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On random processes as an implicit solution of equations

Petr Lachout (2017)

Kybernetika

Random processes with convenient properties are often employed to model observed data, particularly, coming from economy and finance. We will focus our interest in random processes given implicitly as a solution of a functional equation. For example, random processes AR, ARMA, ARCH, GARCH are belonging in this wide class. Their common feature can be expressed by requirement that stated random process together with incoming innovations must fulfill a functional equation. Functional dependence is...

On risk reserve under distribution constraints

Mariusz Michta (2000)

Discussiones Mathematicae Probability and Statistics

The purpose of this work is a study of the following insurance reserve model: R ( t ) = η + 0 t p ( s , R ( s ) ) d s + 0 t σ ( s , R ( s ) ) d W s - Z ( t ) , t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: i n f 0 t T P R ( t ) c γ is considered.

On robust consensus of multi-agent systems with communication delays

Jiangping Hu (2009)

Kybernetika

In this paper, two robust consensus problems are considered for a multi-agent system with various disturbances. To achieve the robust consensus, two distributed control schemes for each agent, described by a second-order differential equation, are proposed. With the help of graph theory, the robust consensus stability of the multi-agent system with communication delays is obtained for both fixed and switching interconnection topologies. The results show the leaderless consensus can be achieved with...

On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model

Jacek Jakubowski, Maciej Wiśniewolski (2013)

Studia Mathematica

We find a probabilistic representation of the Laplace transform of some special functional of geometric Brownian motion using squared Bessel and radial Ornstein-Uhlenbeck processes. Knowing the transition density functions of these processes, we obtain closed formulas for certain expectations of the relevant functional. Among other things we compute the Laplace transform of the exponent of the T transforms of Brownian motion with drift used by Donati-Martin, Matsumoto, and Yor in a variety of identities...

On some problem of A. Rosłanowski

Szymon Plewik (1996)

Colloquium Mathematicae

We present a negative answer to problem 3.7(b) posed on page 193 of [2], where, in fact, A. Rosłanowski asked: Does every set of Lebesgue measure zero belong to some Mycielski ideal?

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