In this post, I give an example of a class of processes which can be expressed as integrals with respect to Brownian motion, but are not themselves martingales. As stochastic integration preserves the local martingale property, such processes are guaranteed to be at least local martingales. However, this is not enough to conclude that they are proper martingales. Whereas constructing examples of local martingales which are not martingales is a relatively straightforward exercise, such examples are often slightly contrived and the martingale property fails for obvious reasons (e.g., doubleloss betting strategies). The aim here is to show that the martingale property can fail for very simple stochastic differential equations which are likely to be met in practice, and it is not always obvious when this situation arises.
Consider the following stochastic differential equation

(1)

for a nonnegative process X. Here, B is a Brownian motion and a,b,c,x are positive constants. This a common SDE appearing, for example, in the constant elasticity of variance model for option pricing. Now consider the following question: what is the expected value of X at time t?
The obvious answer seems to be that , based on the idea that X has growth rate b on average. A more detailed argument is to write out (1) in integral form

(2)

The next step is to note that the first integral is with respect to Brownian motion, so has zero expectation. Therefore,
This can be differentiated to obtain the ordinary differential equation , which has the unique solution .
In fact this argument is false. For there is no problem, and as expected. However, for all the conclusion is wrong, and the strict inequality holds.
The point where the argument above falls apart is the statement that the first integral in (2) has zero expectation. This would indeed follow if it was known that it is a martingale, as is often assumed to be true for stochastic integrals with respect to Brownian motion. However, stochastic integration preserves the local martingale property and not, in general, the martingale property itself. If then we have exactly this situation, where only the local martingale property holds. The first integral in (2) is not a proper martingale, and has strictly negative expectation at all positive times. The reason that the martingale property fails here for is that the coefficient of dB grows too fast in X.
In this post, I will mainly be concerned with the special case of (1) with a=1 and zero drift.

(3)

The general form (1) can be reduced to this special case, as I describe below. SDEs (1) and (3) do have unique solutions, as I will prove later. Then, as X is a nonnegative local martingale, if it ever hits zero then it must remain there (0 is an absorbing boundary).
The solution X to (3) has the following properties, which will be proven later in this post.
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