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Ornstein-Uhlenbeck Equation 📂Stochastic Differential Equations

Ornstein-Uhlenbeck Equation

Definition 1

dXt=aXtdt+σdWt d X_{t} = a X_{t} dt + \sigma d W_{t} Let a,σ∈Ra , \sigma \in \mathbb{R}. The stochastic differential equation given above is called the Ornstein-Uhlenbeck Equation, and its solution, the stochastic process XtX_{t}, is called the Ornstein-Uhlenbeck Process. Xt=X0eat+σ∫0tea(t−s)dWs 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<0a < 0, then the sign of XtX_{t} and aXtdta X_{t} dt reverses, resulting in stationary movement. When Xt>0X_{t} > 0, it tends to decrease, and when Xt<0X_{t} < 0, it tends to increase, meaning XtX_{t} always tries to return to 00. Generalizing this for μ∈R\mu \in \mathbb{R}, we obtain an SDE that has a Mean-reverting Ornstein-Uhlenbeck Process as its solution. dXt=a(Xt−μ)dt+σdWt,a<0 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 μ\mu.

Solution

dXt=aXtdt+σdWt d X_{t} = a X_{t} dt + \sigma d W_{t} Multiply both sides by e−ate^{-a t} to obtain the following. e−atdXt=e−atXt+σe−atdWt e^{-a t}d X_{t} = e^{-a t} X_{t} + \sigma e^{-a t} d W_{t} If we compute d(e−atXt)d \left( e^{-a t} X_{t} \right), then d(e−atXt)dt=−ae−atXt+e−atdXtdt  ⟹  d(e−atXt)=−ae−atXtdt+e−atdXt {{ 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, e−atdXt=ae−atXtdt+d(e−atXt)=ae−atXtdt+σe−atdWt \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 ae−atXtdta e^{-a t} X_{t} dt and taking ∫0t\int_{0}^{t} results in the following. d(e−atXt)=σe−atdWt  ⟹  ∫0td(e−asXs)=∫0tσe−asdWs  ⟹  e−atXt−X0=∫0tσe−asdWs  ⟹  e−atXt=X0+∫0tσe−asdWs  ⟹  Xt=eatX0+∫0tσea(t−s)dWs \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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  1. Øksendal. (2003). Stochastic Differential Equations: An Introduction with Applications: p74. ↩︎

  2. Panik. (2017). Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling: p144. ↩︎