It\^o's formula, the stochastic exponential and change of measure on general time scales

We provide an It\^{o}'s formula for stochastic dynamical equation on general time scales. Based on this It\^{o}'s formula we give a closed form expression for stochastic exponential on general time scales. We then demonstrate a 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 (q-time scale).


Introduction
The theory of dynamical equation on time scales ( [3]) 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 ( [5], [4], [10], [8], [7]). In the work [4] the authors mainly work with a discrete time scale; in [5] 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 [10] the authors make use of their results on the quadratic variation of a Brownian motion ( [9]) on time scales and, based on this, they define the stochastic integral via a generalized version of the Itô isometry; in [7] the authors introduce the so called ∇-stochastic integral via the backward jump operator and they also derive an Itô's 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 [5], in that we define our stochastic integral using the definition given in [5]. We then present a general Itô's formula for stochastic dynamical equations under the framework of [5]. Our Itô's formula works for general time scales, and thus fills the gap left in [4], which deals with only discrete time scales. By making use of the Itô's formula we obtain a closed-form expression for the stochastic exponential on general times 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 [2] in that we are working with the case that the time parameter of the process is running on a time scale, whereas in [2] and related works (e.g. [11], [1], [6]) the authors are working with the case that the state space of the process is a time scale.
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 [5] and define the stochastic integral and stochastic dynamical equation on time scales. In Section 4 we present and prove our Itô's formula. In Section 5 we discuss the formula for stochastic exponential. In Section 6 we prove the change of measure (Girsanov's) formula.
2 Set-up: Basics of time scales calculus.
A time scale T is an arbitrary nonempty closed subset of the real numbers R, where we assume that T has the topology that it inherits from the real numbers R with the standard topology.
We define the forward jump operator by σ(t) = inf{s ∈ T : s > t} for all t ∈ T such that this set is non-empty , and the backward jump operator by ρ(t) = sup{s ∈ T : s < t} for all t ∈ T such that this set is non-empty .
Let t ∈ T. If σ(t) > t, then t is called right-scattered. If σ(t) = t, then t is called right dense. If ρ(t) < t, then t is called left-scattered. If ρ(t) = t, then t is called left-dense. Moreover, the sets T κ and T κ are derived from T as follows: If T has a left-scattered maximum, then T κ is the set T without that left-scattered maximum; otherwise, T κ = T. If T has a right-scattered minimum, then T κ is the set T without that right-scattered minimum; otherwise, T κ = T. The graininess function is defined by Notice that since T is closed, for any t ∈ T, the points σ(t) and ρ(t) are belonging to T.
For a set A ⊂ R we denote the set A T = A ∩ T.
3 Stochastic integrals and stochastic differential equations on time scales.
We will adopt the definitions introduced in [5] as our definition of a Brownian motion and Itô's stochastic integral on time scales. In the next section we will derive an Itô's formula corresponding to the stochastic integral defined in such a way. Definition 1. A Brownian motion indexed by a time scale T is an adapted stochastic process {W t } t∈T on a filtered probability space (Ω, F t , P) such that (1) P(W 0 = 0) = 1; (2) If 0 ≤ s < t ≤ 1 and s, t ∈ T then the increment W t − W s is independent of F s and is normally distributed with mean 0 and variance t − s; (3) The process W t is almost surely continuous on T.
Note that the property (3) is proved in the work [8].
For a random function f : for all t ∈ [0, ∞). We shall make use of the definitions given in [5] for the classical Lebesgue and Riemann integral. For any random function f : [0, ∞) T × Ω → R and T < ∞ 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 [5] defines a stochastic integral for an L 2 ([0, T ] T )-progressively measurable random function f (t, ω) as where again the right hand side of the above equation is interpreted as a standard Itô's stochastic integral. Note that the way (1) that we define the extension guarantees that the function f (t, ω) is progressively measurable.
In [5] the authors then defined the solution of the ∆-stochastic differential equation indicated by the notation as the process 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 t-variable and uniform Lipschitz continuity in the x-variable of the functions b(t, x) and σ(t, x), together with being no worse than linear growth in x-variable, existence and pathwise uniqueness of strong solution to (2) are proved in [5].
4 Itô's formula for stochastic integrals on time scales.
We will make use of the following fact that is simple to prove.
Proposition 1. The set of all left-scattered or right-scattered points of T is at most countable.
Proof. If x ∈ T is a right-scattered point, then I x = (x, σ(x)) is an open interval such that I x ∩ T = ∅. Similarly, if x ∈ T is a left-scattered point, then I x = (ρ(x), x) is an open interval such that I x ∩ T = ∅. Suppose x < y and x, y ∈ T. We then distinguish four different cases.
