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Estimating a Geometric Brownian Motion with Stochastic Mean

Article Quant Q&A · Author: Elle
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Summary

The document considers a geometric Brownian motion whose drift itself follows a mean-reverting stochastic process. It highlights an identification challenge: observed changes in the price process may come from changing drift or from the price process's own Brownian shock. The question assumes the long-run mean of the drift is known and asks how to estimate the remaining parameters.

The responses suggest taking logarithms of the price equation to remove its direct dependence on the price level, then representing the system as a state-space model and estimating parameters with a Kalman filter or maximum likelihood. One response reports solving a linear, normally distributed state-space formulation with maximum likelihood, but provides no data, equations for the likelihood, or empirical results. The choice depends on whether a discretized model is adequate and on the model's assumptions; the notes point toward estimation methods rather than giving a complete implementation or proof of identifiability.

Key ideas

  • Taking logarithms of the price equation removes its direct dependence on the price level.
  • A state-space formulation can represent the latent, time-varying drift alongside observed prices.
  • Kalman filtering and maximum likelihood are proposed for parameter estimation under linearity and normality assumptions.
  • Price shocks and drift variation can both explain observed changes, so model assumptions affect what can be identified.
  • The document gives methodological suggestions but no worked estimation or data-based validation.

Tags

StatisticsMachine learningBacktesting
Full text
# How to estimate parameters of geometric brownian motion with time-varying mean?


# How to estimate parameters of geometric brownian motion with time-varying mean?












Does anyone know how to estimate $A$, $\sigma_1$,$\sigma_2$ from the following system?

$$dx = \mu_t x dt + \sigma_1 x dB_x$$

$$d\mu = A(\bar\mu - \mu) dt + \sigma_2 dB_\mu$$

Variation in $x$ could be either attributed to variation in $\mu$, or variation in $dB_x$, right?

Suppose I know $\bar \mu$, but need to estimate all the rest of the parameters.

## Answer by Kiwiakos (score 3)

https://quant.stackexchange.com/a/16866

I would say

- Take log of first equation to get rid of dependence on $x_t$

- Apply Kalman filter equations to estimate parameters

I believe Conrad and Kaul (1988) J of Business do exactly what you describe.

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/16858

It depends on the use of your model as pointed out in the comments. If a discretized version is sufficient then state space models could be a solution.

You can check out the free online textbook by Athana­sopou­los and Hyndman. State space model describe time series in terms of level/trend (and seasonality) on an additive or multiplicative way. There are nice procedure and packages to estimate and forecast such models.

## Answer by Elle (score 1)

https://quant.stackexchange.com/a/16941

Thank you guys. Sorry for the late reply, I just solved it in matlab using maximum likelihood estimation. Turns out that all we need to do is to specify a state space model, then estimate the coefficient using MLE. The linearity and normality here makes things pretty simple.

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