Stochastic Stability

  • Zeev Schuss
Part of the Applied Mathematical Sciences book series (AMS, volume 170)


The notion of stability in deterministic and stochastic systems is not the same. The solution \(\xi(t)\) of a deterministic system of differential equations
$$\dot{x}= b(x, t)$$
is stable if for any positive number \(\varepsilon\) there exist two numbers, \(\delta > 0\) and \(T\), such that for any solution \(x(t)\) of (11.1)
$$|x(t) -\xi(t)| < \varepsilon\ {\rm for}\ t \geq T,$$
$$|x(t_0) - \xi(t_0)| < \delta$$
for some \(t_0 \leq T\). The solution \(\xi(t)\) is said to be asymptotically stable if it is stable and, in addition,
$$\mathop{\lim}\limits_{t\to1} |x(t) - \xi(t)| = 0 $$
for any solution \(x(t)\) satisfying (11.2). If eq. (11.3) holds for all solutions of eq. (11.1), then \(\xi(t)\) is said to be globally stable.


Drilling Autocorrelation Librium 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Zeev Schuss
    • 1
  1. 1.Department of Applied MathematicsSchool of Mathematical Science Tel Aviv UniversityTel AvivIsrael

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