Case 1: both x and y are right-scattered. We argue that in this case we have I x ∩ I y = ∅. Suppose this is not the case, then we must have σ(x) > y. But we see that σ(x) = inf{s > x : s ∈ T} and y ∈ T. So we must have σ(x) ≤ y. We arrive at a contradiction; Case 2: both x and y are left-scattered. This case is similar to Case 1 and we conclude that I x ∩ I y = ∅; Case 3: x is left-scattered, y is right-scattered. In this case we see that I x = (ρ(x), x) and I y = (y, σ(y)), as well as x < y. This implies that I x ∩ I y = ∅; Case 4: x is right-scattered, y is left-scattered. In this case I x = (x, σ(x)) and That implies further σ(x) = y and ρ(y) = x, i.e., Thus we see that for all points x ∈ T being left or right-scattered, the set of all open intervals of the form I x are disjoint subsets of R. Henceforth there are at most countably many such intervals. Each such interval corresponds to one or two endpoints in T that are either left or right-scattered. Thus the total number of left or right-scattered points in T are at most countably many.
Let C be the (at most) countable set of all left-scattered or right-scattered points of T. As we have already seen in the proof of the previous Proposition, the set C corresponds to at most countably many open intervals I = {I 1 , I 2 , ...} such that (1) for any k = l, I k ∩ I l = ∅; (2) either the left endpoint or right endpoint or both endpoints of any of the I k 's are in T, and are left or right-scattered; (3) I k ∩ T = ∅ for any k = 1, 2, ...; (4) any point in C is a left or right-endpoint of one of the I k 's.
We will denote I k = (s − I k , s + I k ). Since for any x ∈ T, the points σ(x) and ρ(x) are in T, we further infer that for any such interval I k , we have s − I k and s + I k are in T, so that s − I k is right-scattered and s + I k is left-scattered. We then establish the following Itô's formula. For any two points t 1 , t 2 ∈ T, t 1 ≤ t 2 , and any open interval I k ∈ I, such that . This is because if not the case, then t 1 or t 2 will belong to I k , contradictory to the fact that I k ∩ T = ∅. We conclude that Proof. We will make use of the following version (Peano form) of Taylor's theorem: for any function f ∈ C (2) (R + , R) and any s 1 , s 2 , x 1 , x 2 we have where and r : R + → R + is an increasing function with lim u↓0 r(u) = 0.
Consider a partition π (n) : Notice that by definition ρ(s i ) = sup{s < s i : s ∈ T}, so that we can always find s i−1 ∈ T so that ρ(s i ) − s i−1 is sufficiently small.
Let the sets C and I be defined as before. Let us fix a partition π (n) , and consider a classification of its corresponding intervals (s i−1 , s i ) , i = 1, 2, ..., n. We will classify all intervals (s i−1 , s i ) such that for all I k ∈ I we have I k ∩ (s i−1 , s i ) = ∅ as class (a); and we classify all intervals (s i−1 , s i ) such that there exist some We also see from the above analysis that all I k 's are contained in intervals (s i−1 , s i ) that belong to class (b). On the other hand, each interval (s i−1 , s i ) is either entirely one of the I k 's, or it contains an interval (ρ(s i ), s i ) that is one of the I k 's. For the latter case, i.e., when s i−1 < ρ(s i ) < s i , the set of intervals of the form (s i−1 , ρ(s i )) are disjoint open intervals such that Now we have We apply (4) term by term in part (I) of (6), and we get Convergence Result 2. We have, with probability 1, that In fact, by the Kolmogorov-Čentsov theorem proved in Theorem 3.1 of [8] we know that for almost all trajectory of W t on T, for each fixed trajectory W t (ω), there exist an n 0 = n 0 (ω) such that for all n ≥ n 0 , for a partition π (n) with a classification of its intervals (s i−1 , s i ) into classes (a) and (b) as above, sup Convergence Result 3. Let We claim that we have P(|A n (ω) − B n (ω)| → 0 as n → ∞) = 1 .
In fact, from the analysis that leads to the estimate (5) we see that we can write A n −B n as From (5), the Kolmogorov-Čentsov theorem proved in Theorem 3.1 of [8], as well as the fact that f ∈ C (2) (R + , R), we immediately see the claim (11).

Lemma 1. (Convergence of ∆-deterministic and stochastic intergarals)
Given a time scale T and t 1 , t 2 ∈ T, t 1 < t 2 ; a probability space (Ω, F, P); a Brownian motion {W t } t∈T on the time scale T, for any progressively measurable random function f ∈ C([t 1 , t 2 ]), viewed as a L 2 ([t 1 , t 2 ] T )-progressively measurable random function f (t, ω) on T, and the families of partitions π (n) : t 1 = s 0 < s 1 < ... < s n = t 2 , Proof. As we have seen in the proof of the Itô's formula, for a given partition π (n) : t 1 = s 0 < s 1 < ... < s n = t 2 , such that s i ∈ T for i = 0, 1, ..., n, and max Let us form a family of partitions σ (n) : t 1 = r 0 < r 1 < ... < r m = t 2 , so that the partition σ (n) is the partition π (n) together with all points in T that are of the form r j = ρ(s i ) for some s i in the partition π (n) . Note that under this construction we have r 0 , r 1 , ..., r m ∈ T. In fact, for any interval (s i−1 , s i ) in (a), there is an identical interval (r j−1 , r j ) in the partition σ (n) corresponding to it; for any interval (s i−1 , s i ) in (b), there are two intervals (r j−2 , r j−1 ) and (r j−1 , r j ) corresponding to it, so that r j−1 = ρ(s i ).
