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Ito's Formula, the Stochastic Exponential, and Change of Measure on General Time Scales.

1. Introduction

The theory of dynamical equation on time scales ([1]) 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 [3] the authors mainly work with a discrete time scale; in [2] 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 [4] the authors make use of their results on the quadratic variation of a Brownian motion ([7]) on time scales and, based on this, they define the stochastic integral via a generalized version of the Ito isometry; in [6] the authors introduce the so-called [nabla]-stochastic integral via the backward jump operator and they also derive an Ito 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 [2], in that we define our stochastic integral using the definition given in [2]. We then present a general Ito's formula for stochastic dynamical equations under the framework of [2]. Our Ito formula works for general time scales and thus fills the gap left in [3], which deals with only discrete time scales. By making use of Ito'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 [8] in that we are working with the case that the time parameter of the process is running on a time scale, whereas in [8] 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 q-Brownian motion has been considered in [12-14]. As an application, we will also work our Ito formula for a Brownian motion on the quantum time scale (q-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 [2] and define the stochastic integral and stochastic dynamical equation on time scales. In Section 4 we present and prove our Ito 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 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

[sigma](t) = inf {s [member of] T : s > t} (1)

[for all]t [member of] T such that this set is nonempty,

and the backward jump operator by

[rho](t) = sup {s [member of] T : s < t} (2)

[for all]t [member of] T such that this set is nonempty.

Let t [member of] T. If [sigma](t) > t, then t is called right-scattered. If [sigma](t) = t, then t is called right-dense. If [rho](t) < t, then t is called left-scattered. If [rho](t) = t, then t is called left-dense. Moreover, the sets [T.sup.[kappa]] and [T.sub.[kappa]] are derived from T as follows: if T has a left-scattered maximum, then [T.sup.[kappa]] is the set T without that left-scattered maximum; otherwise, [T.sup.[kappa]] = T. If T has a right-scattered minimum, then [T.sub.[kappa]] is the set T without that right-scattered minimum; otherwise, [T.sub.[kappa]] = T. The graininess function is defined by [mu](t) = [sigma](t) - t for all t [member of] [T.sup.[kappa]].

Notice that since T is closed, for any t [member of] T, the points [sigma](t) and [rho](t) are belonging to T.

For a set A [subset] R we denote the set [A.sub.T] = A [intersection] T.

Given a time scale T and a function f: T [right arrow] R, the delta (or Hilger) derivative [f.sup.[DELTA]](t) of f at t [member of] T is defined as follows ([1, Definition 1.10]).

Definition 1. Assume f : T [right arrow] R is a function and let t [member of] [T.sup.[kappa]]. Then we define [f.sup.[DELTA]](t) to be the number (provided that it exists) with the property that, given any [epsilon] > 0, there is a neighborhood U of t (i.e., U = (t - [delta], t + [delta]) [intersection] T for some [delta] > 0) such that

[absolute value of ([f([sigma] (t)) - f(s)]- [f.sup.[DELTA]](t)[[sigma](t) - s])] [less than or equal to] [epsilon] [absolute value of ([sigma](t) - s)]

[for all]s [member of] U. (3)

The delta derivative is characterized by the following theorem [1, Theorem 1.16].

Theorem 2. Assume that f : T [right arrow] R is a function and let t [member of] [T.sup.[kappa]]. Then one has the following:

(i) if f is differentiable at t, then f is continuous at t.

(ii) if f is continuous at t and t is right-scattered, then f is differentiable at t with

[f.sup.[DELTA]](t) = f([sigma](t)) - f(t)/[sigma](t) - t. (4)

(iii) If t is right-dense, then f is differentiable at t if and only if the limit

[mathematical expression not reproducible] (5)

exists as a finite number. In this case

[mathematical expression not reproducible] (6)

(iv) If f is differentiable at t, then

f([sigma](t)) = f(t) + [mu](t) [f.sup.[DELTA]](t). (7)

3. Stochastic Integrals and Stochastic Differential Equations on Time Scales

We will adopt the definitions introduced in [2] as our definition of a Brownian motion and Ito's stochastic integral on time scales. In the next section we will derive an Ito formula corresponding to the stochastic integral defined in such a way.

Definition 3. A Brownian motion indexed by a time scale T [subset] R is an adapted stochastic process [{[W.sub.t]}.sub.t[member of]TU{0}] on a filtered probability space ([OMEGA], [F.sub.t], P) such that

(1) P([W.sub.0] = 0) = 1;

(2) if s < t and s, t [member of] T, then the increment [W.sub.t] - [W.sub.s] is independent of [F.sub.s] and is normally distributed with mean 0 and variance t - s;

(3) the process [W.sub.t] is almost surely continuous on T.

Note that property (3) is proved in the work [5].

