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1
Content available The impact of two independent gaussian white noises on the behavior of a stochastic epidemic model
Yavuz Mehmet, Boulaasair Lahcen, Bouzahir Hassane, Diop Mamadou Abdoul, Benaid Brahim
Journal of Applied Mathematics and Computational Mechanics
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2024
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Vol. 23, nr 1
121-134
EN
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.
2
Content available remote Continuous-state branching processes with migration
Vidmar Matija
Probability and Mathematical Statistics
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2022
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Vol. 42, Fasc. 2
227--249
EN
Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties of these processes are described. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.
3
Content available Stochastic model of drug concentration level during IV-administration
Dzhalladova Irada, Růžičková Miroslava
Opuscula Mathematica
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2022
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Vol. 42, no. 6
833--847
EN
A stochastic model describing the concentration of the drug in the body during its IV-administration is discussed. The paper compares a deterministic model created with certain simplifications with the stochastic model. Fluctuating and irregular patterns of plasma concentrations of some drugs observed during intravenous infusion are explained. An illustrative example is given with certain values of drug infusion rate and drug elimination rate.
4
Content available remote Distribution Tails for Solutions of Sde Driven By An Asymmetric Stable Lévy Process
Eon Richard, Gradinaru Mihai
Probability and Mathematical Statistics
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2020
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Vol. 40, Fasc. 2
317--330
EN
The behaviour of the tails of the invariant distribution for stochastic differential equations driven by an asymmetric stable Lévy process is obtained. We generalize a result by Samorodnitsky and Grigoriu [8] where the stable driving noise was supposed to be symmetric.
5
Content available remote A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion
Jum E., Kobayashi K.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 2
201--220
EN
This paper establishes a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift, where the time change is given by a general inverse subordinator. The scheme involves two types of errors: one generated by application of the Euler-Maruyama scheme and the other ascribed to simulation of the inverse subordinator. With the two errors carefully examined, the orders of strong and weak convergence are established. In particular, an improved error estimate for the Euler-Maruyama scheme is derived, which is required to guarantee the strong convergence. Numerical examples are attached to support the convergence results.
6
Content available Ruin probability in a risk model with variable premium intensity and risky investments
Mishura Y., Perestyuk M., Ragulina O.
Opuscula Mathematica
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2015
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Vol. 35, no. 3
333--352
EN
We consider a generalization of the classical risk model when the premium inten­sity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.
7
Content available On potential kernels associated with random dynamical systems
Hmissi M., Mokchaha F., Hmissi A.
Opuscula Mathematica
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2015
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Vol. 35, no. 4
499--515
EN
Let (Θ, φ) be a continuous random dynamical system defined on a probability space (Ω, F, P) and taking values on a locally compact Hausdorff space E. The associated potential kernel V is given by [formula]. In this paper, we prove the equivalence of the following statements: 1. The potential kernel of (Θ, φ) is proper, i.e. V ƒ is x-continuous for each bounded, x-continuous function with uniformly random compact support. 2. (Θ, φ) has a global Lyapunov function, i.e. a function L : Ω x E → (0, ∞) which is x-continuous and L,(Θ tω, φ(t, ω)x) ↓0 as t ↑ ∞. In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems.
8
Content available remote Application of Stochastic Differential Equations in Second-Order Electrical Circuits Analysis
Kolářová E., Brančík L.
Przegląd Elektrotechniczny
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2012
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R. 88, nr 7a
103-107
EN
The paper deals with an unconventional approach to the analysis of electrical circuits with randomly varying parameters based on stochastic differential equations (SDE). A response of the electrical circuit is computed in the form of a sample mean with a particular confidence interval to provide credible estimate of the result. The method is applied to get voltage and current responses of the second-order RLGC network excited from a voltage source with a noise term. The results are compared with the classical deterministic state-variable approach.
PL
W artykule przedstawiono niekonwencjonalna metodę analizy obwodu elektrycznego o przypadkowo zmienianych parametrach. Metoda bazuje na stochastycznych równaniach różnicowych SDE. Metodę sprawdzono na przykładzie określania napięcia i prądu w sieci drugiego rzędu RLGC zasilanej napięciem z szumami.
9
Content available remote Discrete approximations of strong solutions of reflecting SDEs with discontinuous coefficients
Semrau A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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Vol. 57, no 2
169-180
EN
We study Lp convergence for the Euler scheme for stochastic differential equations reflecting on the boundary of a general convex domain D ⊆ Rd. We assume that the equation has the pathwise uniqueness property and its coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. In the case D = [0, ∞) new sufficient conditions ensuring pathwise uniqueness for equations with possibly discontinuous coefficients are given.
10
Content available remote Euler's approximations of weak solutions of reflecting SDEs with discontinuous coefficients
Semrau A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
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Vol. 55, no 2
171-182
EN
We study convergence in law for the Euler and Euler-Peano schemes for stochastic differential equations reflecting on the boundary of a general convex domain. We assume that the coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. The proofs are based on new estimates of Krylov's type for the approximations considered.
11
Content available remote Existence of strong solutions to stochastic differential equations in conuclear spaces
Jakubowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
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Vol. 48, no 3
247-252
EN
The existence of an unique strong solutions to stochastic differential equations with respect to a generalized non-homogeneous Wiener process in the dual of a nuclear space is proved under monotonicity condition and conditions which guarantee e~stence of weak solutions.
12
Content available remote Unicité Trajectorielle des Équations Différentielles Stochastiques Avec Temps Local
Ouknine Y.
Probability and Mathematical Statistics
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1999
|
Vol. 19, Fasc. 1
55--62
EN
We study the pathwise uniqueness of a one-dimensional stochastic differential equation driven by white noise and involving local time of the unknown process. We introduce a very weak condition on the diffusion term which is sufficient for the pathwise uniqueness if one considers an equation of the form [formula] where v stands for a signed Radon measure on R.
13
Content available remote Sur l’unicité forte des solutions d’une equation différentielle stochastique
Zili M.
Probability and Mathematical Statistics
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1998
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Vol. 18, Fasc. 2
335--350
EN
In this paper we prove the pathwise uniqueness of solutions of a stochastic differential equation with a singular drift which depends on time. Our method is of probabilistic nature, and it is based on an Al-Hussaini and Elliott result.
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