An Optimal Treatment Strategy for Diabetes

2003 ◽  
Vol 11 (04) ◽  
pp. 419-425 ◽  
Author(s):  
Farai Nyabadza ◽  
Edward M. Lungu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Stochastic Control ◽  
Treatment Strategy ◽  
Sufficient Conditions ◽  
Optimal Treatment ◽  
The Body ◽  
Optimal Treatment Strategy

Consider a system on n variables involved in the regulation of glucose in the body, whose concentrations are given by stochastic differential equations driven by m-dimensional Brownian motion. We formulate a stochastic control problem and give sufficient conditions for the existence of an optimal treatment strategy. We study the following problem: what treatment strategy for the n variables, maximizes the expected benefit from treatment.

scholarly journals Maxima of stochastic processes driven by fractional Brownian motion

2005 ◽  
Vol 37 (03) ◽  
pp. 743-764 ◽  
Author(s):  
Boris Buchmann ◽  
Claudia Klüppelberg
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Maximum Domain Of Attraction

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

scholarly journals T-stability of the Euler method for impulsive stochastic differential equations driven by fractional Brownian motion

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6493-6503
Author(s):  
Hui Yu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Euler Method ◽  
Variable Step Size ◽  
Step Size ◽  
Theoretical Results

Due to the fact that a fractional Brownian motion (fBm) with the Hurst parameter H ? (0,1/2) U(1/2, 1) is neither a semimartingale nor a Markov process, relatively little is studied about the T-stability for impulsive stochastic differential equations (ISDEs) with fBm. Here, for such linear equations with H ? (1/3, 1/2), by means of the average stability function, sufficient conditions of the T-stability are presented to their numerical solutionswhich are established fromthe Euler-Maruyama method with variable step-size. Moreover, some numerical examples are presented to support the theoretical results.

scholarly journals Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Caibin Zeng ◽  
Qigui Yang ◽  
YangQuan Chen
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Stochastic Stabilization ◽  
Lyapunov Techniques ◽  
Moment Exponential Stability ◽  
Stochastic Lyapunov Function

Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

scholarly journals Maxima of stochastic processes driven by fractional Brownian motion

2005 ◽  
Vol 37 (3) ◽  
pp. 743-764 ◽  
Author(s):  
Boris Buchmann ◽  
Claudia Klüppelberg
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Maximum Domain Of Attraction

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

Evidence-based optimal treatment strategy of patients with angina pectoris

2008 ◽  
Vol 149 (7) ◽  
pp. 293-298
Author(s):  
András Jánosi
Keyword(s):  
Angina Pectoris ◽  
Treatment Strategy ◽  
Optimal Treatment ◽  
Evidence Based ◽  
Optimal Treatment Strategy

A szerző összefoglalja a stabil angina pectoris optimális kezelésével kapcsolatos evidenciákat. Az invazív kezelési stratégia (katéterterápia) térnyerése a stabil angina pectoris esetében is megfigyelhető. Számos országban – így hazánkban is – a percutan intervenciók száma meghaladja a műtéti beavatkozások gyakoriságát. A percutan intervenció, illetve a revascularisatiós műtét helyének meghatározása az angina pectoris kezelésében igen fontos és sokszor vitatott klinikai probléma. A szerző áttekinti a lehetséges három kezelési mód (gyógyszeres kezelés, percutan intervenció, revascularisatiós műtét) eredményességét vizsgáló randomizált tanulmányokat, illetve az ezekből levonható következtetéseket. A rendelkezésre álló adatok azt igazolják, hogy – a diagnózis objektív módszerrel is alátámasztott felállítása után első lépésben – a rizikófaktorok korrekciója, az életmódrendezés és az optimális gyógyszeres kezelés a választandó kezelési stratégia. Optimális gyógyszeres kezelésnek tekintjük a statin-, aszpirin-, ACE-gátló-terápiát és a tünetek befolyásolására irányuló antianginás kezelést, amelyben a béta-blokkoló alkalmazásának elsőrendű jelentősége van. A percutan intervenció első terápiás eszközként történő alkalmazása nem indokolt, mivel nincs adat arra, hogy javítaná az életkilátásokat, illetőleg a beavatkozással megelőzhető lenne a szívinfarktus. Amennyiben a panaszok gyógyszeres kezeléssel nem vagy nem eléggé befolyásolhatók, indokolt a revascularisatio (percutan intervenció vagy műtét) elvégzése, mivel e beavatkozásokkal a gyógyszeres kezelésnél jobban javítható a funkcionális stádium. A revascularisatiós műtét bizonyos esetekben (pl. főtörzsszűkület, háromérbetegség és csökkent balkamra-funkció) a panaszok kedvező befolyásolásán túlmenően a betegek életkilátásait is javítja. A rendelkezésre álló kezelési lehetőségek optimális megválasztása nemcsak a betegek számára fontos, hanem komoly gazdasági jelentősége is van.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

scholarly journals Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 988
Author(s):  
Pengju Duan
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Mild Solution ◽  
Intermittent Control

The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefficients, when intermittent intervals and coefficients satisfy suitable conditions; by use of the G-Lyapunov function, the p-th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.

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