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Two inequalities for the first moments of a martingale, its square function and its maximal function

100%

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

tom Vol. 53, no 4

441-449

Given a Hilbert space valued martingale (Mn), let (M∗n) and (Sn (M)) denote its maximal function and square function, respectively. We prove that E|Mn |≤ 2ESn (M), n = 0,1,2,…, EM∗n ≤ E|Mn| + 2ESn (M), n = 0,1,2,…. The first inequality is sharp, and it is strict in all nontrivial cases.

Information pricing for portfolio optimization

86%

Banek T. , Kulikowski R.

Control and Cybernetics

2003

tom Vol. 32, no 4

867-882

We consider the following problem: is there a rational or fair price for the reports made by analysts, experts, investor advisers concerning the rate of return (RR) of investments? We define the notion of the value of information included in the family of probability distributions of the RR. Next, we illustrate this notion for a linear-quadratic utility function.

Invariance principle for a diffusion in a Markov field

86%

Fannjiang A. , Komorowski T.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

tom Vol. 49, no 1

45--65

We study a diffusion with a random, time Markovian drift and non-vanishing diffusivity. We prove the invariance principle when the drift possesses certain decorrelation properties both in time and space.

Sharp norm inequalities for martingales and their differential subordinates

72%

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom Vol. 55, no 4

373-385

Suppose ƒ = (ƒn), g = (gn) are martingales with respect to the same filtration, satisfying |ƒn-ƒn-i| ≤ |gn -gn-1|, n = 1,2,..., with probability 1. Under some assumptions on ƒo, go and an additional condition that one of the processes is nonnegative, some sharp inequalities between the pth norms of ƒ and g, 0 < p < ∞, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.

The impact of two independent gaussian white noises on the behavior of a stochastic epidemic model

58%

Yavuz M. , Boulaasair L. , Bouzahir H. , Diop M. A. , Benaid B.

Journal of Applied Mathematics and Computational Mechanics

2024

tom Vol. 23, nr 1

121-134

The aim of this paper is to investigate a stochastic SIS (Susceptible, Infected, Susceptible) epidemic model in which the disease transmission coefficient and the death rate are subject to random disturbances. Using the convergence theorem for local martingales and solving the Fokker-Planck equation associated with the one-dimensional stochastic differential equation, we demonstrate that the disease will almost surely persist in the mean. In the case of global asymptotic stability of the endemic equilibrium for a SIS deterministic epidemic model, we formulate suitable conditions guaranteeing that the stochastic SIS model has a unique ergodic stationary distribution. Furthermore, we deal with the exponential extinction of the disease. Finally, some numerical simulations are provided to illustrate the obtained analytical results.

Weak predictable processes in finite von Neumann algebras

58%

Mohammed A.

Demonstratio Mathematica

2006

tom Vol. 39, nr 3

553-570

We investigate the predictable processes as a strong limit of a sequence of simple processes and their stochastic integrals as L2-limit of stochastic integrals of a sequence of simple processes. In the second case new class of predictable processes is defined and characterized, the mutual relations between the von Neumann algebras generated by the various classes of predictable processes are examined.