uniqueness of solutions are stated under some suitable conditions.
Reviewer: M.Chaleyat-Maurel

MSC:
60J65 Brownian motion
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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