In this paper we present a martingale related to the exit measures of super Brow-nian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem ...

The martingale optimal transport problem is motivated by model-independent bounds for the pricing and hedging exotic options. In the simplest one-period model, the dual formulation of the robust superhedg-ing cost differs from the standard optimal transport problem by the presence of a martingale constraint on the set of coupling measures. The one-dimensional Brenier theorem has a natural exten...

We study optimal hedging of barrier options using a combination of a static position in vanilla options and dynamic trading of the underlying asset. The problem reduces to computing the Fenchel-Legendre transform of the utility-indifference price as a function of the number of vanilla options used to hedge. Using the well-known duality between exponential utility and relative entropy, we provid...

This thesis investigates the employment of different entropic measures, including Rényi entropy, in the context of image registration. Specifically, we focus on the entropy estimation problem for image registration and provide theoretical and experimental comparisons of two important entropy estimators: the plug-in estimator and minimal entropic graphs. We further develop an image registration ...

In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ Theta $. The relative entropy of a sub-$ sigma_Theta $-algebra having finite atoms is defined and then   the ergodic properties of relative  semi-dynamical systems are investigated.  Also,  a relative version of Kolmogorov-Sinai theorem  is given. Finally, it is proved  that the relative entropy of a...

In the present paper we study the entropy gain H(Φ[ρ]) − H(ρ) for infinite-dimensional channels Φ. We show that unlike finite-dimensional case where the minimal entropy gain is always nonpositive [1], there is a plenty of channels with positive minimal entropy gain. We obtain the new lower bound and compute the minimal entropy gain for a broad class of Bosonic Gaussian channels by proving that ...

There are numerous characterizations of Shannon entropy and Tsallis entropy as measures of information obeying certain properties. Using work by Faddeev and Furuichi, we derive a very simple characterization. Instead of focusing on the entropy of a probability measure on a finite set, this characterization focuses on the “information loss”, or change in entropy, associated with a measure-preser...

For a set-valued stochastic sequence (Gn)Nn=0 with relatively open convex values Gn(ω) we give a criterion for the existence of an adapted sequence (xn)Nn=0 of selectors, admitting an equivalent martingale measure. Mentioned criterion is expressed in terms of supports of the regular conditional upper distributions of the elements Gn. This result is a refinement of the main result of author’s pr...