# Discretisation of OU (mean reverting) process with a jump process

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 ($$\int_s^t e^{-\int_s^t \alpha dv} dY_u$$) ?

• What is $Y$? Hasnt been defined – Sanjay Aug 8 at 4:12
• Sorry, I forgot to define Y. I have modified my question. – Geoff Chen Aug 8 at 4:24
• 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. – alexbougias Aug 10 at 19:37