Expected value of a mean-reverting process

MediumStochastic Processes~4m

An Ornstein–Uhlenbeck process obeys $\mathrm{d}X_t = \theta(\mu - X_t)\,\mathrm{d}t + \sigma\,\mathrm{d}W_t$ with reversion speed $\theta = 0.5$ , long-run level $\mu = 10$ , and volatility $\sigma = 3$ . It starts at $X_0 = 4$ . What is $E[X_2]$ (the expected value at time $t=2$ , to four decimals)?

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Take expectations: $\mathrm{d}m/\mathrm{d}t=\theta(\mu-m)$ (noise drops out).
$m(t)=\mu+(X_0-\mu)e^{-\theta t}$ ; here $\theta t=1$ .

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7.7927 (± 0.001)

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Expected value of a mean-reverting process

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