For a random function f : [[0, [infinity]).sub.T] x [OMEGA] [right arrow] R we define the extension [??] : [0, [infinity]) x [OMEGA] [right arrow] R by

[??](t, [omega]) = f(sup [[0,t].sub.T], [omega])) (8)

for all t [member of] [0, [infinity]).

We shall make use of the definitions given in [2] for the classical Lebesgue and Riemann integral. For any random function f : [[0, [infinity]).sub.T] x [OMEGA] [right arrow] R and T < [infinity] we define its [DELTA]-Riemann (Lebesgue) integral as

[intergral.sup.T.sub.0] f(t, [omega]) [DELTA]t = [intergral.sup.T.sub.0] [??](t, [omega])dt, (9)

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 [2] defines a stochastic integral for an [L.sup.2][([0, T].sub.T])-progressively measurable random function f(t, [omega]) as

[intergral.sup.T.sub.0]f(t, [omega])[DELTA][W.sub.t] = [intergral.sup.T.sub.0] [??](t,[omega])d[W.sub.t], (10)

where again the right-hand side of the above equation is interpreted as a standard Ito stochastic integral. Note that the way (8) in which we define the extension guarantees that the function [??](t, [omega]) is progressively measurable.

In [2] the authors then defined the solution of the [DELTA]-stochastic differential equation indicated by the notation

[DELTA][X.sub.t] = b(t, X)[DELTA]t + [sigma](t, X) [DELTA][W.sub.t], (11)

as the process [mathematical expression not reproducible] such that

[mathematical expression not reproducible] (12)

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 invariable and uniform Lipschitz continuity in the t-variable of the functions b(t, x) and [sigma](t, x), together with being no worse than linear growth in x-variable, existence and pathwise uniqueness of strong solution to (11) are proved in [2].

4. Ito'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 T is at most countable.

Proof. If x [member of] T is a right-scattered point, then [I.sub.x] = (x, [sigma](x)) is an open interval such that [I.sub.x] [intersection] T = 0. Similarly, if x [member of] T is a left-scattered point, then [I.sub.x] = ([rho](x), x) is an open interval such that [I.sub.x] [intersection] T = 0. Suppose x < y and x, y [member of] 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.sub.x] [intersection] [I.sub.y] = 0. Suppose this is not the case, then we must have [sigma](x) > y. But we see that [sigma](x) = inf{s > x : s [member of] T} and y [member of] T. So we must have [sigma](x) [less than or equal to] 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.sub.x] [intersection] [I.sub.y] = 0.

Case 3 (x is left-scattered; y is right-scattered). In this case we see that [I.sub.x] = ([rho](x), x) and [I.sub.y] = (y, [sigma](y)), as well as x < y. This implies that [I.sub.x] [intersection] [I.sub.y] = 0.

Case 4 (x is right-scattered; y is left-scattered). In this case [I.sub.x] = (x, [rho](x)) and [I.sub.y] = ([rho](y), y). If [sigma](x) [less than or equal to] [rho](y), then [I.sub.x] [intersection] [I.sub.y] = 0. If [sigma](x) > [rho](y), then we see that (x, y) = [I.sub.x] [union] [I.sub.y] so that (x, y) [intersection] T = 0. That implies further that [sigma](x) = y and [rho](y) = x; that is, [I.sub.x] = [I.sub.y].

Thus we see that for all points x [member of] T being left- or right-scattered, the set of all open intervals of the form [I.sub.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 is 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.sub.1], [I.sub.2], ...} such that (1) for any k [not equal to] l, [I.sub.k] [intersection] [I.sub.l] = 0; (2) either the left-endpoint or right-endpoint or both endpoints of any of the [I.sub.k]'s are in T and are left- or right-scattered; (3) [I.sub.k] [intersection] T = 0 for any k = 1,2, ...; (4) any point in C is a left- or right-endpoint of one of the [I.sub.k]'s.

We will denote [mathematical expression not reproducible]. Since, for any x [member of] T, the points [sigma](x) and [rho](x) are in T, we further infer that, for any such interval [I.sub.k], we have the fact that [mathematical expression not reproducible] and [mathematical expression not reproducible] are in T, so that [mathematical expression not reproducible] is right-scattered and [mathematical expression not reproducible] is left-scattered.

We then establish the following Ito formula.

