Abstract
Conditions for non-explosion, boundedness in probability and stability in probability of stochastic processes defined by the system of ODE with random coefficients are proven in this chapter.
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Notes
- 1.
Sometimes (see Chap. 3), but only when this is explicitly mentioned, we shall find it convenient to consider random variables which can take on the values ±∞ with positive probability.
- 2.
General conditions for every solution to be unboundedly continuable have been obtained by Okamura and are described in [178]. These results imply Theorem 1.3.
- 3.
See [285].
- 4.
The author’s exposition of this example in [121] contains an error. The following corrected version is due to Nevelson.
- 5.
Throughout this chapter we shall consider stability and asymptotic stability in the weak sense (compare Chap. 5, where stability in the strong sense will be discussed).
- 6.
We denote (see Sect. 1.1)
$$K(s,t)=((K^{ij}(s,t)))=\operatorname{cov}(\xi(s),\xi(t)).$$ - 7.
Theorem 1.16 generalizes a result of Shur [259].
- 8.
It is clear that when α=α ∗ the argument of the exponential function in (1.106) attains its minimum.
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Khasminskii, R. (2012). Boundedness in Probability and Stability of Stochastic Processes Defined by Differential Equations. In: Stochastic Stability of Differential Equations. Stochastic Modelling and Applied Probability, vol 66. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23280-0_1
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