On continuity properties for option prices in exponential L\'evy models

For a converging sequence of exponential L\'evy models, we give conditions under which the associated sequence of option prices converges. We also study the behaviour of the prices when no such convergence holds. We then consider two special cases, first when the martingale measure is chosen by minimisation of entropy and then when it minimises Hellinger integrals.Comment: 24 pages, no figure

On regularity properties of Bessel flow

We study the differentiability of Bessel flow $\rho : x \to \rho ^x_t$, where $(\rho ^x_t)_{t\geq 0}$ is BES $^x(\delta$) process of dimension $\delta >1$ starting from $x$. For $\delta \geq 2$ we prove the existence of bicontinuous derivatives in P-a.s. sense at $x\geq 0$ and we study the asymptotic behaviour of the derivatives at $x=0$. For $1< \delta <2$ we prove the existence of a modification of Bessel flow having derivatives in probability sense at $x\geq 0$. We study the asymptotic behaviour of the derivatives at $t=\tau_0(x)$ where $\tau_0(x)$ is the first zero of $(\rho ^x_t)_{t\geq 0}$.Comment: 21 pages, no figure

Reflection principle and Ocone martingales

Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $\sigma$-field of the invariant events by all reflections at levels $a_n$, $n\ge1$ is trivial. We establish similar results for skip free $\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretisation in space

Methodology for the identification of bioactive and marker peptides in the organs of cattle and pigs

The development of general conception methodology for the meat-based functional food compositions is especially relevant today due to the growing consumers’ interest and attention to their health. This category of these food-products is intended for personal  ized nutrition of various age groups in the population, taking into account fortification of the food with nutraceuticals and with functional and metabolically active ingredients obtained from animal and vegetable source. Therefore, it was necessary to develop a certain tool for reliable identification of free peptides from the offals (by-products like hearts and aorta from Sus scrofa and Bos taurus) and from the ready-to-consume meat food (canned food) based on the free peptides, which food is potentially targeted to help with some issues in the human body. The authors proposed the methodology for identification of peptides weighing less than 5 kDa. This methodology has a row of significant advantages, such as a short time of analysis (90 minutes) and the possibility to prepare a large number of samples simultaneously (n=16). Analysis of bioactive peptides (BAPs) was performed by liquid chro  matography combined with time-of-flight mass spectrometry (Agilent 6545XT AdvanceBio LC/Q-TOF). The marker peptides were detected by a triple quadrupole mass spectrometer (Agilent 6410 Triple Quadrupole LC/MS). All peptide sequences were defined with the help of mass spectrometric data processing databases like PepBank, BioPep, AHTPDB. In this work from 39 to 269 peculiar soluble peptides were found, with an extraction level of 0.17–0.23%. The main fraction consisted of short peptides less than 1000 Da (71.0–98.0%). In experimental samples of pork hearts and arteries 7 peculiar marker peptides were identified. FFESFGDL  SNADAVMGNPK peptide obtained from the β-hemoglobin protein is of a special interest, as this peptide showed the maximum intensity of a signal. Presumably, this peptide can serve as an indicator of the blood presence in the finished food product. So it can serve as an assessment tool of bleeding degree of meat raw. For pork aortas a specific peptide TVLGNFAAFVQK was isolated from serum albumin, which turned out to be stable during heat treatment. This is also important for assessment of meat food that are subjected to high thermal exposure.The development of general conception methodology for the meat-based functional food compositions is especially relevant today due to the growing consumers’ interest and attention to their health. This category of these food-products is intended for personal  ized nutrition of various age groups in the population, taking into account fortification of the food with nutraceuticals and with functional and metabolically active ingredients obtained from animal and vegetable source. Therefore, it was necessary to develop a certain tool for reliable identification of free peptides from the offals (by-products like hearts and aorta from Sus scrofa and Bos taurus) and from the ready-to-consume meat food (canned food) based on the free peptides, which food is potentially targeted to help with some issues in the human body. The authors proposed the methodology for identification of peptides weighing less than 5 kDa. This methodology has a row of significant advantages, such as a short time of analysis (90 minutes) and the possibility to prepare a large number of samples simultaneously (n=16). Analysis of bioactive peptides (BAPs) was performed by liquid chro  matography combined with time-of-flight mass spectrometry (Agilent 6545XT AdvanceBio LC/Q-TOF). The marker peptides were detected by a triple quadrupole mass spectrometer (Agilent 6410 Triple Quadrupole LC/MS). All peptide sequences were defined with the help of mass spectrometric data processing databases like PepBank, BioPep, AHTPDB. In this work from 39 to 269 peculiar soluble peptides were found, with an extraction level of 0.17–0.23%. The main fraction consisted of short peptides less than 1000 Da (71.0–98.0%). In experimental samples of pork hearts and arteries 7 peculiar marker peptides were identified. FFESFGDL  SNADAVMGNPK peptide obtained from the β-hemoglobin protein is of a special interest, as this peptide showed the maximum intensity of a signal. Presumably, this peptide can serve as an indicator of the blood presence in the finished food product. So it can serve as an assessment tool of bleeding degree of meat raw. For pork aortas a specific peptide TVLGNFAAFVQK was isolated from serum albumin, which turned out to be stable during heat treatment. This is also important for assessment of meat food that are subjected to high thermal exposure

On the Stability of Prices of Contingent Claims in Incomplete Models Under Statistical Estimations

In exponential semi-martingale setting for risky asset we estimate the difference of prices of options when initial physical measure P and corresponding martingale measure Q change to ̃ P and ̃Q respectively. Then, we estimate PL 1-distance of option prices for corresponding parametric models with known and estimated parameters. The results are applied to exponential Lévy models with special choice of martingale measure as Esscher measure, minimal entropy measure and Pfq-minimal martingale measure. We illustrate our results by considering GMY and CGMY models

Revisiting integral functionals of geometric Brownian motion

Research Federation of “Mathématiques des Pays de la Loire