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Boundedness and exponential stability of highly nonlinear stochastic differential equations. (English) Zbl 1186.34081

Summary: Let \(B(t)=(B_1(t),B_2(t),\dots,B_m(t))^T\) be an \(m\)-dimensional standard Brownian motion defined on a complete probability space \((\Omega,\mathcal {F,P})\). We consider the \(n\)-dimensional stochastic differential equation
\[ dx(t) = f(x(t), t)dt + g(x(t), t)dB(t),\quad t\geq 0,\tag{2.1} \]
with initial condition \(x(t_0) = x_0\in\mathbb R^n\).
We use Lyapunov functions to study the boundedness and exponential asymptotic stability of solutions. We provide several examples in which we consider stochastic systems with unbounded terms.

MSC:

34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34D20 Stability of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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