Bearwing's Blog: Closed Form Solution of Orstein-Uhlenbeck Process

The Ornstein-Uhlenbeck Process is a stochastic process defined by:

${\mathrm{dx_{t}} }=\theta \left ( \mu -x_t \right ){\mathrm{dt}}+\sigma{\mathrm{dW_t}}$

where W_t

is the Wiener Process, or the Standard Brownian Motion

The term $\mu$ is the long-term mean, whereas $\theta$ is the speed for returning the mean. $\sigma$ is the volatility caused by shocks from the Brownian Motion.

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To obtain the closed form solution for the Ornstein-Uhlenbeck Process, we applied the Ito's lemma to

$f(x_t,t)=x_te^{\theta t}$

(which is a very common technique for solving first order differential equation)

This yields

$\mathrm{d}f(x_t,t)=\theta x_te^{\theta t}\mathrm{d}t + e^{\theta t}\mathrm{d}x_t$

\\ There is no ${\mathrm{d} x_t^2}$ term because it becomes zero after differentiation.

$\mathrm{d}f(x_t,t)=\theta x_te^{\theta t}\mathrm{d}t + e^{\theta t}(\theta(\mu - x_t)\mathrm{d}t + \sigma \mathem{d}W_t)$

\\ Just by substituting the original equation into the new stochastic differential equation

$\mathrm{d}f(x_t,t)=e^{\theta t}\theta\mu\mathrm{d}t + \sigma e^{\theta t}\mathrm{d}W_t$

\\ Rearranging and subtracting terms

Then, we take integration on both sides of the equation:

$\int_{0}^{T} \mathrm{d}f(x_t,t)=\int_{0}^{T} e^{\theta t}\theta\mu\mathrm{d}t +\int_{0}^{T} \sigma e^{\theta t}\mathrm{d}W_t$

\\ For the first integral, we can integrate it directly since we have the anti-derivative of it (don't forget how we obtain the SDE, we just differentiate it!!!)

\\ For the second integral, just do normal integration since there is no stochastic term

\\ Leave the third integral and don't do anything

$x_T e^{\theta T} - x_0 e^{0} = \mu (e^{\theta T}-e^{0})+\sigma\int_{0}^{T} e^{\theta t} \mathrm{d}W_t$
\\ Calculate e^0 s and rearranging terms

Hence we obtain the closed form solution:

$x_T = x_0 + \mu (1-e^{-\theta T})+\sigma\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t$

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Mean

$E(x_T) = x_0 + \mu (1-e^{-\theta T})+\sigma E(\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t)$

\\ The problem is the mean of the last term. In a vague sense, the expected value can be taken as an integral and we may exchange the two integral.

$E(\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t) = \int_{0}^{T} e^{\theta (t-T)} E(\mathrm{d}W_t)$

\\ We transform the question so now we only need to know what is the expected value of the differential of the Wiener Process.

Note that

$\mathrm{d} W_t = W_{t+\mathrm{d}t} - W_t$

\\ We consider the change of the Wiener Process in a very short period of time, say, dt

$E(\mathrm{d} W_t) = E(W_{t+\mathrm{d}t} - W_t)$

\\ Note that the mean of Wiener Process is zero

$E(\mathrm{d} W_t) = 0$

Hence,

$E(x_T) = x_0 +\mu(1-e^{\theta T})$

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Variance

$Var(x_T) = Var\left (\sigma\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t \right )$

\\ Constant terms become zero after taking variance

$Var\left (\sigma\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t \right ) = \sigma^2 \int_{0}^{T} e^{2\theta (t-T)} Var(\mathrm{d}W_t) \right$

\\ Applying similar technique to exchange the integral. Don't forget the coefficient have to be squared after taking it out from variance

Note that

$\mathrm{d} W_t = W_{t+\mathrm{d}t} - W_t$

\\ We consider the change of the Wiener Process in a very short period of time, say, dt

$Var(\matherm{d}W_t) = Var(W_{t+\mathrm{d}t}-W_t)$

Since

$W_{t+\mathrm{d}t}-W_t \sim N(0,t+\mathrm{d}t-T) \sim N(0,\mathrm{d}t)$

Hence

$Var(\mathrm{d}W_t) = dt$

Finally we have

$Var\left (\sigma\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t \right ) = \sigma^2 \int_{0}^{T} e^{2\theta (t-T)} \mathrm{d}t$

After integration we have

$Var\left (\sigma\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t \right ) = \sigma^2 \left ( \frac{e^{2\theta (t-T)}}{2\theta} \right )_{0}^{T}$

Substituting into the integral

$Var\left (\sigma\int_{0}^{T} e^{\theta (t-T)} \mathrm{d}W_t \right ) = \sigma^2 \left ( \frac{1-e^{-2\theta T}}{2\theta} \right )$

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2013年4月27日星期六

Closed Form Solution of Orstein-Uhlenbeck Process

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2013年4月27日 星期六

Closed Form Solution of Orstein-Uhlenbeck Process

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張貼留言

2013年4月27日星期六