For nonnegative local martingales, there is an interesting symmetry between the failure of the martingale property and the possibility of hitting zero, which I will describe now. I will also give a necessary and sufficient condition for solutions to a certain class of stochastic differential equations to hit zero in finite time and, using the aforementioned symmetry, infer a necessary and sufficient condition for the processes to be proper martingales. It is often the case that solutions to SDEs are clearly local martingales, but is hard to tell whether they are proper martingales. So, the martingale condition, given in Theorem 4 below, is a useful result to know. The method described here is relatively new to me, only coming up while preparing the previous post. Applying a hedging argument, it was noted that the failure of the martingale property for solutions to the SDE for is related to the fact that, for , the process hits zero. This idea extends to all continuous and nonnegative local martingales. The Girsanov transform method applied here is essentially the same as that used by Carlos A. Sin (Complications with stochastic volatility models, Adv. in Appl. Probab. Volume 30, Number 1, 1998, 256-268) and B. Jourdain (Loss of martingality in asset price models with lognormal stochastic volatility, Preprint CERMICS, 2004-267).
Consider nonnegative solutions to the stochastic differential equation
where , B is a Brownian motion and the fixed initial condition is strictly positive. The multiplier X in the coefficient of dB ensures that if X ever hits zero then it stays there. By time-change methods, uniqueness in law is guaranteed as long as a is nonzero and is locally integrable on . Consider also the following SDE,
Being integrals with respect to Brownian motion, solutions to (1) and (2) are local martingales. It is possible for them to fail to be proper martingales though, and they may or may not hit zero at some time. These possibilities are related by the following result.
As an example, consider for some fixed exponent , so that . In the previous post it was shown that X fails to be a proper martingale when and hits zero when . Theorem 1 shows that these two statements are equivalent.
We can be a bit more precise than the statement in Theorem 1. Being nonnegative local martingales, the processes X, Y are automatically supermartingales. For times , is nonnegative and, hence, will be almost surely zero if and only if it has zero expectation. So, we see that the martingale condition is satisfied whenever . Furthermore, the supermartingale condition gives . It follows that X is a martingale over a time interval if and only if . The following theorem shows that this is equivalent to being strictly positive with probability one, giving a more precise statement than Theorem 1 above.
To apply these results, the following necessary and sufficient condition for solutions to the SDE (1) to hit zero after a finite time can be used. This is a special case of Feller’s test for explosions, and a proof is given further below.
Theorem 3 Suppose that a is nonzero and is locally integrable on . Then, solutions X to (1) hit zero with positive probability if and only if
for . In this case, X hits zero almost surely.
Using Theorem 1, this can be transformed into a condition for the process X to be a martingale. In particular, X satisfying (1) will be a proper martingale if and only if the solution Y to (2) has zero probability of hitting zero. By Theorem 3 this is equivalent to
So, we have arrived at a necessary and sufficient condition for X to be a martingale.
Theorem 4 Suppose that a is nonzero and is locally integrable on . Then, solutions X to (1) are proper martingales if and only if
Looking again at the SDE , we take ,
This is infinite if and only if , in which case X is a martingale and, for , Theorem 4 shows that it fails to be a martingale. Consider, also, the following SDE, whose coefficient grows very slightly faster than linearly in X,
Again, c is a fixed positive constant. In this case, we take . Up to a finite scaling factor, the integral (4) gives
This is finite if and only if . So, solutions to (6) are martingales whenever and are local martingales, but not proper martingales, for all .
Proof of Theorem 2
I will now give a proof of Theorem 2 using local Girsanov transforms. As is standard, we work with respect to a filtered probability space . However, in order for the Girsanov transform method to be successfully applied, we do not assume that the filtration is complete. We will also work in the more general setting of continuous local martingales, not necessarily defined by an SDE.
For the remainder of this section, let X be a continuous local martingale taking values in the extended nonnegative real numbers , and with the fixed initial condition . The local martingale property implies that X is a supermartingale and that it is almost surely finite. Let us also set and Y=1/X, which explodes when X hits zero and hits zero in the (zero probability) event that X explodes. Let also define to be the first time at which X hits zero and to be the first time at which Y hits zero or, equivalently, X hits infinity. From this setup, is almost-surely infinite.
The idea is to use X to define a change of measure . However, Girsanov transform theory would only be applicable when X is a uniformly integrable and positive martingale. To get around this restriction, we instead apply the change of measure locally. That is, if is a stopping time such that is a uniformly integrable martingale and , then we define the restriction of the probability measure to by
That is, for any bounded -measurable random variable Z. There is a subtle issue here though, as the measure need not exist at all. This is an issue which was encountered previously in my stochastic calculus notes, in the application of measure changes to stochastic differential equations. This problem can arise because, although and are defined to be equivalent on , they need not be equivalent on . For example, being a local martingale, X is almost surely bounded under . However, if it is not a proper martingale, then we will see that Y=1/X hits zero in a finite time, at which X explodes. If we did not include such events in the probability space to start with, then defining the transformed measure would be impossible. Similarly, problems would be caused by including such zero probability events in the initial sigma algebra . This is the reason for considering X to lie in the extended nonnegative real numbers in the setup above, and also for not assuming that the filtration satisfies the usual completeness properties. These are not real problems though, and just require a bit of care with the construction of the underlying filtered probability space. For now, we ignore these issues and assume that the measure exists, in which case it is essentially unique. We can prove the following much more general version of Theorem 2, applying to arbitrary continuous and nonnegative local martingales. Actually, Theorem 2 will follow as a corollary of Lemma 5 when applied to solutions of the SDE (1).
