Reducing Two Correlated Brownian Drivers to One Effective SDE Factor

Summary

The document considers estimating parameters in a process driven by two Brownian motions, with constant drift and separate diffusion coefficients. The accepted response points out that the two drivers can be combined into a single effective Brownian factor when their correlation is accounted for. The resulting diffusion scale is determined by both individual coefficients and their correlation.

This reduction means the observed process, as written, has the same one-factor form as a process with a single Brownian motion. The response therefore questions why two drivers are needed before discussing estimation. The document does not give a parameter-estimation procedure, data requirements, or empirical results, and it does not explore cases where the drivers have distinct effects on multiple observed state variables. Its contribution is a model-identification insight: for this scalar process, the two noise terms are not separately identifiable from the combined diffusion without additional structure or information.

Key ideas

• Two correlated Brownian terms in a scalar process can be combined into one effective Brownian driver.
• The effective diffusion scale depends on both coefficients and the correlation between the drivers.
• The stated process has a one-factor representation under the assumptions in the response.
• Separate estimation of the two coefficients requires additional structure beyond this single observed process.

Tags

# SDE Parameter Estimation


# SDE Parameter Estimation












Have a question about "How to estimate parameters for SDE with multiple Brownian Motions ?" Let's say $X_t$ follows the process: $dX_t=\mu dt+\sigma_1 dW_t^1 + \sigma_2 dW_t^2 $

I think I've checked Sim.DiffProc for R and SDE Toolbox for MATLAB. Could someone lead me on this matter please ? Thanks for your kind attention.

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/49191

> Thought to add this as a comment, but it appears too long.

Your question does not appear complete, that is, the rationale for using two Brownian motions is not clear. Note that \begin{align*} dX_t &= \mu dt + \sigma_1 dW^1_t + \sigma_2 dW^2_t \\ &=\mu dt + \sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}\frac{\sigma_1 dW^1_t + \sigma_2 dW^2_t}{\sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}}\\ &= \mu dt + \sigma dW_t, \end{align*} where $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}$ and $\Big\{W_t = \frac{\sigma_1 dW^1_t + \sigma_2 dW^2_t}{\sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}}, \, t \ge0\Big\}$ is a standard Brownian. That is, $X_t$ is completely described by a one factor model.

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