Adding Poisson Jumps to an Ornstein–Uhlenbeck Process

Summary

This note asks how to extend a Vasicek-style Ornstein–Uhlenbeck process with a compound Poisson jump component for Monte Carlo path simulation. It states the standard mean-reverting diffusion setup, identifies the mean-reversion speed, long-run mean, and diffusion scale, then proposes jumps arriving at a Poisson rate with independent normally distributed sizes. The author wants to know how to incorporate the jump component's expectation and variance into the drift and diffusion calculations.

A second question concerns modeling positive and negative jumps separately, potentially with distinct jump components. The document presents the model specification and questions but does not derive the resulting conditional moments, discretization, or simulation algorithm. It also does not clarify whether the jump process should be compensated or how jump-size distributions should be calibrated. Thus it serves as a prompt for stochastic-process modeling rather than a complete recipe; implementation requires resolving those assumptions and checking the resulting path behavior.

Key ideas

• The baseline process mean-reverts toward a long-run level and includes a Brownian diffusion term.
• The proposed extension adds compound Poisson jumps with a specified arrival rate and random jump sizes.
• The author asks how jumps change the process expectation and variance for Monte Carlo simulation.
• Separate jump components are proposed as a possible way to represent positive and negative jumps.
• The note leaves compensation conventions, moment derivations, calibration, and implementation unresolved.

Tags

# Ornstein-Uhlnbeck Process with Jumps


# Ornstein-Uhlnbeck Process with Jumps












I am trying to simulate an OU Process (Vasicek version) with jumps and I would like to derive the drift and diffusion term when jumps are incorporated, which will enable me to perform monte carlo simulations.

In the absence of jumps we have the following drift:

$$ X_0e^{-kT}+v(1-e^{-kT}) $$ where $k$ mean reversion and $v$ long term mean and the following diffusion:

$$ σ\sqrt{\frac{(1-e^{-kT})}{2κ}} $$

with the multiplication of a standard normal we can simulate via discretization the paths as shown below (having solved Ito's Lemma etc.)

$$ X_0e^{-kT}+v(1-e^{-kT}) + σ\sqrt{\frac{(1-e^{-kT})}{2κ}} N(0,1) $$ Lets leave aside any parameter estimation, say we have estimated $k,v,σ$.

Now instead of the above which comes from the equation

$$ dX_t=k(v-X_t)dt+σdW_t $$ we want to estimate the drift and diffusion of the same process but with added Poisson jumps

$$ dX_t=k(v-X_t)dt+σdW_t+dJ_t $$

where J is the jump component deriving from a Poisson process with intensity $λ$ (say 20 jumps/year) and a jump size distribution $N(μ_{jump},σ_{jump})$.

I have seen that the $J$ component is defined as: $$ J_t=\sum_{i=1}^{N_i} Z_i $$

where $N_i$ is a Poisson process with mean rate $λ$ and $Z_i$ are i.i.d jump sizes.

So, the ask - to my understanding - is to get some help on how to derive the expectation and variance of the Poisson process and add them to the vanilla case drift and diffusion respectively.

And as a second question - is there any way to model negative and positive only jumps seperately? Maybe we would need two Jump components say $J_1$ and $J_2$, the former with negative $μ_{jump}$ and the latter with positive only $μ_{jump}$?

Thanks in advance

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