Lemma 5 If it exists, the measure defined by (7) is uniquely defined on . Furthermore, , is a -local martingale with quadratic variation
(under ) and, for any time ,
Proof: Let be a sequence of stopping times increasing to and such that are uniformly integrable martingales with (almost surely). For example, we could take to be the first time that X hits either n or 1/n. Then, (7) defines the restriction of to . Letting n go to infinity this uniquely defines on . Once it is shown that -almost surely, then this will also uniquely determine on .
By Ito’s lemma,
so that and . However, applying the Girsanov theorem to the local martingale X shows that
is a -local martingale over the intervals and, therefore, so is . Letting n go to infinity shows that is a -local martingale. In particular, by the supermartingale property, so that is finite and -almost surely.
So, we have shown that , is uniquely defined on and that is a -local martingale. Using (10), the quadratic variation of Y is given by
on , giving (8).
The first of the identities in (9) comes from the following sequence of equalities.
The first equality is using the fact that, -almost surely, Y hits zero at time whenever this is finite. The second equality is just using the change of measure definition (7). The third equality is using the condition that are martingales and -almost surely. In a similar way, the second of the identities in (9) comes from the following,
Now, let’s move on to showing that with the underlying filtered probability space set up correctly, measures defined by local Girsanov transforms do indeed exist. The idea is to construct locally via equation (7), and apply the Kolmogorov extension theorem to extend to a measure on . Kolmogorov’s theorem states that if we have a consistent set of probability measures defined on the finite products of some underlying measurable spaces (in particular, Polish spaces), then they extend uniquely to a measure on the infinite product.
Lemma 6 Let be the set of continuous functions , X be the coordinate process
be the natural filtration
and set .
Then, for any probability measure on making X a local martingale with almost surely, a measure defined by (7) exists.
Proof: Let each n, let be the stopping time
These increase to a limit , which is the first time that X hits either zero or infinity. Define the measure on according to (7),
This can be extended to a measure on the sigma algebra by supposing that X remains constant after time . That is, the -expectation of any -measurable and bounded function is given by,
This defines a sequence of measures on such that, for all , and agree when restricted to . Denote the infinite product space as with product sigma algebra , and let be the projection onto the n‘th component. Then, applying the Kolmogorov extension theorem, there exists a measure on with respect to which has distribution and, for any , and are equal up until the first time that they exceed the level m or drop below 1/m, -almost surely. With respect to , then, the limit
exists and, up until the first time at which it passes the upper level n or drops below 1/n, this agrees with the distrubution of X under the measure . It can also be seen that is almost surely continuous. So, we can define to be the measure on with respect to which X has the same distribution as does under . Finally, for a stopping time for which is a -uniformly integrable martingale and , it needs to be shown that (7) holds. For any finite time t and bounded -measurable random variable Z, is -measurable. So, using the martingale property for X,
The last equality here uses the martingale property to replace by , and the fact that for large n, almost surely, which follows from the property that is almost surely bounded and nonzero. Using Z=1 in (11) shows that -almost surely and, letting t increase to infinity,
for all -measurable and bounded random variables Z. ⬜
Now that it has been shown that the measure change is well-defined, assuming the setup of Lemma 6, we can finally move on to the proof of Theorem 2. Rather than defining the process X via the SDE (1), however, it helps to rewrite it in a more intrinsic form without reference to a driving Brownian motion. Any such process is a local martingale with quadratic variation
Conversely, by enlarging the probability space to add a Brownian motion if required, any local martingale satisfying (12) and the initial condition solves the SDE (\ref) for some Brownian motion B}. So, assuming uniqueness in law of (2), Theorem 2 is a consequence of equations (9) and the following result.
Then, letting be the measure defined by Lemma 5, the process is a -local martingale with quadratic variation
Proof: Lemma 5 states that is a -local martingale with quadratic variation
As after time , this gives (13). ⬜
Proof of the zero-hitting condition
Theorem 3, providing a necessary and sufficient condition for solutions to the SDE (1) to hit zero, can be proven by applying a well-chosen transformation to the local martingale X. Then, martingale convergence will be used — with probability one, whenever a continuous local martingale is bounded above or below then it converges to a finite value as . The result also follows from Feller’s test for explosions (see Karatzas and Shreve, Brownian Motion and Stochastic Calculus, Chapter 5).
As is assumed to be locally integrable, a convex function can be defined by
This is continuously differentiable with second order derivative defined in the sense of distributions. Ito’s lemma gives
Although Ito’s lemma only directly applies in the twice differentiable case, where a is continuous, (14) extends to all a with locally integrable by taking limits (using the dominated convergence theorem to take the limits, and the monotone class theorem to extend to locally integrable). Then, as X satisfies the SDE (1) and a is assumed to be nonzero, its quadratic variation is given by for nonzero up until the time at which X hits zero. So
where is the first time at which X hits zero. In particular, is a local martingale. The limiting value of F at zero is
which is just the integral (4). If this is infinite, X cannot hit zero in finite time, as it would imply that the local martingale M explodes. Alternatively, suppose that the integral (4) is finite, so that is finite. As X is a nonnegative local martingale, it converges almost surely to a finite value at infinity. So, is bounded above and, by martingale convergence, tends to a finite limit. This shows that converges as , which can only be the case if is finite and X hits zero.