For any two points [t.sub.1], [t.sub.2] [member of] T, [t.sub.1] [less than or equal to] [t.sub.2], and any open interval [I.sub.k] [member of] I, such that [I.sub.k] [intersection] [[t.sub.1], [t.sub.2]] = 0, we have [I.sub.k] [subset] ([t.sub.1], [t.sub.2]). This is because if that is not the case, then [t.sub.1] or [t.sub.2] will belong to [I.sub.k], contradictory to the fact that [I.sub.k] [intersection] T = 0. We conclude that

{[I.sub.k] [member of] I: [I.sub.k] [intersection] [[t.sub.1], [t.sub.2]] [not equal to] 0} = {[I.sub.k] [member of] I: [I.sub.k] [subset] ([t.sub.1], [t.sub.2])}. (13)

Let us consider a function f(t, x) : T x R [right arrow] R. Let [f.sup.[DELTA]](t,x), [mathematical expression not reproducible] be the first- and second-order delta (Hilger) derivatives of f with respect to time variable t at (t, x) and let ([partial derivative]f/[partial derivative]x)(t,x) and ([[partial derivative].sup.2]f/[partial derivative][x.sup.2])(t, x) be the first- and second-order partial derivatives of f with respect to space variable x at (t, x).

Theorem 5 (Ito's formula). Let any function f : T x R [right arrow] R be suchthat [f.sup.[DELTA]](t,x), [mathematical expression not reproducible], ([partial derivative]f/[partial derivative]x)(t, x), ([[partial derivative].sup.2]f/[partial derivative][x.sup.2])(t,x), ([partial derivative][f.sup.[DELTA]]/[partial derivative]x)(t, x), and ([[partial derivative].sup.2][f.sup.[DELTA]]/[partial derivative][x.sup.2])(t, x) are continuous on T x R. Set any [t.sub.1] [less than or equal to] [t.sub.2], [t.sub.1], [t.sub.2] [member of] [[0, [infinity]).sub.T]; then we have

[mathematical expression not reproducible] (14

Proof. We will make use of the following classical version (Peano form) of Taylor's theorem: for any function f : T x R [right arrow] R such that ([partial derivative]f/[partial derivative]x)(t,x) and ([[partial derivative].sup.2]/[partial derivative][x.sup.2](t,x) are continuous on T x R, and any s [member of] T and [x.sub.1], [x.sub.2] [member of] R, we have

[mathematical expression not reproducible] (15)

where

[absolute value of ([R.sup.f.sub.C](s; [x.sub.1], [x.sub.2]))] [less than or equal to] r([absolute value of ([x.sub.2] - [x.sub.1])]) [([x.sub.2] - [x.sub.1]).sup.2], (16)

and r : [R.sub.+] [right arrow] [R.sub.+] is an increasing function with [lim.sub.u[down arrow]0]r(u) = 0.

We will also make use of the time scale Taylor formula (see [1, Theorem 1.113] as well as [15]) applied to f(t,x) up to first order in t: for any [s.sub.2] > [s.sub.1] and [s.sub.1], [s.sub.2] [member of] T; we have

f([s.sub.2], x) - f([s.sub.1], x) = [f.sup.[DELTA]]([s.sub.1], x) ([s.sub.2] - [s.sub.1]) + [R.sup.f.sub.TS](x; [s.sub.1], [s.sub.2]), (17)

where

[mathematical expression not reproducible] (18)

with r(*) as before.

Combining (15) and (17) we see that we have

[mathematical expression not reproducible] (19)

with

[absolute value of (R([s.sub.1], [s.sub.2]; [x.sub.1], [x.sub.2]))] [less than or equal to] r([absolute value of ([s.sub.2] - [s.sub.1])]) [absolute value of ([s.sub.2] - [s.sub.1])] + r([absolute value of ([x.sub.2] - [x.sub.1])])[([x.sub.2] - [x.sub.1]).sup.2] (20)

for another function r : [R.sub.+] [right arrow] [R.sub.+] increasing with [lim.sub.u[down arrow]0]r(M) = 0.

Consider a partition [[pi].sup.(n)] : [t.sub.1] = [s.sub.0] < [s.sub.1] < ... < [s.sub.n] = [t.sub.2], such that (1) each [s.sub.i] [member of] T; (2) [max.sub.i]([rho]([s.sub.i]) - [s.sub.i-1]) [less than or equal to] [1/2.sup.n] for i = 1, 2, ..., n. Notice that by definition [rho]([s.sub.i]) = sup{s < [s.sub.i] : s [member of] T}, so that we can always find [s.sub.i-1] [member of] T so that [rho]([s.sub.i]) - [s.sub.i-1] is sufficiently small.

