Euler Discretization for Lévy-Driven Ornstein–Uhlenbeck Processes

Summary

The document asks how to simulate a mean-reverting Ornstein–Uhlenbeck process driven by a Lévy process. It proposes a basic Euler step: approximate the drift over each time interval using the current process value, then add an increment sampled from the Lévy process over that interval. This frames a central distinction from ordinary diffusion simulation: the driving noise increment should reflect the Lévy process’s distribution over the step, including its possible jumps.

The question also asks about convergence rates and alternative schemes, but supplies no answer, analysis, or empirical comparison. It therefore serves as a starting point for studying numerical simulation of jump or other Lévy-driven stochastic differential equations, rather than as a complete implementation guide. The suitability and accuracy of the proposed Euler approximation depend on the driver and the convergence criterion, none of which are examined here.

Key ideas

• The proposed Euler update uses the current state to approximate the Ornstein–Uhlenbeck drift over a time step.
• The stochastic term is an increment drawn from the Lévy driver over that interval.
• Lévy-driven simulation must account for the driver’s increment distribution, which may include jumps.
• The document raises convergence rates and alternative schemes as open questions without resolving them.

Tags

# How are Levy driven SDE simulated?


# How are Levy driven SDE simulated?












Do you just use an Euler scheme as before?

E.g. take this process, OU process with a Levy driver.

\begin{equation} \text{d}V_t = -\lambda V_t\text{d}t + dZ_t \end{equation}

Do you just have

$V_{t_{i+1}} = V_{t_i} - V_{t_i}\lambda \delta t + \delta Z$?

where $\delta_t = t_{i+1}-t_i$ and $\delta Z$ is drawn from the distribution of $Z_{\delta t} - Z_0$?

Does anyone have a reference for the convergence rate for the Euler scheme/reference for any other related scheme?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.