Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales
We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (-time scale).
The theory of dynamical equation on time scales () has attracted many researches recently. In particular, attempts of extension to stochastic dynamical equations and stochastic analysis on general time scales have been made in several previous works ([2–6]). In the work  the authors mainly work with a discrete time scale; in  the authors introduce an extension of a function and define the stochastic as well as deterministic integrals as the usual integrals for the extended function; in  the authors make use of their results on the quadratic variation of a Brownian motion () on time scales and, based on this, they define the stochastic integral via a generalized version of the Itô isometry; in  the authors introduce the so-called -stochastic integral via the backward jump operator and they also derive an Itô formula based on this definition of the stochastic integral. We notice that different previous works adopt different notions of the stochastic integral and there lacks a uniform and coherent theory of a stochastic calculus on general time scales.
The purpose of the present article is to fill in this gap. We will be mainly working under the framework of , in that we define our stochastic integral using the definition given in . We then present a general Itô’s formula for stochastic dynamical equations under the framework of . Our Itô formula works for general time scales and thus fills the gap left in , which deals with only discrete time scales. By making use of Itô’s formula we obtain a closed-form expression for the stochastic exponential on general time scales. We will then demonstrate a change of measure (Girsanov’s) theorem for stochastic dynamical equation on time scales.
We would like to point out that our change of measure formula is different from the continuous process case in that the density function is not given by the stochastic exponential but rather is found by the fact that the process on the time scale can be extended to a continuous process simply by linear extension.
It is also worth mentioning that our construction is different from  in that we are working with the case that the time parameter of the process is running on a time scale, whereas in  and related works (e.g., [9–11]) the authors are working with the case that the state space of the process is a time scale.
We note that stochastic calculus on the so-called -Brownian motion has been considered in [12–14]. As an application, we will also work our Itô formula for a Brownian motion on the quantum time scale (-time scale) case at the last section of the paper.
The paper is organized as follows. In Section 2 we discuss some basic set-up for time scales calculus. In Section 3 we will briefly review the results in  and define the stochastic integral and stochastic dynamical equation on time scales. In Section 4 we present and prove our Itô formula. In Section 5 we discuss the formula for stochastic exponential. In Section 6 we prove the change of measure (Girsanov’s) formula. Finally in Section 7 we consider an example of Brownian motion on a quantum time scale.
2. Set-Up: Basics of Time Scales Calculus
A time scale is an arbitrary nonempty closed subset of the real numbers , where we assume that has the topology that it inherits from the real numbers with the standard topology.
We define the forward jump operator byand the backward jump operator by
Let . If , then is called right-scattered. If , then is called right-dense. If , then is called left-scattered. If , then is called left-dense. Moreover, the sets and are derived from as follows: if has a left-scattered maximum, then is the set without that left-scattered maximum; otherwise, . If has a right-scattered minimum, then is the set without that right-scattered minimum; otherwise, . The graininess function is defined by for all .
Notice that since is closed, for any , the points and are belonging to .
For a set we denote the set .
Given a time scale and a function , the delta (or Hilger) derivative of at is defined as follows ([1, Definition ]).
Definition 1. Assume is a function and let . Then we define to be the number (provided that it exists) with the property that, given any , there is a neighborhood of (i.e., for some ) such that
The delta derivative is characterized by the following theorem [1, Theorem ].
Theorem 2. Assume that is a function and let . Then one has the following:(i)if is differentiable at , then is continuous at .(ii)if is continuous at and is right-scattered, then is differentiable at with(iii)If is right-dense, then is differentiable at if and only if the limit exists as a finite number. In this case (iv)If is differentiable at , then
3. Stochastic Integrals and Stochastic Differential Equations on Time Scales
We will adopt the definitions introduced in  as our definition of a Brownian motion and Itô’s stochastic integral on time scales. In the next section we will derive an Itô formula corresponding to the stochastic integral defined in such a way.
Definition 3. A Brownian motion indexed by a time scale is an adapted stochastic process on a filtered probability space such that(1);(2)if and , then the increment is independent of and is normally distributed with mean and variance ;(3)the process is almost surely continuous on .
Note that property (3) is proved in the work .
For a random function we define the extension byfor all .
We shall make use of the definitions given in  for the classical Lebesgue and Riemann integral. For any random function and we define its -Riemann (Lebesgue) integral as where the integral on the right-hand side of the above equation is interpreted as a standard Riemann (Lebesgue) integral. In a similar way, the work  defines a stochastic integral for an -progressively measurable random function aswhere again the right-hand side of the above equation is interpreted as a standard Itô stochastic integral. Note that the way (8) in which we define the extension guarantees that the function is progressively measurable.
In  the authors then defined the solution of the -stochastic differential equation indicated by the notationas the process such that with the deterministic and stochastic integrals on the right-hand side of the above equality interpreted as was just mentioned. Under the condition of continuity in the -variable and uniform Lipschitz continuity in the -variable of the functions and , together with being no worse than linear growth in -variable, existence and pathwise uniqueness of strong solution to (11) are proved in .
4. Itô’s Formula for Stochastic Integrals on Time Scales
We will make use of the following fact that is simple to prove.
Proposition 4. The set of all left-scattered or right-scattered points of is at most countable.
Proof. If is a right-scattered point, then is an open interval such that . Similarly, if is a left-scattered point, then is an open interval such that . Suppose and . We then distinguish four different cases.
Case 1 (both and are right-scattered). We argue that in this case we have . Suppose this is not the case, then we must have . But we see that and . So we must have . We arrive at a contradiction.
