We present and analyze an infinite dimensional Bayesian inference formulation, and its numerical approximation, for the inverse problem of inferring the shape of an obstacle from scattered acoustic waves. Given a Gaussian prior measure on the shape space, whose covariance operator is the inverse of the Laplacian, the Bayesian solution of the inverse problem is the posterior measure given by the...

Dosiev (2008) obtained a Stinespring’s theorem for local completely positive maps (in short: CP-maps) on locally C ∗ -algebras. In this article suitable notion of minimality construction has been identified so as to ensure uniqueness up unitary equivalence the associated representation. Using Radon–Nikodym type proved. Further, unbounded operator valued Hilbert modules over -algebras (also call...

The present study utilizes the Girsanov transformation-based framework for solving a nonlinear stochastic dynamical system in an efficient way comparison with other available approximate methods. In this approach, rejection sampling is formulated to evaluate Radon–Nikodym derivative arising from change of measure due transformation. Rejection applied on Euler–Maruyama approximated sample paths,...

A measure-preserving endomorphism is one-sided Bernoulli if it is isomorphic to a noninvertible Bernoulli shift. We show that in piecewise smooth settings this property is very strong and far more subtle than the weak Bernoulli property, by extending of results of W. Parry and P. Walters and proving new results based on continuity of the Radon-Nikodym derivative. In particular, we provide tests...

Example 1. Change of measure on Ω = [0, 1]. Example 2. Change of measure for normal random variables. • We can also perform change of measure for a whole process rather than for a single random variable. Suppose that there is a probability space (Ω,F) and a filtration {Ft, 0 ≤ t ≤ T}. Suppose further that FT -measurable Z is an almost surely positive random variable satisfying E[Z] = 1. We can ...

Stochastic calculus methods are used to estimate the instantaneous frequency of a signal. The frequency is modeled as a polynomial in time. It is assumed that the phase has a Brownian-motion component. Using stochastic calculus, one is able to develop a stochastic differential equation that relates the observations to instantaneous frequency. Pseudo-maximum likelihood estimates are obtained thr...

We study multiple defaults where the global market information is modelled as progressive enlargement of filtrations. We shall provide a general pricing formula by establishing a relationship between the enlarged filtration and the reference defaultfree filtration in the random measure framework. On each default scenario, the formula can be interpreted as a Radon-Nikodym derivative of randommea...

Random finite set provides a rigorous foundation for optimal Bayes multi-target filtering. The major hurdle faced in Bayes multi-target filtering is the inherent computational intractability. Even the Probability Hypothesis Density (PHD) filter, which propagates only the first moment (or PHD) instead of the full multi-target posterior, still involves multiple integrals with no closed forms. In ...

We give improved lower bounds on the size of Kakeya and Nikodym sets over Fq. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in Fq, and show that this conjecture implies an optimal bound on the size of a Nikodym set. Finally, we study the notion of a weak Nikodym set and give improved, and in some special cases optimal, bounds fo...

This paper considers delta estimators of the Radon-Nikodym derivative of a probability function with respect to a a-finite measure. We provide sufficient conditions for universal consistency, which are checked for some wide classes of non-parametric estimators. Let P be a probability measure in the Borel space (]l~d,~d), absolutely continuous with respect to the a-finite measure # and f = dP/d#...