Let the sets C and I be defined as before. Let us fix a partition [[pi].sup.(n)], and consider a classification of its corresponding intervals ([s.sub.i-1], [s.sub.i]), i = 1,2, ..., n. We will classify all intervals ([s.sub.i-1], [s.sub.i]) such that for all [I.sub.k] [member of] I we have [I.sub.k] [intersection] ([s.sub.i-1], [s.sub.i]) = 0 as class (a); and we classify all intervals ([s.sub.i-1], [s.sub.i]) such that there exist some [I.sub.k] [member of] I with ([s.sub.i-1], [s.sub.i]) [intersection] [I.sub.k] [not equal to] 0 as class (b). For an interval ([s.sub.i-1], [s.sub.i]) in class (a), since for all [I.sub.k] [member of] I we have [I.sub.k] [intersection] ([s.sub.i-1], [s.sub.i]) = 0, we see that [rho]([s.sub.i]) = [s.sub.i], because otherwise ([rho]([s.sub.i]), [s.sub.i]) will be one of the [I.sub.k]'s. Thus in this case we have [s.sub.i] - [s.sub.i-1] < [1/2.sup.n]. For an interval ([s.sub.i-1], [s.sub.i]) in class (b), since both [s.sub.i-1] and [s.sub.i] are in T, we see that we have in fact [I.sub.k] [subset or equal to] ([s.sub.i-1], [s.sub.i]). In this case either [I.sub.k] = ([s.sub.i-1], [s.sub.i]), or [I.sub.k] [not equal to] ([s.sub.i-1], [s.sub.i]). If the latter happens, then ([rho]([s.sub.i]), [s.sub.i]) [member of] I is one of the [I.sub.k]'s and [rho]([s.sub.i]) - [s.sub.i-1] < [1/2.sup.n]. We also see from the above analysis that all [I.sub.k]'s are contained in intervals ([s.sub.i-1], [s.sub.i]) that belong to class (b). On the other hand, either each interval ([s.sub.i-1], [s.sub.i]) is entirely one of the [I.sub.k]'s, or it contains an interval ([rho]([s.sub.i]), [s.sub.i]) that is one of the [I.sub.k]'s. For the latter case, that is, when [s.sub.i-1] < [rho]([s.sub.i]) < [s.sub.i], the set of intervals of the form ([s.sub.i-1], [rho]([s.sub.i])) are disjoint open intervals such that

[mathematical expression not reproducible] (21)

Now we have

[mathematical expression not reproducible] (22)

We apply (19) term by term in part (I) of (22), and we get

[mathematical expression not reproducible] (23)

We have the following four convergence results.

Convergence Result 1.1. By Lemma 6 ((35) and (36)) established below we have

[mathematical expression not reproducible] (24)

Convergence Result 1.2. By Lemma 7, (43), and Lemma 6, (35), established below we have

[mathematical expression not reproducible] (25)

Convergence Result 2. We have, with probability one, that

[mathematical expression not reproducible] (26)

as n [right arrow] [infinity].

In fact, by the Kolmogorov-Centsov theorem proved in Theorem 3.1 of [5] we know that for almost all trajectories of [W.sub.t] on T, for each fixed trajectory [W.sub.t]([omega]), there exists an [n.sub.0] = [n.sub.0]([omega]) such that for all n [greater than or equal to] [n.sub.0], for a partition [[pi].sup.(n)] with a classification of its intervals ([s.sub.i-1], [s.sub.i]) into classes (a) and (b) as above, [mathematical expression not reproducible] for some fixed [delta] > 0 and [gamma] > 0. From here we can estimate

[mathematical expression not reproducible] (27)

that is,

[mathematical expression not reproducible] (28)

Convergence Result 3. Let

[mathematical expression not reproducible] (29)

We claim that we have

P ([absolute value of ([A.sub.n]([omega]) - [B.sub.n]([omega]))] [right arrow] 0 as n [right arrow] [infinity]) = 1. (30)

In fact, from the analysis that leads to estimate (17) we see that we can write [A.sub.n] - [B.sub.n] as

[mathematical expression not reproducible] (31)

Here

[mathematical expression not reproducible] (32)

From (21), the Kolmogorov-Centsov theorem proved in Theorem 3.1 of [5], as well as the assumptions about function f, we see that

P([absolute value of ([(VI).sub.1])] + [absolute value of ([(VI).sub.2])] + [absolute value of ([(VI).sub.3])] + [absolute value of ([(VI).sub.4])] + [absolute value of ((VII))] [right arrow] 0 as n [right arrow] [infinity]) = 1. (33)

From here we immediately see the claim (30).

Note that for any interval [mathematical expression not reproducible] we have [mathematical expression not reproducible]; therefore we see that

[mathematical expression not reproducible] (34)

Combining 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 Ito's formula, but they are also of independent interest.

