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.
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.
Here are two nice references:
