Estimating a Linear Brownian-Drift Process with Positive Growth
Summary
The document considers estimating parameters in the stochastic differential equation dλ = aλ dt + σ dW, where the drift coefficient is positive. It clarifies that this is not mean-reverting: without the random term, the state grows exponentially when it starts positive. The answer therefore describes the model as a linear diffusion with an amplifying drift rather than a conventional Ornstein–Uhlenbeck process.
For observations spaced by a fixed time interval, the proposed approximation regresses each change in the state on its previous value. The slope estimates the drift coefficient multiplied by the interval, while the residual standard deviation estimates volatility multiplied by the square root of that interval; rescaling gives parameter estimates. The response also points to external references for related estimation procedures. This is a first-order discretization, not a full derivation of an exact likelihood, and the answer itself retracts an earlier line of reasoning. It gives no data example or empirical assessment of the estimator’s accuracy.
Key ideas
- A positive linear drift does not pull the process toward a long-run mean.
- For positive initial values, the drift-only equation produces exponential growth.
- A discrete regression of state changes on lagged state can estimate the drift term.
- Residual dispersion can estimate diffusion volatility after adjustment for the observation interval.
- The response outlines an approximation and does not establish its finite-sample performance.
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# Calibration of non-mean-reverting OU process
# Calibration of non-mean-reverting OU process
I'm looking for some reference on how to calibrate a non-mean-reverting Ornstein-Uhlenbeck process to historical data using MLE or OLS. The model has the following SDE:
$d\lambda(t)=a\lambda(t)dt+\sigma dW(t)$
with $a>0$ and $\sigma \geq 0$.
Hints on how to adapt the procedure from a mean-reverting OU may be useful too.
## Answer by Richi Wa (score 3, accepted)
https://quant.stackexchange.com/a/18762
EDIT: My reasoning below seems to be wrong. The process as you write it tends to infinity if $a$ is big enough and positive and if $\lambda_0$ is positive. I would not call this process non-meanreverting OU. It is just an Ito process of a simple form. If we remove the stochastic part then we get $$ d\lambda_t = a \lambda_t dt $$ with the solution (if $\lambda_0>0$) $\lambda_t = \lambda_0 \exp(a t)$ which for $a>0$ just grows exponentially. If look at the whole thing then we add a stochastic disturbance at each time step of size $\sigma dB_t$. Thinking about it this way I think that the process above does not have too much in common with an OU-process.
I delete my previous answer.
Concerning the estimation: If you have a process that you have observed on a time grid with width $ \Delta t$ then a discretization of your SDE could look like this: $$ \lambda(t + \Delta t) - \lambda(t) = \theta \lambda(t) \Delta t + \sigma \sqrt{\Delta t} \epsilon_i $$ where $\epsilon_i$ is standard normal. Thus a regression of $\lambda(t + \Delta t) - \lambda(t)$ on $\lambda(t)$ gives you $\theta \Delta t$. The volatility of the residuals gives you an estimate of $\sigma \sqrt{\Delta t}$. Dividing these quantities by the grid width (resp its square-root) gives you the parameters.
Look at a similar question here.
## Answer by Aimin Huang (score 1)
https://quant.stackexchange.com/a/30643
Here are two nice references:
- https://www.sitmo.com/?p=134
- https://commoditymodels.files.wordpress.com/2010/02/estimating-the-parameters-of-a-mean-reverting-ornstein-uhlenbeck-process1.pdfShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.