Lemma 6 (convergence of [DELTA]-deterministic and stochastic integrals). Given a time scale T and [t.sub.1], [t.sub.2] [member of] T, [t.sub.1] < [t.sub.2]; a probability space ([OMEGA], F, P); a Brownian motion [{[W.sub.t]}.sub.t[member of]T] on the time scale T, for any progressively measurable random function f that is continuous on [[t.sub.1], [t.sub.2]] [intersection] T, viewed as a [L.sup.2]([[[t.sub.1], [t.sub.2]].sub.T])- progressively measurable random function f(t, [omega]) on T, and the families of partitions [[pi].sup.(n)] : [t.sub.1] = [s.sub.0] < [s.sub.1] < ... < [s.sub.n] = [t.sub.2], [s.sub.0], [s.sub.1], ..., [s.sub.n] [member of] T, [max.sub.i=1,2, ..., n]([rho]([s.sub.i]) - [s.sub.i-1]) < [1/2.sup.n], one has

[mathematical expression not reproducible] (35)

Proof. As we have seen in the proof of Ito's formula, for a given partition [[pi].sup.(n)] : [t.sub.1] = [s.sub.0] < [s.sub.1] < ... < [s.sub.n] = [t.sub.2], such that [s.sub.i] [member of] T for i = 0, 1, ..., n, and [max.sub.i=1,2, ..., n]([rho]([s.sub.i]) - [s.sub.i-1]) < [1/2.sup.n], we can classify all intervals of the form ([s.sub.i-1], [s.sub.i]) into two classes (a) and (b): class (a) is those open intervals ([s.sub.i-1], [s.sub.i]) such that it does not contain any open intervals [I.sub.k] [member of] I; class (b) is those open intervals ([s.sub.i-1], [s.sub.i]) such that it contains at least one open interval [I.sub.k] [member of] I, the latter of which has endpoints that are left- or right-scattered.

Let us form a family of partitions [[sigma].sup.(n)] : [t.sub.1] = [r.sub.0] < [r.sub.1] < ... < [r.sub.m] = [t.sub.2], so that the partition [[sigma].sup.(n)] is the partition [[pi].sup.(n)] together with all points in T that are of the form [r.sub.j] = [rho]([s.sub.i]) for some [s.sub.i] in the partition [[pi].sup.(n)]. Note that under this construction we have [r.sub.0], [r.sub.1], ..., [r.sub.m] [member of] T. In fact, for any interval ([s.sub.i-1], [s.sub.i]) in (a), there is an identical interval ([r.sub.j-1], [r.sub.j]) in the partition [[sigma].sup.(n)] corresponding to it; for any interval ([s.sub.i-1], [s.sub.i]) in (b), there are two intervals ([r.sub.j-2], [r.sub.j-1]) and ([r.sub.j-1], [r.sub.j]) corresponding to it, so that [r.sub.j-1] = [rho]([s.sub.i]). And by (21) we know that

[mathematical expression not reproducible] (37)

Note that the number m depends on n and the partition [[pi].sup.(n)] : m = m(n, [[pi].sup.(n)]). In particular m [right arrow] [infinity] as n [right arrow] [infinity]. For simplicity we will suppress this dependence later in our proof.

Let us recall the definition of deterministic and stochastic [DELTA]-integrals as defined in Section 2. Let [??] be the extension of f that we have in (8): for any t [member of] T,

[??](t,[omega]) = f(sup [[0,t].sub.T], [omega]). (38)

Note that if t [member of] T is such that [rho](t) = t, then [??](t, [omega]) = f(t, [omega]); otherwise if t [member of] T is such that [rho](t) < t, then [??](t, [omega]) = f([rho](t), [omega]). Thus we see that

[mathematical expression not reproducible] (39)

So it suffices to prove that

[mathematical expression not reproducible] (40)

In fact, for any interval ([s.sub.i-1], [s.sub.i]) in class (a), there exist an interval ([r.sub.j-1], [r.sub.j]) identical to the interval ([s.sub.i-1], [s.sub.i]), so that

[mathematical expression not reproducible] (41)

For any open interval ([s.sub.i-1], [s.sub.i]) in class (b), there are two corresponding intervals ([r.sub.j-2], [r.sub.j-1]) and ([r.sub.j-1], [r.sub.j]) such that [r.sub.j-2] = [s.sub.i-1], [r.sub.j-1] = [rho]([s.sub.i]) and [r.sub.j-] = [s.sub.i]. In this case

[mathematical expression not reproducible] (42)

From the above calculations and the fact that we have (21) and that f is continuous on [[t.sub.1], [t.sub.2]] [intersection] T, together with the fact that [s.sub.j-1], [rho]([s.sub.j]) [member of] T, 0 [less than or equal to] [rho]([s.sub.j]) - [s.sub.j-1] [less than or equal to] [1/2.sup.n], we see the claim as follows.

