For a stochastic process X taking values in a state space E, its local time at a point is a measure of the time spent at x. For a continuous time stochastic process, we could try and simply compute the Lebesgue measure of the time at the level,
For processes which hit the level and stick there for some time, this makes some sense. However, if X is a standard Brownian motion, it will always give zero, so is not helpful. Even though X will hit every real value infinitely often, continuity of the normal distribution gives at each positive time, so that that defined by (1) will have zero expectation.
Rather than the indicator function of as in (1), an alternative is to use the Dirac delta function,
Unfortunately, the Dirac delta is not a true function, it is a distribution, so (2) is not a well-defined expression. However, if it can be made rigorous, then it does seem to have some of the properties we would want. For example, the expectation can be interpreted as the probability density of evaluated at , which has a positive and finite value, so it should lead to positive and finite local times. Equation (2) still relies on the Lebesgue measure over the time index, so will not behave as we may expect under time changes, and will not make sense for processes without a continuous probability density. A better approach is to integrate with respect to the quadratic variation,
which, for Brownian motion, amounts to the same thing. Although (3) is still not a well-defined expression, since it still involves the Dirac delta, the idea is to come up with a definition which amounts to the same thing in spirit. Important properties that it should satisfy are that it is an adapted, continuous and increasing process with increments supported on the set ,
Local times are a very useful and interesting part of stochastic calculus, and finds important applications to excursion theory, stochastic integration and stochastic differential equations. However, I have not covered this subject in my notes, so do this now. Recalling Ito’s lemma for a function of a semimartingale X, this involves a term of the form and, hence, requires to be twice differentiable. If we were to try to apply the Ito formula for functions which are not twice differentiable, then can be understood in terms of distributions, and delta functions can appear, which brings local times into the picture. In the opposite direction, which I take in this post, we can try to generalise Ito’s formula and invert this to give a meaning to (3).
There are two main approaches to local times. One method, which is used in Markov process theory, makes use of dual projections to construct the continuous increasing process . The other method uses stochastic integration of real-valued semimartingales. This post is concerned only with the semimartingale approach and, throughout, all semimartingales will be assumed to be real-valued. As usual, we work with respect to a complete filtered probability space , and the starting point is the simple result below. Every convex function has well-defined left and right-hand derivatives, and I use for the left-hand derivative, although this is just convention and there would not be any significant change to any of the arguments of this post if the right-hand derivatives were used instead.
Lemma 1 Let X be a semimartingale and be convex. Then,
for a cadlag increasing process V.
Proof: First, in the case where is convex and twice continuously differentiable, then (4) is immediately given by Ito’s formula with
By convexity, and are nonnegative, so that V is increasing as required.
We extend to the non-twice-differentiable case by approximating with smooth functions. So, suppose that is convex. We convolve this with any twice continuously differentiable nonnegative function with compact support and unit integral, , to obtain a sequence of smooth approximations,
Convexity and twice continuous differentiability of follows from the corresponding properties of, respectively, and . So, (4) applies to giving
for increasing processes . By continuity of and left-continuity of its left-handed derivative, and converge respectively to and as goes to infinity. So, by bounded convergence, and converge to and respectively. Furthermore, as and its derivative are locally bounded, applying locally bounded convergence shows that converges in probability to . Hence, if we define V by (4) then in probability and, in particular, (almost surely) whenever . As the stochastic integral is cadlag, V must also have cadlag sample paths, so is increasing up to evanescence. ⬜
Before applying lemma 1 to construct local times, I first note some straightforward consequences. In the case where X is a continuous local martingale, then is a local submartingale, and is a local martingale. So, (4) is the Doob-Meyer decomposition. More generally, for any semimartingale X, using the fact that stochastic integrals and FV processes are semimartingales, (4) shows that is also a semimartingale.
Corollary 2 Let X be a semimartingale and be convex. Then, is a semimartingale.
Lemma 1 also has the following simple consequence.
Corollary 3 If X is a semimartingale, then is an FV process for any .
Proof: Set and, for a convex function , set . Then, is convex satisfying and , using for the right-hand derivative. So, applying (4) to gives
for some cadlag increasing process . In particular, taking the difference with (4) gives
which is an FV process. In particular, for any fixed , if we define then , giving the result. ⬜
Corollary 3 can be significantly strengthened for continuous local martingales.
Corollary 4 Let X be a continuous local martingale. Then, for any .
