Ornstein-Uhlenbeck Equation
Definition 1
$$ d X_{t} = a X_{t} dt + \sigma d W_{t} $$ Let $a , \sigma \in \mathbb{R}$. The stochastic differential equation given above is called the Ornstein-Uhlenbeck Equation, and its solution, the stochastic process $X_{t}$, is called the Ornstein-Uhlenbeck Process. $$ X_{t} = X_{0} e^{a t} + \sigma \int_{0}^{t} e^{a (t-s)} d W_{s} $$
Description 2
The Ornstein-Uhlenbeck equation is also known as the Langevin Equation.
If $a < 0$, then the sign of $X_{t}$ and $a X_{t} dt$ reverses, resulting in stationary movement. When $X_{t} > 0$, it tends to decrease, and when $X_{t} < 0$, it tends to increase, meaning $X_{t}$ always tries to return to $0$. Generalizing this for $\mu \in \mathbb{R}$, we obtain an SDE that has a Mean-reverting Ornstein-Uhlenbeck Process as its solution. $$ d X_{t} = a \left( X_{t} - \mu \right) dt + \sigma d W_{t} \qquad , a < 0 $$ This Ornstein-Uhlenbeck process can be seen as a Brownian motion fluctuating around the mean $\mu$.
Solution
$$ d X_{t} = a X_{t} dt + \sigma d W_{t} $$ Multiply both sides by $e^{-a t}$ to obtain the following. $$ e^{-a t}d X_{t} = e^{-a t} X_{t} + \sigma e^{-a t} d W_{t} $$ If we compute $d \left( e^{-a t} X_{t} \right)$, then $$ {{ d \left( e^{-a t} X_{t} \right) } \over { dt }} = -a e^{-a t} X_{t} + e^{-a t} {{ d X_{t} } \over { dt }} \\ \implies d \left( e^{-a t} X_{t} \right) = -a e^{-a t} X_{t} dt + e^{-a t} d X_{t} $$ So, $$ \begin{align*} e^{-a t}d X_{t} =& a e^{-a t} X_{t} dt + d \left( e^{-a t} X_{t} \right) \\ =& a e^{-a t} X_{t} dt + \sigma e^{-a t} d W_{t} \end{align*} $$ Arranging $a e^{-a t} X_{t} dt$ and taking $\int_{0}^{t}$ results in the following. $$ \begin{align*} & d \left( e^{-a t} X_{t} \right) = \sigma e^{-a t} d W_{t} \\ \implies& \int_{0}^{t} d \left( e^{-a s} X_{s} \right) = \int_{0}^{t} \sigma e^{-a s} d W_{s} \\ \implies& e^{-a t} X_{t} - X_{0} = \int_{0}^{t} \sigma e^{-a s} d W_{s} \\ \implies& e^{-a t} X_{t} = X_{0} + \int_{0}^{t} \sigma e^{-a s} d W_{s} \\ \implies& X_{t} = e^{a t} X_{0} + \int_{0}^{t} \sigma e^{a (t-s)} d W_{s} \end{align*} $$
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