We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

In this note we described the robustness properties of optimal risksensitive controllers for quantum systems. We consider a quantum generalization of risk-sensitive criteria using the framework of (James, 2004). The robustness properties are derived by evaluating certain Radon-Nikodym derivatives of the quantum models and of the cost criteria. In addition to induced perturbations in the quantum...

We define a class of higher-order linear maps that transform quantum operations into quantum operations and satisfy suitable requirements of normality and complete positivity. For this class of maps we prove two dilation theorems that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. A structure theorem for probability measures with values in this class of ...

In this article we define measurable multifunctions in nonseparable Banach spaces, prove a weak compactness criterion for the selectors of multifunctions integrably bounded, characterize Banach spaces that have the Radon–Nikodym property by means of convergence of multivalued martingales, generalize some recent results on convergence of set-valued conditional expectations, and give some applica...

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

Consider a stochastic process {X,},,, where JI is either Iw + = [0, cc) or N (the set of natural numbers). For some compact metric space D, let PO, 8 ED, be a family of probability measures on the trajectory (‘canonical’) space of ( X,},EJ, equipped with the appropriate topology and the corresponding Bore1 u-field. Assume that the ‘true’ probability measure is PO,, for some 19, ED. Pe7-, T E ,I...

This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared...