Abstract
The paper deals with a system of nonlinear differential equations under the influence of white noise. This system can be used as a mathematical model of various real problems in finance, mathematical biology, climatology, signal theory and others. Necessary and sufficient conditions for the asymptotic mean square stability of the zero solution of this system are derived in the paper. The paper introduces a new approach to studying such problems - construction of a suitable deterministic system with the use of Lyapunov function.
Highlights
1 Introduction We can come across stochastic behavior while examining many important problems of a global character in various fields of research, for example, in the theory of climate change
Detailed understanding of extreme events in climate, of phenomena that are beyond our normal expectations, is a very important topic in climatology, meteorology and related fields
The idea of replacing the whole deterministic system with a stochastic differential equation was introduced by Hasselmann in his work [ ] on stochastic climate models that appeared in
Summary
We can come across stochastic behavior while examining many important problems of a global character in various fields of research, for example, in the theory of climate change. If there exists a neighborhood O(o), in which the function δ(x) = xT Cf (x) + hT (x)Cdh(x) is positive definite with respect to system ( ), the trivial solution of ( ) is unstable on the interval [ , ∞). If there exists a neighborhood O(o), in which the function n δ(t, x) = xT Cf (t, x) + hTk (t, x)Cdhk(t, x) is positive definite with respect to system ( ), the trivial solution of ( ) is unstable on the interval [ , ∞).
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