A martingale measure is constructed by using a mean correcting transform for the geometric Lévy processes model. It is shown that this measure is the mean correcting martingale measure if and only if, in the Lévy process, there exists a continuous Gaussian part. Although this measure cannot be equivalent to a physical probability for a pure jump Lévy process, we show that a European call option...

During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory (e.g., concentration of measure phenomena in random graphs, random matrices, and percolation), information theory, theoretical computer scienc...

Integrating discrete information extracted from random process solves the impulse cutting off entropy functional (EF) measure on trajectories Markov diffusion process, integrated in information path functional (IPF). Defining the EF via the process additive functional with functions drift and diffusion allows reducing this functional on trajectories to a regular integral functional. Compared to...

For any discrete-time P–local martingale S there exists a probability measure Q ∼ P such that S is a Q–martingale. A new proof for this result is provided. This proof also yields that, for any ε > 0, the measure Q can be chosen so that dQ/dP ≤ 1 + ε.

The purpose of this study is to define the concepts of Tsallis entropy and conditional Tsallis entropy of fuzzy partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for ...

We derive a martingale representation for a contingent claim under a Markov-modulated version of the Black-Scholes economy. The martingale representation for the price of the claim is established with respect to an equivalent martingale measure chosen by the Esscher transform. Under some differentiability conditions for the coefficients of the price processes, we shall identify explicitly the i...