The local time of a semimartingale at a level x is a continuous increasing process, giving a measure of the amount of time that the process spends at the given level. As the definition involves stochastic integrals, it was only defined up to probability one. This can cause issues if we want to simultaneously consider local times at all levels. As x can be any real number, it can take uncountably many values and, as a union of uncountably many zero probability sets can have positive measure or, even, be unmeasurable, this is not sufficient to determine the entire local time ‘surface’
for almost all . This is the common issue of choosing good versions of processes. In this case, we already have a continuous version in the time index but, as yet, have not constructed a good version jointly in the time and level. This issue arose in the post on the Ito–Tanaka–Meyer formula, for which we needed to choose a version which is jointly measurable. Although that was sufficient there, joint measurability is still not enough to uniquely determine the full set of local times, up to probability one. The ideal situation is when a version exists which is jointly continuous in both time and level, in which case we should work with this choice. This is always possible for continuous local martingales.
Theorem 1 Let X be a continuous local martingale. Then, the local times
have a modification which is jointly continuous in x and t. Furthermore, this is almost surely -Hölder continuous w.r.t. x, for all and over all bounded regions for t.
A proof will be given further down. Theorem 1 applies, in particular, to Brownian motion although, in this case, the continuous modification also satisfies the stronger property of joint Hölder continuity.
Theorem 2 Let X be a Brownian motion with arbitrary starting value. Then, the jointly continuous version of the local times are almost surely jointly -Hölder continuous in x and t, for all and bounded intervals for t.
Again, the proof will be given further down. In fact, theorem 2 can be used to determine the joint continuity properties for the local times of any continuous local martingale, giving an improvement over the previous result. We know that any local martingale X can be written as a time-change of a standard Brownian motion B started from . Specifically, , where is the quadratic variation. Also, local times transform in the expected way under continuous time changes. If we write for the local times of B which, by theorem 2, is locally Hölder continuous in x and t, then the local time of X is
This shows that has a version which is a locally -Hölder continuous function of and x for all .
Next, consider more general continuous semimartingales. It turns out that, now, the local times need not have a jointly continuous version. For example, if B is a Brownian motion, then is a reflected Brownian motion. Writing for the local times of B, then the local times of X are given by,
This is jointly continuous when x is away from 0 but, as x passes through 0, then jumps by twice the local time of B. The best that we can hope for is that is jointly continuous in t and cadlag in x. This means that for each then there exists a left-limit such that, for sequences and ,
Equivalently, considered as a set of continuous functions , one for each x, then is cadlag under the topology of uniform convergence on compacts.
Theorem 3 Let X be a continuous semimartingale. Then, its local times have a version which is jointly continuous in t and cadlag in x.
Furthermore, if is the decomposition into a continuous local martingale and FV process V then, with probability one,
for all times t and levels x.
We can further ask whether the semimartingale X needs to be continuous in order that the local times have a modification as in the theorem above. In fact, it is possible to extend to a class of non-continuous processes but, unfortunately, not to all semimartingales. This will be stated in a moment, and theorem 3 will follow from this more general result. We need to restrict to a class of semimartingales whict have only a finite variation coming from the jumps, which can be expressed in a couple of different ways.
Lemma 4 Let X be a semimartingale. Then, the following are equivalent,
- , almost surely for all times t.
- X decomposes as the sum of a continuous local martingale and an FV process.
Furthermore, in this case, X decomposes as
for a continuous local martingale M and continuous FV process V.
Proof: If the first condition holds, then we can define the pure jump process . So, is a continuous semimartingale and, therefore, decomposes into the sum of a continuous local martingale M and continuous FV process V. This gives decomposition (1) and, as is an FV process, also implies the second condition.
Conversely, if the second condition holds, then write for continuous local martingale M and FV process V. As , the sum is equal to . As this is bounded by the variation of V, it is almost surely finite. ⬜
We will restrict to the class of processes identified by the equivalent conditions above. I am not aware of any standard terminology for referring to such semimartingales other than the following definition as used by Protter, although it is a rather unimaginative name.
Definition 5 A semimartingale satisfies Hypothesis A iff the equivalent conditions of lemma 4 hold.
This captures many types of processes that we would like to handle, although there are semimartingales which do not satisfy Hypothesis A. For example, it is not satisfied by Cauchy processes. Theorem 3 can now be generalised.
Theorem 6 Let X be a semimartingale satisfying Hypothesis A. Then, its local times have a version which is jointly continuous in t and cadlag in x.
Furthermore, if V is the process in decomposition (1) then, with probability one, the jump with respect to x is
for all times t and levels x.
As continuous semimartingales trivially satisfy Hypothesis A, theorem 3 is an immediate consequence of this result.
Proof of Continuity
I now give proofs of the local time continuity results above, for which the main tool will be the Kolmogorov continuity theorem. Other than that, the Burkholder-Davis–Gundy (BDG) inequality will play an important part in the proof that the hypotheses of Kolmogorov’s theorem is satisfied, although I will only require the simple case of the right-hand inequality for large exponents. As the first and main step, we show that certain stochastic integrals with respect to a continuous martingale have a jointly continuous modification.
Lemma 7 Let X be a semimartingale decomposing as for a continuous martingale M and FV process A. We suppose that and the variation of A over are -integrable for all positive p. Then,
has a version which is jointly continuous in x and t. Furthermore, with probability one, this version is -Hölder continuous in x for all .
