Abstract
In Chap. 1 we studied problems of stability under random perturbations of the parameters. We noted there that no significant results can be expected unless the random perturbations possess sufficiently favorable mixing properties. Fortunately, in practical applications one may often assume that the “noise” has a “short memory interval.” The natural limiting case of such noise is of course white noise. Thus it is very important to study the stability of solutions of Itô equations since this is equivalent to the study of stability of systems perturbed by white noise. Generalization of well known results on stability and instability for the deterministic ODE in terms of the Lyapunov functions are proven for SDE. Conditions for stability and instability of moments are also proven.
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Notes
- 1.
See [221].
- 2.
- 3.
- 4.
- 5.
An analogous condition for deterministic systems was considered by Krasovskii [155, p. 23] in connection with the inversion of Lyapunov’s theorems on asymptotic stability and instability.
- 6.
We shall continue to use this term in this sense in the sequel, though it is somewhat vague. For example, if a stochastic process X(t) is a “continuous approximation” to a discrete Markov chain which is the solution of a finite-difference equation, it is natural to use an Itô equation (see [251]).
- 7.
If Φ(x 1,…,x l ,t) is a random variable depending on the parameters x 1,…,x l ,t, its partial derivative in mean square with respect to x i is defined as the random variable \(\frac{\partial\varPhi}{\partial x_{i}}(x_{1},\dots ,x_{l},t)\) such that
- 8.
This estimate follows from the obvious inequality (u>0)
$$\frac{|u^{\beta/2}-1-(\beta/2)u^{\beta/2-1}(u-1)|}{(u-1)^2}<k(u^{\beta/2-2}+1),$$if we set u=(y+Δy)/y.
- 9.
After the first edition of this book the study of p-stability became very popular, see Appendix B, and references therein.
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Khasminskii, R. (2012). Stability of Stochastic 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_5
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