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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 $(\Omega,\cal {F,P})$. We consider the $n$-dimensional stochastic differential equation $$dx(t) = f(x(t), t)dt + g(x(t), t)dB(t),\quad t\ge 0,\tag{2.1}$$ with initial condition $x(t_0) = x_0\in\Bbb R^n$. 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.

34F05ODE with randomness
60H10Stochastic ordinary differential equations
34D20Stability of ODE
34C11Qualitative theory of solutions of ODE: growth, boundedness
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