Proof: First note that is a continuous local martingale starting at zero. Although it is not essential for the proof, it simplifies things a bit to use the Ito isometry,
showing that is an bounded martingale and, by martingale convergence, the limit exists. We will let E denote the space of continuous functions , with the topology of uniform convergence. So, for each x, the paths can be considered as a random variable taking values in E. Using d for the supremum metric then, for any and positive ,
This used the BDG inequality, so that is a fixed positive constant. Next, we will apply Ito’s formula to the convex function,
Although this is not twice continuously differentiable, Ito’s formula still applies by approximating with smooth functions giving,
By convexity of , the final summation is positive. Also, noting that
we obtain the inequality
Let V be the variation process of A. Using the fact that is bounded by , this gives
for some process bounded by 1. Raising to the power of and taking expectations,
Once again, the BDG inequality was used for the expectation of the first two terms in the parantheses on the right hand side. So, is a positive constant. Combining with (3) gives,
for the positive constant
Now, Kolmogorov’s continuity theorem can be applied with , so long as . This guarantees the existence of a modification of which, for all , is locally -Hölder continuous . Finally, note that from the definition, can be chosen constant in x over the range and, similarly, over . So, it is constant for x outside of the range and, hence, is globally -Hölder continuous. Letting increase to infinity, this holds for all . ⬜
Localization extends the result above extends to all semimartingales satisfying Hypothesis A.
Lemma 8 Let X be a semimartingale satisfying Hypothesis A, and M be as in decomposition (1). Then, , defined by (2) has a version which is jointly continuous in t and x. Furthermore, over any bounded range for t, this version is almost surely -Hölder continuous in x for all .
Proof: As it satisfies Hypothesis A, we can decompose for a continuous local martingale M and FV process A. Next, choose a sequence of stopping times, , increasing to infinity and such that and the variation of the pre-stopped processes are all bounded. For example, letting V be the variation process of A, we can take
Then, we can decompose
These pre-stopped processes satisfy the conditions of lemma 7 and, hence, there exists jointly continuous versions of the processes
However, if we define by (2) then, by optional stopping of stochastic integrals, almost surely, whenever . In particular, this means that almost surely, whenever and . By continuity, this holds simultaneously for all x and all , with probability one. Restricting to the event where this holds, we can therefore define the modification
for all n such that . For any positive time T then, almost surely, we can choose n such that , in which case is -Hölder continuous in x, over . ⬜
Applying lemma 8 to the definition of the local time for a continuous local martingale immediately provides a jointly continuous modification.
Proof of Theorem 1: By definition, the local times of a continuous semimartingale are given by,
As X is a continuous local martingale, we can take in lemma 8, so that the integral above as a version which is jointly continuous in t and x, and which is -Hölder continuous in x for all and over all bounded intervals for t. We use this version to define the local times . As all the terms on the right hand side of the above equality satisfy these continuity conditions, the same is true for . ⬜
In the case of Brownian motion, the proof of lemma 7 can be extended to give joint Hölder continuity.
Lemma 9 Let X be a standard Brownian motion with arbitrary starting value. Then,
has a version which is jointly continuous in x and t. Furthermore, with probability 1, this is jointly -Hölder continuous over all finite time intervals for t.
Proof: For any and , the BDG inequality gives,
Choosing and , the Kolmogorov continuity theorem provides a jointly continuous version of over , which is locally -Hölder continuous for all . Letting go to infinity, this holds for all . Also, as argued in the proof of lemma 7, the fact that is constant in x for large positive, and large negative, values of x means that it will be globally -Hölder continuous. Finally, the result follows by letting T go to infinity. ⬜
Proof of Theorem 2: We again express the local time using (5). Since we know that the path of a Brownian motion is locally almost surely -Hölder continuous for all , it follows that is jointly -Hölder continuous in t and x, over bounded time intervals. By lemma 9, the same is true of and, hence, of . ⬜
I finally complete the proof of theorem 6, showing that Hypothesis A is sufficient for local times to have a modification that is jointly continuous in time and cadlag in the level. The idea is similar to that given above for continuous local martingales but, now, we have additional terms to account for the jumps of the process and the drift V. In particular, the drift can introduce discontinuities, explaining why we only obtain cadlag versions in x.
Proof of Theorem 6: Let us define . By definition, the local times are given by
We let M and V be as in decomposition (1).
Each of the terms on the right hand side can be defined pathwise, except for the stochastic integral which, by lemma 8, has a jointly continuous version. So, it only remains to check joint continuity for A and B.
Starting with A, for each fixed x, this is a cadlag process with jump , so is continuous. Joint continuity will follow if we can prove uniform continuity in x for t restricted to any bounded interval . Choosing , then the function is bounded by and has derivative bounded by 1. Hence,
gives the bound
By Hypothesis A, is almost surely bounded and then, by dominated convergence, tends uniformly to zero as . So is almost surely uniformly continuous in x over the range , as required.
Finally, we look at , which we consider to be defined in a pathwise sense. That is, for each fixed define it as the pathwise Lebesgue-Stieltjes integral with respect to the locally finite variation path . For each x, it is an integral with respect to a continuous FV process, so is continuous in t. To complete the proof, we need to show that it is jointly continuous in t and cadlag in x. For this, it is sufficient to show that, over each bounded time interval , the paths are cadlag in x under uniform convergence. Fix and choose . Then,
The integrand tends to zero as y decreases to x so, by bounded convergence, tends uniformly to zero over . This gives right-continuity in x. To show that the left limits exist, define
Then, for we similarly obtain,
Again, as y increases to x, bounded convergence shows that tends to zero uniformly over . Hence, is almost surely jointly continuous in t and cadlag in x, so the same holds for . Finally, using the expression above for we obtain that the jump w.r.t. x is,
as required. ⬜