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I am wondering how we can solve the SDE:

$$ dX_t = -\beta(X_t-\alpha)dt + \sigma dW_t $$

where $W_t$ is a Wiener process, and $\beta >0$, $\alpha \in \mathbb{R}, \sigma>0$ are constants. It seems that we can define it according to the OU process, but with a substitution. Is a substitution doable?

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1 Answer 1

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HINT :

Introduce the process $Y_t=e^{\beta t}X_t$. When you apply Ito's lemma , you have

$$dY_t=\beta Y_tdt+e^{\beta t}dX_t=\beta Y_tdt+ e^{\beta t}(-\beta(X_t-\alpha)dt + \sigma dW_t)$$

Thus, $$dY_t=\alpha e^{\beta t}+e^{\beta t}\sigma dW_t$$

$$Y_t=Y_0+\int_{0}^{t}\alpha e^{\beta u}du+\int_{0}^{t}{\sigma e^{\beta u}dW_u}$$

Finally, you rewrite it in terms of $X_t$

$$X_t=X_0e^{-\beta t} +\int_{0}^{t}\alpha e^{-\beta(t- u)}du+\int_{0}^{t}{\sigma e^{-\beta(t-u)}dW_u}$$

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