Proof: By corollary 3, is an FV process and, as it is also a continuous local martingale then it must be constant. ⬜
I note that corollary 4 can be shown directly in the case where X is Brownian motion. Since has a continuous probability distribution at all positive times, and applying the Ito isometry,
An alternative proof of the corollary follows by representing the local martingale as a time-changed Brownian motion.
Moving on, we want use Lemma 1 to construct the local times of a semimartingale. If X is continuous, then so is the process V. However, if X is discontinuous then, since local times are defined to be continuous, we decompose V into its continuous and pure-jump components. Recall that the jumps of a cadlag process V are denoted by which, for an increasing process, are nonnegative. The following can be viewed as a generalisation of Ito’s formula where the term has been replaced by a continuous increasing process A.
Lemma 5 Let X be a semimartingale and be convex. Then,
for a continuous increasing process A starting from zero.
Proof: Defining the increasing cadlag process V by (4), it can be decomposed as
where A is continuous and increasing. Clearly and, by properties of the stochastic integral, we have
as required. ⬜
By Ito’s lemma, if is twice continuously differentiable then the process A is equal to . If we try applying this equality for the (not twice differentiable) function then we obtain
where is the Dirac delta function, although this is clearly not rigorous and is not a well defined expression. However, lemma 5 still applies and gives a continuous increasing process which is called the local time of X at zero. I use to denote the function equal to -1 for and equal to 1 for . Note the lack of symmetry at where it takes the value -1, but this convention is convenient as it is equal to the left derivative of . I also use and for, respectively, the positive and negative parts of the real number .
Definition 6 (Tanaka’s Formula) The local time of a semimartingale X at zero, denoted by , is the unique process satisfying either (and then, all) of the following identities,
where J is the pure jump process .
Then, the local time at , denoted by , is defined to be the local time of at zero.
The equivalence of the three statements (5) is straightforward. Note that taking the sum of the first identity with the following and halving gives the second identity.
Similarly, taking this away from the second identity gives the third, and adding it to twice the third identity gives the first. So, all three identities of (5) are equivalent. The first of these identities is sometimes referred to as the Meyer-Tanaka formula.
By lemma 5, the local time is a continuous increasing process starting from zero and, we can show that it only increases when .
Lemma 7 Let X be a semimartingale and . Then, is a continuous increasing process starting from zero such that is supported on the set .
Proof: It is sufficient to prove this for , so we just consider the local time at zero. Consider the process
By definition, the local time L is the continuous part of V. The idea is simple enough — over intervals where X does not hit zero or change sign then V is constant. To make this rigorous, fix a time and define the stopping time
As is constant on the interval we obtain,
For , the right hand side is equal to and, hence, , showing that V is almost surely constant on . Hence, L is also constant on this interval and, by continuity, is almost surely constant on .
Now, choose a sequence which is dense in . For example, enumerate the positive rational numbers. As X is cadlag, for any with then the sign of X will not change sign on a sufficiently small interval , so whenever is in this interval, giving
almost surely. Finally, as X is cadlag, everywhere except for a countable set. So,
almost surely, as required. ⬜
The local time of a continuous semimartingale can be expressed compactly as an integral with respect to , which also makes clear that its increments are supported by the set .
Lemma 8 Let X be a continuous semimartingale. Then, its local time at zero satisfies
Proof: Integrating with respect to the second of identities (5),
with the second equality from lemma 7. This is the first equality of (6). The second and third equalities then follow from . ⬜
We obtain particularly simple expressions for the the local time of a continuous local martingale.
Lemma 9 Let X be a continuous local martingale. Then, its local time at zero satisfies
Proof: Combine lemma 8 with corollary 4. ⬜
The local time of a continuous semimartingale can also be constructed as a running maximum. I use the notation for the running maximum of process X, which is automatically increasing and is continuous whenever X is.
Lemma 10 Let X be a continuous semimartingale. Then, , where Y is a continuous semimartingale and L is the both the local time at zero of X and is equal to .
Proof: First, if L is the local time then, from the definition, , which is a continuous semimartingale.
Now, consider a fixed time . On the event that for all then, for , lemma 7 states that and, so, . This gives .
