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I have a question about how to apply the Euler approximation on OU process with a jump process. The stochastic process $X_t$ has dynamic

$$dX_t=\alpha(\beta-X_t)dt+\sigma dW_t+dY_t$$ where $dY_t=JdN_t$,

$J$~$N(\mu_J,\sigma_J)$, and $N_t$ is a Poisson process with constant intensity $\lambda$

After applying the Ito's lemma, I have

$$X_t=X_s e^{-\int_s^t \alpha du}+\int_s^te^{-\int_u^t \alpha dv} \alpha \beta du+ \int_s^t e^{-\int_u^t \alpha dv} \sigma dW_u+\int_s^t e^{-\int_s^t \alpha dv} dY_u$$

However, I don't know how to discretize the Jump part ($\int_s^t e^{-\int_s^t \alpha dv} dY_u$) ?

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    $\begingroup$ What is $Y$? Hasnt been defined $\endgroup$ – Sanjay Aug 8 at 4:12
  • $\begingroup$ Sorry, I forgot to define Y. I have modified my question. $\endgroup$ – Geoff Chen Aug 8 at 4:24
  • $\begingroup$ I would work with the first equation if you have to discritisize and perform a Monte Carlo valuation, since the term $dN_t$ can be seen for each step as a rv taking 1 with probability $1-\lamda dt$ and 0 otherwise. $\endgroup$ – alexbougias Aug 10 at 19:37

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