# Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales

Authors: Wenqing Hu
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

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 (q-time scale).


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## Body

## 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 Itô isometry; in [6] 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 [2], in that we define our stochastic integral using the definition given in [2]. We then present a general Itô’s formula for stochastic dynamical equations under the framework of [2]. Our Itô 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 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 [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-calledq-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 (q-time scale) case at the last section of the paper.The paper is organized as follows. In Section2 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 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

Atime 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 theforward jump operator by(1)σt=inf⁡s∈T:s>t∀t∈Tsuchthatthissetisnonempty,and thebackward jump operator by(2)ρt=sup⁡s∈T:s<t∀t∈Tsuchthatthissetisnonempty.Lett∈T. If σ(t)>t, then t is calledright-scattered. If σ(t)=t, then t is calledright-dense. If ρ(t)<t, then t is calledleft-scattered. If ρ(t)=t, then t is calledleft-dense. Moreover, the sets Tκ and Tκ are derived from T as follows: if T has a left-scattered maximum, then Tκ is the set T without that left-scattered maximum; otherwise, Tκ=T. If T has a right-scattered minimum, then Tκ is the set T without that right-scattered minimum; otherwise, Tκ=T. Thegraininess function is defined by μ(t)=σ(t)-t for all t∈Tκ.Notice that sinceT is closed, for any t∈T, the points σ(t) and ρ(t) are belonging to T.For a setA⊂R we denote the set AT=A∩T.Given a time scaleT and a function f:T→R, thedelta (orHilger)derivativefΔ(t) of f at t∈T is defined as follows ([1, Definition 1.10]).Definition 1.
Assumef:T→R is a function and let t∈Tκ. Then we define fΔ(t) to be the number (provided that it exists) with the property that, given any ε>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩T for some δ>0) such that (3)fσt-fs-fΔtσt-s≤εσt-s∀s∈U.The delta derivative is characterized by the following theorem [1, Theorem 1.16].Theorem 2.
Assume thatf:T→R is a function and let t∈Tκ. Then one has the following:(i)
iff is differentiable at t, then f is continuous at t.(ii)
iff is continuous at t and t is right-scattered, then f is differentiable at t with(4)fΔt=fσt-ftσt-t.(iii)
Ift is right-dense, then f is differentiable at t if and only if the limit(5)lims→tft-fst-sexists as a finite number. In this case(6)fΔt=lims→tft-fst-s.(iv)
Iff is differentiable at t, then(7)fσt=ft+μtfΔt.


## 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 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 scaleT⊂R is an adapted stochastic process {Wt}t∈T∪{0} on a filtered probability space (Ω,Ft,P) such that(1)
P(W0=0)=1;(2)
ifs<t and s,t∈T, then the increment Wt-Ws is independent of Fs and is normally distributed with mean 0 and variance t-s;(3)
the processWt is almost surely continuous on T.Note that property (3) is proved in the work [5].For a random functionf:[0,∞)T×Ω→R we define theextensionf~:[0,∞)×Ω→R by(8)f~t,ω=fsup⁡0,tT,ωfor all t∈[0,∞).We shall make use of the definitions given in [2] for the classical Lebesgue and Riemann integral. For any random function f:[0,∞)T×Ω→R and T<∞ we define its Δ-Riemann (Lebesgue) integral as (9)∫0Tft,ωΔt=∫0Tf~t,ωdt,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 L2([0,T]T)-progressively measurable random function f(t,ω) as(10)∫0Tft,ωΔWt=∫0Tf~t,ωdWt,where 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 f~(t,ω) is progressively measurable.In [2] the authors then defined the solution of the Δ-stochastic differential equation indicated by the notation(11)ΔXt=bt,XΔt+σt,XΔWt,as the process {Xt}t∈[0,T]T such that (12)Xt2-Xt1=∫t1t2bt,XtΔt+∫t1t2σt,XtΔWt,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 (11) are proved in [2].