Case 2 (both and are left-scattered). This case is similar to Case 1 and we conclude that .
Case 3 ( is left-scattered; is right-scattered). In this case we see that and , as well as . This implies that .
Case 4 ( is right-scattered; is left-scattered). In this case and . If , then . If , then we see that so that . That implies further that and ; that is, .
Thus we see that for all points being left- or right-scattered, the set of all open intervals of the form are disjoint subsets of . Henceforth there are at most countably many such intervals. Each such interval corresponds to one or two endpoints in that are either left- or right-scattered. Thus the total number of left- or right-scattered points in is at most countably many.
Let be the (at most) countable set of all left-scattered or right-scattered points of . As we have already seen in the proof of the previous proposition, the set corresponds to at most countably many open intervals such that (1) for any , ; (2) either the left-endpoint or right-endpoint or both endpoints of any of the ’s are in and are left- or right-scattered; (3) for any ; (4) any point in is a left- or right-endpoint of one of the ’s.
We will denote . Since, for any , the points and are in , we further infer that, for any such interval , we have the fact that and are in , so that is right-scattered and is left-scattered.
We then establish the following Itô formula.
For any two points , , and any open interval , such that , we have . This is because if that is not the case, then or will belong to , contradictory to the fact that . We conclude that
Let us consider a function . Let , be the first- and second-order delta (Hilger) derivatives of with respect to time variable at and let and be the first- and second-order partial derivatives of with respect to space variable at .
Theorem 5 (Itô’s formula). Let any function be such that , , , , , and are continuous on . Set any , ; then we have
Proof. We will make use of the following classical version (Peano form) of Taylor’s theorem: for any function such that and are continuous on , and any and , we havewhereand is an increasing function with .
We will also make use of the time scale Taylor formula (see [1, Theorem ] as well as ) applied to up to first order in : for any and ; we have wherewith as before.
Combining (15) and (17) we see that we havewith for another function increasing with .
Consider a partition , such that (1) each ; (2) for . Notice that by definition , so that we can always find so that is sufficiently small.
Let the sets and be defined as before. Let us fix a partition , and consider a classification of its corresponding intervals . We will classify all intervals such that for all we have as class ; and we classify all intervals such that there exist some with as class . For an interval in class , since for all we have , we see that , because otherwise will be one of the ’s. Thus in this case we have . For an interval in class , since both and are in , we see that we have in fact . In this case either , or . If the latter happens, then is one of the ’s and . We also see from the above analysis that all ’s are contained in intervals that belong to class . On the other hand, either each interval is entirely one of the ’s, or it contains an interval that is one of the ’s. For the latter case, that is, when , the set of intervals of the form are disjoint open intervals such thatNow we haveWe apply (19) term by term in part of (22), and we getWe have the following four convergence results.
Convergence Result 1.1. By Lemma 6 ((35) and (36)) established below we haveConvergence Result 1.2. By Lemma 7, (43), and Lemma 6, (35), established below we haveConvergence Result 2. We have, with probability one, thatas .
In fact, by the Kolmogorov–Čentsov theorem proved in Theorem 3.1 of  we know that for almost all trajectories of on , for each fixed trajectory , there exists an such that for all , for a partition with a classification of its intervals into classes and as above, for some fixed and . From here we can estimate that is,Convergence Result 3. LetWe claim that we have In fact, from the analysis that leads to estimate (17) we see that we can write asHereFrom (21), the Kolmogorov–Čentsov theorem proved in Theorem 3.1 of , as well as the assumptions about function , we see thatFrom here we immediately see the claim (30).
Note that for any interval we have ; therefore we see thatCombining the convergence results (24), (25), (28), and (30), together with (22) and (23) and (34), we establish (14).
The next two lemmas are used in the above proof of Itô’s formula, but they are also of independent interest.
Lemma 6 (convergence of -deterministic and stochastic integrals). Given a time scale and , ; a probability space ; a Brownian motion on the time scale , for any progressively measurable random function that is continuous on , viewed as a -progressively measurable random function on , and the families of partitions , , , one has
Proof. As we have seen in the proof of Itô’s formula, for a given partition , such that for , and , we can classify all intervals of the form into two classes and : class is those open intervals such that it does not contain any open intervals ; class is those open intervals such that it contains at least one open interval , the latter of which has endpoints that are left- or right-scattered.
Let us form a family of partitions , so that the partition is the partition together with all points in that are of the form for some in the partition . Note that under this construction we have . In fact, for any interval in , there is an identical interval in the partition corresponding to it; for any interval in , there are two intervals and corresponding to it, so that . And by (21) we know thatNote that the number depends on and the partition . In particular as . For simplicity we will suppress this dependence later in our proof.
Let us recall the definition of deterministic and stochastic -integrals as defined in Section 2. Let be the extension of that we have in (8): for any ,Note that if is such that , then ; otherwise if is such that , then . Thus we see that So it suffices to prove that In fact, for any interval in class , there exist an interval identical to the interval , so thatFor any open interval in class , there are two corresponding intervals and such that , , and . In this caseFrom the above calculations and the fact that we have (21) and that is continuous on , together with the fact that , , we see the claim as follows.
Lemma 7 (convergence of quadratic variation of Brownian motion on time scale). Given a time scale and , ; a probability space ; a Brownian motion on the time scale , let any -progressively measurable random function on be defined such that is uniformly bounded on . Consider the families of partitions , , . One classifies all the intervals into two classes and as before. Then one has
Proof. We notice that for all intervals we have and thus . Let us denote that Since is progressively measurable, we see that is independent of . ThusFurthermoreIf , then