Discretizing an Ornstein–Uhlenbeck Process with Poisson Jumps

Summary

The document poses a numerical modeling question about discretizing a mean reverting Ornstein–Uhlenbeck process that combines continuous Brownian noise with random jumps. The jump component is modeled as a normally distributed jump size multiplied by a Poisson counting process with constant intensity. The author writes an integral solution and asks how to handle the jump term in a time step approximation.

No answer or discretization scheme is included, so the document does not establish a particular method, compare Euler approximation with an exact transition, or give empirical evidence. Its useful content is the setup: jump arrivals are discrete events, and their contribution must be incorporated alongside the mean reversion and diffusion components. Any implementation would need to account for the number and timing of jumps within each interval, details that remain unresolved in the text.

Key ideas

• The process combines Ornstein–Uhlenbeck mean reversion, Brownian diffusion, and Poisson jumps.
• Jump sizes are specified as normally distributed, with arrivals governed by constant Poisson intensity.
• The author asks how to represent the jump integral in a discrete time step.
• The document provides no answer, numerical scheme, or evidence for a preferred discretization.

Tags

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


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

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