The Optimal Filtering of Markov Jump Processes in Additive White Noise
This note is based on Wonham \cite{Wonham}. The differences between this note and [Wonham] are discussed in Section VIII.
math.PR↗
arXiv subjects
M. Zakai
Publications and source records attributed to M. Zakai.
The Clark-Ocone formula for vector valued random variables in abstract Wiener space
The classical representation of random variables as the Ito integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron-Martin space. The Ito integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense. However, the published paper contains a mistake which restricts the validity of the main result, as described in the added Erratum.
math.PR↗
The divergence of Banach space valued random variables on Wiener space
The domain of definition of the divergence operator δon an abstract Wiener space (W, H, μ) is extended to include W-valued and W\otimesW-valued "integrands". The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ's extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields "are responsible" for generating measure preserving flows.
math.PR↗
Correction to "The divergence of Banach space valued random variables on Wiener space", Prob. Th. Rel. Fields 132, 291-320 (2005)
As a result of some mistakes discovered in the paper mentioned in the title, Corollaries 3.5 and 3.17a) are withdrawn and a new proof is provided for Proposition 3.14, under the added assumption that the second dual of the underlying Banach space Y possesses the Radon-Nykodim property.
math.PR↗
The realization of positive random variables via absolutely continuous transformations of measure on Wiener space
Let $μ$ be a Gaussian measure on some measurable space $\{W=\{w\},{\mathcal{B}}(W)\}$ and let $ν$ be a measure on the same space which is absolutely continuous with respect to $ν$. The paper surveys results on the problem of constructing a transformation $T$ on the $W$ space such that $Tw=w+u(w)$ where $u$ takes values in the Cameron-Martin space and the image of $μ$ under $T$ is $μ$. In addition we ask for the existence of transformations $T$ belonging to some particular classes.
math.PR↗
Rotations and Tangent Processes on Wiener Space
The paper considers (a) Representations of measure preserving transformations (``rotations'') on Wiener space, and (b) The stochastic calculus of variations induced by parameterized rotations $\{T_θw, 0 \le θ\le \eps\}$: ``Directional derivatives'' $(dF(T_θw)/d θ)_{θ=0}$, ``vector fields'' or ``tangent processes'' $(dT_θw /dθ)_{θ=0}$ and flows of rotations.
math.PR↗
Tangent Processes on Wiener Space
This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space, (i.e. transformations induced by general measure preserving transformations, called `rotations', and H-valued shifts) and the associated flows on abstract Wiener spaces
math.PR↗
Some Ergodic Theorems for Random Rotations on Wiener Space
In this paper we study ergodicity and mixing property of some measure preserving transformations on the Wiener space (W,H,μ) which are generated by some random unitary operators defined on the Cameron-Martin space H.
math.PR↗
Some measure-preserving point transformations on the Wiener space and their ergodicity
Suppose that T is a map of the Wiener space into itself, of the following type: T=I+u where u takes its values in the Cameron-Martin space H. Assume also that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work we prove that if T preserves the Wiener measure, then necessarily u is in the first Wiener chaos and the transformation corresponding to it is a rotation in the sense of [9]. Afterwards the ergodicity and mixing of such transformations, which are second quantizations of the unitary operators on the Cameron-Martin space, are characterized. Finally, the ergocity of the transformation dY_t=gamma(t)dW_t, 0 \le t \le 1 where W is n-dimensional Wiener and gamma is non random is characterized
math.PR↗