On the other hand, on the event that for some , we can let be the last time in the interval at which . Then, and lemma 7 states that for all . As L is increasing, for all so,
This shows that . Therefore, as required. ⬜
Lemma 10 is particularly helpful when applied to Brownian motion as, in that case, it can be shown that the semimartingale Y is itself a Brownian motion. Recall that B is a Brownian motion with respect to a filtration if it is a continuous process such that has a centered normal distribution with variance and is independent of , for all times . For standard Brownian motion, it is common to also require that it starts from zero, although that will not be required here. An example plot demonstrating theorem 11 is shown in figure 1 above.
Theorem 11 Let B be a Brownian motion. Then, for a Brownian motion W with , where L is both the local time at zero of B and is equal to .
Proof: We can apply lemma 10, so that it just needs to be shown that W is a Brownian motion. Since we know that , which is a local martingale,
The first equality is applying the quadratic variation of stochastic integrals, the second is plugging in the quadratic variation of Brownian motion , and the third is using . So, by Lévy’s characterisation, W is a Brownian motion. ⬜
Theorem 11 has the following immediate consequence, which allows us to completely determine the distribution of Brownian local times.
Theorem 12 Let B and W be standard Brownian motions such that and have the same distribution. If L is the local time of B at zero, then and have the same joint distribution.
Proof: Theorem 11 says that for some Brownian motion W with . As the distribution of a Brownian motion only depends on its starting distribution, we can replace W by any Brownian motion such that has the same distribution as . ⬜
For example, if B is a Brownian motion starting from zero, then it is well-known that has the same distribution as . This is commonly shown as an application of the reflection principle. Theorem 12 then states that the local time of B at zero, , also has the same distribution as . That is, it is distributed as the absolute value of a centred normal of variance t.
Continuing our investigation of local times of general semimartingales, we show that they behave as expected under optional stopping. As usual, for a random time , represents the process X stopped at time . We also use for the pre-stopped process, defined to be for and for . I also use the notation to denote the local time of a semimartingale X at level and time .
Lemma 13 Let X be a semimartingale and . Then,
for all stopping times .
Proof: It is sufficient to prove the result for . Stopping (4) at time , with , gives
where we have used the fact that stopping the stochastic integral at time gives the integral with respect to . As the continuous part of V is just , the continuous part of is , from which we obtain . Similarly, stopping just before time gives
from which we obtain which, by continuity of the local time, is equal to . ⬜
Local times are also stable under continuous time changes which, in particular, allows us to compute local times of any continuous local martingale in terms of Brownian local times.
Lemma 14 Let X be a semimartingale with local time at , and be a collection of stopping times which are continuous and increasing in t. Then, is a semimartingale with respect to the filtration , and has local time at given by
Proof: From the post on time changes, we know that is a semimartingale with respect to the time-changed filtration. We just need to prove the time-change property for the local time at , and the general case follows from applying this to . Let us define the cadlag increasing processes V and by
Again, applying lemma 4 of the post on time changes,
So, is the time change of V, shifted to start from zero. It follows that the same holds for the continuous and pure jump components of and V and, in particular, taking the continuous component gives (8). ⬜
I finish by looking at an important property of local times of continuous local martingales. That is, whenever the process hits a level x then, unless it remains constant at that level, the local time has to increase. This is similar to the property of the quadratic variation which, as we saw in a previous post, must always increase whenever X moves. The difference here is that we are localising to a level . I start by looking at time zero.
Lemma 15 Let X be a continuous local martingale with . Then, with probability one, for all times at which .
Proof: By continuity of X and L, it is sufficient to prove the result for t restricted to a countable dense subset of the positive reals. Then, by countable additivity, it is enough to prove the result for each fixed positive time t. Now, fix and define the stopping time
We use the fact that is a local martingale. then is a local martingale bounded below by and, hence, is a supermartingale. So,
at each positive time t. We then compute
Letting decrease to zero gives and, then, letting go to zero gives . ⬜
We extend the previous result to all positive times.
Lemma 16 Let X be a continuous local martingale and . Then, with probability one, the local time strictly increases whenever X hits level without remaining there.
More precisely, with probability one, for all times for which then .
Proof: Without loss of generality, we take . Then, for any time t define the stopping time
Then, is a continuous local martingale starting from zero, and its local time at zero satisfies . This is straightforward, and follows from lemma 14 restricted to . Applying lemma 15 gives for any time at which .
Now choose a sequence of times which are dense in . With probability one, we have at all times for which . Now, for any such sample path, consider times for which . As the sequence is dense, we have for some n and, then, . As , this gives as required. ⬜