Narrow Results

In this paper, we show that all local martingales with respect to the initially enlarged natural filtration of a vector of multivariate point processes can be weakly represented up to the minimum among the explosion times of the components. We also prove that a strong representation holds if any multivariate point process of the vector has almost surely infinite explosion time and discrete marks space. Then we provide a condition under which the components of the multidimensional local martingale driving the strong representation are pairwise orthogonal.

In this paper, we study stochastic partial differential equations with two reflecting smooth walls h¹ and h², driven by space-time white noise with non-constant diffusion coefficients. The existence and uniqueness of the solutions are established.

In this paper, we study elliptic stochastic partial differential equations with two reflecting walls h¹ and h², driven by multiplicative noise. The existence and uniqueness of the solutions are established.

Random field with paths given as restrictions of holomorphic functions to Euclidean space-time can be Wick-rotated by pathwise analytic continuation. Euclidean symmetries of the correlation functions then go over to relativistic symmetries. As a concrete example, convoluted point processes with interactions motivated from quantum field theory are discussed. A general scheme for the construction of Euclidean invariant infinite volume measures for systems of continuous particles with ferromagnetic interaction is given and applied to the models under consideration. Connections with Euclidean quantum field theory, Widom-Rowlinson and Potts models are pointed out. For the given models, pathwise analytic continuation and analytically continued correlation functions are shown to exist and to expose relativistic symmetries.

In the present paper we study multiply selfdecomposable probability measures (SDPM) and processes and prove their integral representations. Similarly, the multiple s-selfdecomposability case is treated. Our results extend some of known results due to Urbanik, K., Jurek, Z., Rosinski, J. and Rajput, B.S. As an application, following Cartea and Howinson ([1]) we introduce the Damped-Lévy-mixed - stable process which leads to a mathematical model for option pricing.

Since the first works laying its foundations as a subfield of Complex Analysis, the theory of reproducing kernels has proved to be a powerful tool in many fields of Pure and Applied Mathematics. The aim of this paper is to give some idea of how and why this theory interacts with Probability and Statistics.