And by (5) we know that Note that the number m depends on n and the partition π (n) : m = m(n, π (n) ). In particular m → ∞ as n → ∞. 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. Letf be the extension of f that we have in (1): for any t ∈ T, So it suffices to prove that In fact, for any interval (s i−1 , s i ) in class (a), there exist an interval (r j−1 , r j ) identical to the interval (s i−1 , s i ), so that For any open interval (s i−1 , s i ) in class (b), there are two corresponding intervals (r j−2 , r j−1 ) and (r j−1 , r j ) such that r j−2 = s i−1 , r j−1 = ρ(s i ) and r j = s i . In this case From the above calculations and the fact that we have (5) and f ∈ C([t 1 , t 2 ]), we see the claim follows.
Lemma 2. Given a time scale T and t 1 , t 2 ∈ T, t 1 < t 2 ; a probability space (Ω, F, P); a Brownian motion {W t } t∈T on the time scale T. Let any L 2 ([t 1 , t 2 ] T )progressively measurable random function f (t, ω) on T be defined such that Ef 2 (t, ω) is uniformly bounded on [t 1 , t 2 ]. Consider the families of partitions π (n) : We classify all the intervals (s i−1 , s i ), i = 1, 2, ..., n into two classes (a) and (b) as before. Then we have Proof. We notice that for all intervals (s i−1 , s i ) ∈ (a) we have ρ(s i ) = s i−1 and thus s i − s i−1 < 1 2 n . Let us denote that as n → ∞. This together with the fact that EV n = 0 for any n implies the claim (14) of the Lemma.
The argument above leads us to an Itô's formula for f (t, W t ). Making use of the same methods, one can derive a more general Itô's formula for the solution X t to the ∆-stochastic differential equation (2). We will not repeat the proof, but we will claim the following Theorem.
Our target in this section is to establish a closed-form formula for the stochastic exponential in the case of general time scales T. Definition 2. We say an adapted stochastic process A(t) defined on the filtered probability space (Ω, F t , P) is stochastic regressive with respect to the Brownian motion W t on the time scale T if and only if for any right-scattered point t ∈ T we have (1 + A(t))(W σ(t) − W t ) = 0 , a.s. for all t ∈ T .
The set of stochastic regressive processes will be denoted by R W .
The following definition of a stochastic exponential was also introduced in [4].
Definition 3. (Stochastic Exponential) Let t 0 ∈ T and A ∈ R W , then the unique solution of the ∆-stochastic differential equation is called the stochastic exponential and is denoted by We note that E A (t, t 0 ) as a solution to the equation (16) can be written into an integral equation We can apply the previous arguments and classify the intervals (s i−1 , s i ) into classes (a) and (b). Notice that on each interval (s − I k , s + I k ), the function V (t, t 0 ) remains constant and the function U (t, t 0 ) has a jump, and on each interval (s i−1 , s i ) in class (a) the function U (t, t 0 ) is a constant. This observation and similar arguments (which we leave to the reader) as in the previous section will enable us to prove that with probability 1, as n → ∞, we will have So we proved (22) and thus (21).
6 Change of measure and Girsanov's theorem on time scales.
We demonstrate in this section a change of measure formula (Girsanov's formula) for Brownian motion on time scales. Our analysis is based on the method of extension that was introduced in Section 3 (originally from [5]).
Let us consider two processes, the standard Brownian motion {W t } t∈T on (Ω, F t , P) on the time scale T, and the process on the time scale t ∈ T.
Let us consider an extension of the (probably random) function A(s) as in (1). Let us define the so obtained extension function to be A(s). Recall that (1) implies that A(s, ω) = A(sup[0, s] T , ω) .
Let W t be a standard Brownian motion on [0, ∞). If we define then the process B t agrees with B t for any time point t ∈ T.
For any t, t 0 ∈ T, t > t 0 , let It is easy to see that the function G A (t, t 0 ) is the standard Girsanov's density function for the process B t with respect to the standard Brownian motion W t . Since B t and W t have the same distributions as B t and W t on the time scale T, we conclude with the following two Theorems.
then for every t ≥ 0 we have EG A (t, t 0 ) = 1 .
Let (24) be satisfied. Let T > 0 and pick T > t 0 , t 0 , T ∈ T. Consider a new measure P B on (Ω, F t ), defined by it Radon-Nikodym derivative with respect to P W , as dP B dP W = G A (T, t 0 ) . Theorem 5. (Girsanov's change of measure on time scales) Under the measure P B the process B t , t ∈ [0, T ] T is a standard Brownian motion on T.