Ornstein-Uhlenbeck-Cauchy Process

We combine earlier investigations of linear systems with L\'{e}vy fluctuations [Physica {\bf 113A}, 203, (1982)] with recent discussions of L\'{e}vy flights in external force fields [Phys.Rev. {\bf E 59},2736, (1999)]. We give a complete construction of the Ornstein-Uhlenbeck-Cauchy process as a fully computable model of an anomalous transport and a paradigm example of Doob's stable noise-supported Ornstein-Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, relevant (pseudodifferential) evolution equations. Finally we prove that this random dynamics is not only mixing (hence ergodic) but also exact. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense.Comment: Latex fil

Coupling and Strong Feller for Jump Processes on Banach Spaces

By using lower bound conditions of the L\'evy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical) L\'evy processes on a Banach space. Unlike in the finite-dimensional case where these properties have also been confirmed for L\'evy processes without drift, in the infinite-dimensional setting the appearance of a drift term is essential to ensure the quasi-invariance of the process by shifting the initial data. Gradient estimates and exponential convergence are also investigated. The main results are illustrated by specific models on the Wiener space and separable Hilbert spaces.Comment: 31 page

Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already been publishe

O warunkowaniach dla zbieżnych do zera ciągów wektorów losowych w przestrzeniach Banacha

https://doi.org/10.26485/0459-6854/2018/68.3/1

Wycena opcji w modelu CRR z parametrami zależnymi od czasu dla dwóch jednostek czasu – cześć II

In the second part of the paper we prove the convergence of option prices in the presented model to the price that is given by some formula corresponding to the Black-Scholes formula

Wycena opcji w modelu CRR z parametrami zależnymi od czasu dla dwóch jednostek czasu – cześć I

In this paper we present some generalization of the Cox-Ross-Rubinstein (CRR) option pricing model. We assume that two parameters of the model (an interest rate of a bank account and a volatility of the logarithm of the stock price’s changes) are different in each of two analyzed periods of time.W pracy przedstawiono pewien uogólniony model Coxa-Rossa-Rubinsteina na wycenę opcji. Założono, że dwa parametry modelu (stopa procentowa oraz współczynnik zmienności logarytmu cen akcji-volatility) zmieniają się w każdej z dwóch jednostek czasu