Say that we have the Vasicek model

$dY_{t} = \alpha(\beta-Y_{t})dt+\sigma dX_{t}$

where $X_{t}$ is an infinite activity Levy process, $\alpha$,$\beta$ and $\sigma$ are constants.

I know that in the simple case when it is driven by Brownian motion instead, one can solve it simply by using an integrating factor.

However, I am not very experienced with Levy driven SDE's.

1) Is it possible to say that $Y_{t}$ is an Ito-Levy process and use the Ito formula for Ito-Levy processes? Or is this only possible for jump diffusions? 2) Alternatively, is $Y_{t}$ a semi-martingale, such that I can proceed in a similar way as for the case of Brownian motion, but use a formula for semi-martingales instead?


By the Lévy-Ito decomposition, we have

$$X_t = \gamma t + \varrho B_t + \int_0^t \!\!\! \int_{|z|<1} z \tilde{N}(dz,ds) + \int_0^t \!\!\! \int_{|z| \geq 1} N(dz,ds)$$

for some constants $\gamma \in \mathbb{R}$, $\varrho \geq 0$, a Brownian motion $(B_t)_{t \geq 0}$ where $N$ ($\tilde{N}$) denotes the (compensated) jump measure of $X$. Consequently, the SDE

$$Y_t -Y_0 = \int_0^t \alpha(\beta-Y_{s-}) \, ds + \sigma X_t$$

is equivalent to

$$Y_t = \int_0^t\bigg(\alpha (\beta-Y_{s-})+\gamma \sigma \bigg) \, ds + \sigma \varrho B_t +\int_0^t \!\!\! \int \sigma z 1_{|z| <1} \tilde{N}(dz,ds) + \int_0^t \!\!\! \int\sigma z 1_{|z| \geq 1} N(dz,ds).$$

This shows that $(Y_t)_{t \geq 0}$ is a Lévy-Ito process (and, hence, in particular a semimartingale). This means that you may use any version of Itô's formula which applies to Lévy-Ito processes or semimartingales.

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