Modeling Correlated Geometric Brownian and Ornstein–Uhlenbeck Processes

Summary

The document poses a stochastic modeling question: how to represent a geometric Brownian motion and an Ornstein–Uhlenbeck process whose innovations have a specified correlation. The first process models a quantity with proportional drift and diffusion, while the second mean-reverts toward a long-run level. Their dependence is expressed through correlated Brownian drivers.

This setup is relevant when modeling a price-like variable alongside a mean-reverting state variable, but the document itself provides no derivation, simulation scheme, references, or application. It states the intended dynamics and asks for a source, so it is best read as a modeling prompt rather than a worked method. In practice, specifying correlation between the driving noises is distinct from asserting a fixed correlation between the process levels, which also depends on their dynamics and initial conditions.

Key ideas

• A geometric Brownian motion and an Ornstein–Uhlenbeck process can be modeled with correlated Brownian innovations.
• The geometric Brownian process has proportional drift and volatility, while the Ornstein–Uhlenbeck process mean-reverts.
• Correlation between the noise drivers does not by itself specify a constant correlation between the process levels.
• The document poses the setup but does not provide a derivation or solution.

Tags

# Correlated GBM and OU processes


# Correlated GBM and OU processes












I want to model two different stochastic processes, such that:

$X_t , V_t$ are correlated with coefficient $\rho$. Where:

$\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\sigma_2 dW_{2,t}$.

Is there any source(book or paper) to address this problem?

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