Consider a geometric Brownian motion driven by the Stochastic Differential Equation (SDE)
where is a canonical Brownian motion on a probability space . The process is often used as the most basic model of stock prices in real time. Here we will motivate the problem by considering as the population of an invasive species that has found its way into a new habitat where resources are plentiful. The lack of scarcity means that the population will not come close to the carrying capacity of its habitat on the time scale we are interested in, hence we can use Eqn. () to model its growth instead of a more complicated logistic type model.
The solution to Eqn. () is given by Since the deterministic growth rate of is , it is natural to consider the quantity . is the ratio between and the population that would result from a model of deterministic exponential growth. We may then ask, what is the probability that ever becomes twice, three times, or times as large as a its deterministically governed counterpart? In other words, we would like to characterise the function paying especial attention to the limit . We can obtain an easy bound on by noting that is a non-negative martingale with for all , hence Doob’s martingale inequality for reads
For this bound is trivial, but,as we will see, it is the best constant-in- bound for when .
If we define then is equivalently written as
Here let us state two very useful results without proof. The first gives us the density of the running maximum of Brownian motion. This is not so difficult to prove, and I hope to make a separate post about it very soon. The second is a simplified version of Girsanov’s theorem which will serve our current needs well enough.
To state the next result we introduce the notation for the quadratic variation of up to time , and we write for the covariation of andNotice that if we set , then
so that . We will now use these two results to calculate . First note that since a.s. for all , and hence exists and is given by Now we have
Below is a plot of the analytic solution from Eqn. () in the case with numerical results obtained by sampling the process at regular intervals. The numerics systematically underestimate the true probability because sampling the process underestimates the maximum . This raises another question: Does the discrete time process converge to ? In what sense? And how quickly? A topic for another day.
It appears from this plot that as . I will leave it here for now, but I will post an edit in the next few days to rigorously establish this limit.