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Abstract

Definition 4.1. Let \( (W_t )_{t \in \mathbb{R}_{\text{ + }} } \) be a Wiener process on the probability space \( \left( {\Omega ,\mathcal{F},P} \right) \) , equipped with the filtration \( \left( {\mathcal{F}_t } \right)t \in \mathbb{R}_ + ,\mathcal{F}_t = \sigma \left( {W_s ,0 \leqslant s \leqslant t} \right) \) . Furthermore, let a(t,x), b(t,x) be measurable functions in [0, T] × ℝ and \( (u(t))t \in \left[ {0,T} \right] \) a stochastic process. Now u(t) is said to be the solution of the stochastic differential equation
$$ du\left( t \right) = a\left( {t,u\left( t \right)} \right)dt + b\left( {t,u\left( t \right)} \right)dW_t $$
(1)
with the initial condition u(0) = uO a.s. (u0 a random variable), (4.2) if
  1. 1.

    u(0) is \( \mathcal{F}_0 \) -measurable;

     
  2. 2.

    \( \left| {a\left( {t,u\left( t \right)} \right)} \right|^{\frac{1} {2}} ,b\left( {t,u\left( t \right)} \right) \in \mathcal{C}_1 \left( {\left[ {0,T} \right]} \right) \)

     
  3. 3.

    u(t) is differentiable and du(t) = a(t,u(t))dt + b(t,u(t))dWt, thus \( u\left( t \right) = u\left( 0 \right) + \int_0^t {a\left( {s,u\left( s \right)} \right)} ds + \int_0^t b \left( {s,u\left( s \right)} \right)dW_{s,} t \in \left] {0,T} \right] \) .

     

Keywords

Brownian Motion Lyapunov Function Wiener Process Exit Time Global Asymptotic Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2005

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