Lemma 7 (convergence of quadratic variation of Brownian motion on time scale). Given a time scale T and [t.sub.1], [t.sub.2] [member of] T, [t.sub.1] < [t.sub.2]; a probability space ([OMEGA], F, P); a Brownian motion [{[W.sub.t]}.sub.t[member of]T] on the time scale T, let any [L.sup.2]([[[t.sub.1],[t.sub.2]].sub.T])-progressively measurable random function f(t, [omega]) on T be defined such that E[f.sup.2](t, [omega]) is uniformly bounded on [[t.sub.1], [t.sub.2]]. Consider the families

of partitions [[pi].sup.(n)] : [t.sub.1] = [s.sub.0] < [s.sub.1] < ... < [s.sub.n] = [t.sub.2], [s.sub.0], [s.sub.1], ..., [s.sub.n] [member of] T, [max.sub.i=1,2, ..., n]([rho]([s.sub.i]) - [s.sub.i-1]) < [1/2.sup.n]. One classifies all the intervals ([s.sub.i-1], [s.sub.i]), i = 1,2, ..., n into two classes (a) and (b) as before. Then one has

[mathematical expression not reproducible] (43)

Proof. We notice that for all intervals ([s.sub.i-1], [s.sub.i]) [member of] (a) we have [rho]([s.sub.i]) = [s.sub.i-1] and thus [s.sub.i] - [s.sub.i-1] < [1/2.sup.n]. Let us denote that

[mathematical expression not reproducible] (44)

Since f(t, [omega]) is progressively measurable, we see that f([s.sub.i-1], [omega]) is independent of [mathematical expression not reproducible]. Thus

[mathematical expression not reproducible] (45)

Furthermore

[mathematical expression not reproducible] (46)

If i < j, then [mathematical expression not reproducible] is independent of [mathematical expression not reproducible], so we have [mathematical expression not reproducible]. Similarly, for i > j we also have [mathematical expression not reproducible]. This implies that

[mathematical expression not reproducible] (47)

as n [right arrow] [infinity]. This together with the fact that E[V.sub.n] = 0 for any n implies claim (43) of the lemma.

The argument above leads us to an ItO formula for f(t, [W.sub.t]). Making use of the same methods, one can derive a more general Ito formula for the solution [X.sub.t] to the [DELTA]-stochastic differential equation (11). We will not repeat the proof, but we will claim the following theorem.

Theorem 8 (general Ito's formula). Let [X.sub.t] be the solution to the [DELTA]-stochastic differential equation (11). Let any function f: T x R [right arrow] R be such that [mathematical expression not reproducible] are continuous on T x R. For any [t.sub.1] [less than or equal to] [t.sub.2], [t.sub.1], [t.sub.2] [member of] [[0, [infinity]).sub.T] one has

[mathematical expression not reproducible] (48)

5. The Stochastic Exponential on Time Scales

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 9. One says an adapted stochastic process A(t) defined on the filtered probability space ([OMEGA], [F.sub.t], P) is stochastic regressive with respect to the Brownian motion [W.sub.t] on the time scale T if and only if for any right-scattered point t [member of] T one has

(1 + A(t)) ([W.sub.[sigma](t)] - [W.sub.t]) [not equal to] 0, a.s. [for all]t [member of] T. (49)

The set of stochastic regressive processes will be denoted by [R.sub.W].

The following definition of a stochastic exponential was also introduced in [3].

Definition 10 (stochastic exponential). Let [t.sub.0] [member of] T and A [member of] [R.sub.W]; then the unique solution of the [DELTA]-stochastic differential equation

[DELTA][X.sub.t] = A(t)[X.sub.t][DELTA][W.sub.t],

X([t.sub.0]) = 1,

t [member of] T (50)

is called the stochastic exponential and is denoted by

[X.sub.*] = [E.sub.A](*, [t.sub.0]). (51)

We note that [E.sub.A](t, [t.sub.0]) as a solution to (50) can be written into an integral equation

[mathematical expression not reproducible] (52)

We will be making use of the set-up we have in Section 4 about Ito's formula. Let [t.sub.0] < t and [t.sub.0], t [member of] T. Let the sets C and I be defined as in Section 4 corresponding to the interval [[t.sub.1], [t.sub.2]] = [[t.sub.0], t]. Let [I.sub.k] [member of] I and [mathematical expression not reproducible]. We note that [mathematical expression not reproducible]. Let

[mathematical expression not reproducible] (53)

We define

[mathematical expression not reproducible] (54)

Theorem 11 (stochastic exponential on time scales). The stochastic exponential has the closed-form expression

[E.sub.A](t, [t.sub.0]) = U(t, [t.sub.0])V(t, [t.sub.0]). (55)

Proof. Consider the process

[mathematical expression not reproducible] (56)

Let us introduce another function [alpha](t) such that

[mathematical expression not reproducible] (57)

We see now that the process [Y.sub.t] is a solution to the [DELTA]-stochastic differential equation

[mathematical expression not reproducible] (58)

Notice that [mathematical expression not reproducible] for any [mathematical expression not reproducible]. Taking this into account, as well as the fact that [alpha](s) = 0 whenever [mathematical expression not reproducible], we can apply the general Ito formula (48) to the function V(t, [t.sub.0]) = exp([Y.sub.t]) and we will get

[mathematical expression not reproducible] (59)

Thus

[mathematical expression not reproducible] (60)

or in other words

[DELTA]V(t, [t.sub.0]) = [alpha](t)V(t, [t.sub.0]) [DELTA][W.sub.t]. (61)

Let us now consider the function [E.sub.A](t, [t.sub.0]) = U(t, [t.sub.0])V(t, [t.sub.0]). We claim that

[mathematical expression not reproducible] (62)

Notice that

[mathematical expression not reproducible] (63)

that is,

[mathematical expression not reproducible] (64)

Using this fact, the above claimed identity (62) can be written as

[mathematical expression not reproducible] (65)

In fact, with respect to the partition [[pi].sup.(n)] : [t.sub.0] = [s.sub.0] < [s.sub.1] < ... < [s.sub.n-1] < [s.sub.n] = t that we have been using, we have

[mathematical expression not reproducible] (66)

Here

[mathematical expression not reproducible] (67)

We can apply the previous arguments and classify the intervals ([s.sub.i-1], [s.sub.i]) into classes (a) and (b). Notice that, on each interval [mathematical expression not reproducible], the function V(t, [t.sub.0]) remains constant and the function U(t, [t.sub.0]) has a jump, and on each interval ([s.sub.i-1], [s.sub.i]) in class (a) the function U(t,[t.sub.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 one, as n [right arrow] [infinity], we will have

[mathematical expression not reproducible] (68)

So we proved (65) and thus (62).

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 [2]).

Let us consider two processes: the standard Brownian motion [{[W.sub.t]}.sub.t[member of]T] on ([OMEGA], [F.sub.t], P) on the time scale T and the process

[B.sub.t] = [W.sub.t] - [[integral].sup.t.sub.0] A(s) [DELTA]s, (69)

on the time scale t [member of] T.

Let us consider an extension of the (probably random) function A(s) as in (8). Let us define the so obtained extension function to be [??](s). Recall that (8) implies that

[??](s, [omega]) = A(sup [[0, s].sub.T], [omega]). (70)

Let [[??].sub.t] be a standard Brownian motion on [0, [infinity]). If we define

[mathematical expression not reproducible] (71)

then the process [[??].sub.t] agrees with [B.sub.t] for any time point t [member of] T.

For any t, [t.sub.0] [member of] T, t > [t.sub.0], let

[mathematical expression not reproducible] (72)

It is easy to see that the function [G.sub.A](t, [t.sub.0]) is the standard Girsanov's density function for the process [[??].sub.t] with respect to the standard Brownian motion [[??].sub.t]. Since [[??].sub.t] and [[??].sub.t] have the same distributions as [B.sub.t] and [W.sub.t] on the time scale T, we conclude with the following two Theorems.

Theorem 12 (Novikov's condition on time scales). If for every t [greater than or equal to] 0 one has

E exp ([[integral].sup.t.sub.0] [A.sup.2](s) [DELTA]s) < [infinity], (73)

then for every t [greater than or equal to] 0 one has

E[G.sub.A](t, [t.sub.0]) = 1. (74)

Let (73) be satisfied. Let T > 0 and pick T > [t.sub.0], [t.sub.0], T [member of] T. Consider a new measure [P.sup.B] on ([OMEGA], [F.sub.t]), defining by it Radon-Nikodym derivative with respect to [P.sup.W], as

d[P.sup.B]/d[P.sup.W] = [G.sub.A](T, [T.sub.0]). (75)

Theorem 13 (Girsanov's change of measure on time scales). Under the measure [P.sup.B] the process [B.sub.t], t [member of] [[0, T].sub.T], is a standard Brownian motion on T.

7. Application to Brownian Motion on a Quantum Time Scale

In this section we are going to apply our result to a quantum time scale (q-time scale, see [1, Example 1.41]). Let q > 1 and

[q.sup.Z] := {[q.sup.k] : k [member of] Z},

[bar.[q.sup.Z]] := [q.sup.Z] [union] {0}. (76)

The quantum time scale (q-time scale) is defined by T = [bar.[q.sup.Z]]. Given the quantum time scale T, one can then construct a Brownian motion [W.sub.t] on T according to Definition 3.

We have

[sigma](t) = inf {[q.sup.n] : n [member of] [m + 1, [infinity])} = [q.sup.m+1] = q[q.sup.m] = [q.sup.t] (77)

if t = [q.sup.m] [member of] T and obviously [sigma](0) = 0. So we obtain

[sigma](t) = qt,

[rho](t) = t/q,

[for all]t [member of] T (78)

and consequently

[mu](t) = [sigma](t) - t = (q - 1)t [for all]t [member of] T. (79)

Here 0 is a right-dense minimum and every other point in T is isolated. For a function f: T [right arrow] R we have

[mathematical expression not reproducible] (80)

provided the limit exists.

The open intervals [I.sub.k] that we have constructed in Section 4 have the form [I.sub.k] = ([q.sup.k], [q.sup.k+1]) where k [member of] Z. For any two points [t.sub.1] < [t.sub.2], [t.sub.1], [t.sub.2] [member of] T, if [t.sub.1], [t.sub.2] [not equal to] 0, then [mathematical expression not reproducible] and [mathematical expression not reproducible] for two integers [k.sub.1] < [k.sub.2]. In this case we can apply (14) and we get

[mathematical expression not reproducible] (81)

Since T \ {0} is a discrete time scale, we have

[mathematical expression not reproducible] (82)

Moreover, we have

[mathematical expression not reproducible] (83)

Therefore (81) becomes

[mathematical expression not reproducible] (84)

which is a trivial telescoping identity. This justifies (14) in the case away from 0.

Let us consider now the case when [t.sub.1] = 0 and [t.sub.2] = [q.sup.k] > 0 for some k [member of] Z. In this case we have, according to (14), that

[mathematical expression not reproducible] (85)

One can justify that in this case we have

[mathematical expression not reproducible] (86)

Moreover, we have

[mathematical expression not reproducible] (87)

Therefore (81) becomes

[mathematical expression not reproducible] (88)

which is also a telescoping identity. This justifies (14) in the case including 0.

Making use of Theorem 11, it is easy to write down the stochastic exponential for the quantum time scale:

[mathematical expression not reproducible] (89)

https://doi.org/10.1155/2017/9140138

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The author would like to express his sincere gratitude to Professor Martin Bohner for inviting him to present this work at the Time Scales Seminar, Missouri S&T, on October 5, 2016, and also for pointing out the use of delta (Hilger) derivatives. He would also like to thank Professor David Grow for an inspiring discussion on the topic. Special thanks are dedicated to Missouri University of Science and Technology (formerly University of Missouri, Rolla) for a startup fund that supports this work.

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[3] M. Bohner and S. Sanyal, "The stochastic dynamic exponential and geometric Brownian motion on isolated time scales," Communications in Mathematical Analysis, vol. 8, no. 3, pp. 120-135, 2010.

[4] S. Sanyal and D. Grow, "Existence and uniqueness for stochastic dynamic equations," International Journal of Statistics and Probability, vol. 2, no. 2, 2013.

[5] D. Grow and S. Sanyal, "Brownian motion indexed by a time scale," Stochastic Analysis and Applications, vol. 29, no. 3, pp. 457-472, 2011.

[6] N. H. Du and N. T. Dieu, "The first attempt on the stochastic calculus on time scale," Stochastic Analysis and Applications, vol. 29, no. 6, pp. 1057-1080, 2011.

[7] D. Grow and S. Sanyal, "The quadratic variation of Brownian motion on a time scale," Statistics and Probability Letters, vol. 82, no. 9, pp. 1677-1680, 2012.

[8] S. Bhamidi, S. N. Evans, R. Peled, and P. Ralph, "Brownian motion on time scales, basic hypergrometric functions, and some continued fractions of Ramanujan," IMS Collections, vol. 2, pp. 42-75, 2008.

[9] T. Kumagai, "Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals," Publications of the Research Institute for Mathematical Sciences, vol. 33, no. 2, pp. 223-240, 1997.

[10] G. Ben Arous and T. Kumagai, "Large deviations for Brownian motion on the Sierpinski gasket," Stochastic Processes and Their Applications, vol. 85, no. 2, pp. 225-235, 2000.

[11] R. Cont and D.-A. Fournie, "Change of variable formulas for non-anticipative functionals on path space," Journal of Functional Analysis, vol. 259, no. 4, pp. 1043-1072, 2010.

[12] E. Haven, "Quantum calculus (q-calculus) and option pricing: a brief introduction," in Quantum Interaction, vol. 5494 of Lecture Notes in Computer Science, pp. 308-314, Springer, Berlin, Germany, 2009.

[13] E. Haven, "Ito's lemma with quantum calculus (q-calculus): some implications," Foundations of Physics, vol. 41, no. 3, pp. 529-537, 2011.

[14] W. Bryc, "On integration with respect to the q-Brownian motion," Statistics & Probability Letters, vol. 94, pp. 257-266, 2014.

[15] R. P Agarwal and M. Bohner, "Basic calculus on time scales and some of its applications," Results in Mathematics, vol. 35, no. 1-2, pp. 3-22, 1999.

Wenqing Hu

Department of Mathematics and Statistics, Missouri University of Science and Technology (Formerly University of Missouri, Rolla), Rolla, MO, USA

Correspondence should be addressed to Wenqing Hu; huwen@mst.edu

Received 8 December 2016; Revised 28 March 2017; Accepted 29 March 2017; Published 18 April 2017

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