## 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 ofT is at most countable.Proof.
Ifx∈T is a right-scattered point, then Ix=(x,σ(x)) is an open interval such that Ix∩T=∅. Similarly, if x∈T is a left-scattered point, then Ix=(ρ(x),x) is an open interval such that Ix∩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 Ix∩Iy=∅. 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 Ix∩Iy=∅.
Case
3 (x is left-scattered; y is right-scattered). In this case we see that Ix=(ρ(x),x) and Iy=(y,σ(y)), as well as x<y. This implies that Ix∩Iy=∅.
Case
4 (x is right-scattered; y is left-scattered). In this case Ix=(x,σ(x)) and Iy=(ρ(y),y). If σ(x)≤ρ(y), then Ix∩Iy=∅. If σ(x)>ρ(y), then we see that (x,y)=Ix∪Iy so that (x,y)∩T=∅. That implies further that σ(x)=y and ρ(y)=x; that is, Ix=Iy.
Thus we see that for all pointsx∈T being left- or right-scattered, the set of all open intervals of the form Ix 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.LetC 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={I1,I2,…} such that (1) for any k≠l, Ik∩Il=∅; (2) either the left-endpoint or right-endpoint or both endpoints of any of the Ik’s are in T and are left- or right-scattered; (3) Ik∩T=∅ for any k=1,2,…; (4) any point in C is a left- or right-endpoint of one of the Ik’s.We will denoteIk=(sIk-,sIk+). Since, for any x∈T, the points σ(x) and ρ(x) are in T, we further infer that, for any such interval Ik, we have the fact that sIk- and sIk+ are in T, so that sIk- is right-scattered and sIk+ is left-scattered.We then establish the following Itô formula.For any two pointst1,t2∈T, t1≤t2, and any open interval Ik∈I, such that Ik∩[t1,t2]≠∅, we have Ik⊂(t1,t2). This is because if that is not the case, then t1 or t2 will belong to Ik, contradictory to the fact that Ik∩T=∅. We conclude that (13)Ik∈I:Ik∩t1,t2≠∅=Ik∈I:Ik⊂t1,t2.Let us consider a functionf(t,x):T×R→R. Let fΔ(t,x), fΔ2(t,x) be the first- and second-order delta (Hilger) derivatives of f with respect to time variable t at (t,x) and let ∂f/∂x(t,x) and ∂2f/∂x2(t,x) be the first- and second-order partial derivatives of f with respect to space variable x at (t,x).Theorem 5 (Itô’s formula).
Let any functionf:T×R→R be such that fΔ(t,x), fΔ2(t,x), ∂f/∂x(t,x), ∂2f/∂x2(t,x), ∂fΔ/∂x(t,x), and ∂2fΔ/∂x2(t,x) are continuous on T×R. Set any t1≤t2, t1,t2∈[0,∞)T; then we have(14)ft2,Wt2-ft1,Wt1=∫t1t2fΔs,WsΔs+∫t1t2∂f∂xs,WsΔWs+12∫t1t2∂2f∂x2s,WsΔs+∑Ik∈I,Ik⊂t1,t2fsIk+,WsIk+-fsIk+,WsIk--∂f∂xsIk-,WsIk-WsIk+-WsIk--12∂2f∂x2sIk-,WsIk-sIk+-sIk-.Proof.
We will make use of the following classical version (Peano form) of Taylor’s theorem: for any functionf:T×R→R such that ∂f/∂x(t,x) and ∂2f/∂x2(t,x) are continuous on T×R, and any s∈T and x1,x2∈R, we have(15)fs,x2-fs,x1=∂f∂xs,x1x2-x1+12∂2f∂x2s,x1x2-x12+RCfs;x1,x2,where(16)RCfs;x1,x2≤rx2-x1x2-x12,and r:R+→R+ is an increasing function with limu↓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 s2>s1 and s1,s2∈T; we have(17)fs2,x-fs1,x=fΔs1,xs2-s1+RTSfx;s1,s2, where(18)RTSfx;s1,s2=∫s1ρs2s2-σsfΔ2sΔs≤rs2-s1s2-s1with r(∙) as before.
Combining (15) and (17) we see that we have(19)fs2,x2-fs1,x1=fs2,x2-fs1,x2+fs1,x2-fs1,x1=fΔs1,x2s2-s1+∂f∂xs1,x1x2-x1+12∂2f∂x2s1,x1x2-x12+RTSfx2;s1,s2+RCfs1;x1,x2=fΔs1,x1+∂fΔ∂xs1,x1x2-x1+12∂2fΔ∂x2s1,x1x2-x12+RCfΔs1;x1,x2s2-s1+∂f∂xs1,x1x2-x1+12∂2f∂x2s1,x1x2-x12+RTSfx2;s1,s2+RCfs1;x1,x2=fΔs1,x1s2-s1+∂f∂xs1,x1x2-x1+12∂2f∂x2s1,x1x2-x12+Rs1,s2;x1,x2,with (20)Rs1,s2;x1,x2≤rs2-s1s2-s1+rx2-x1x2-x12for another function r:R+→R+ increasing with limu↓0⁡r(u)=0.
Consider a partitionπ(n):t1=s0<s1<⋯<sn=t2, such that (1) each si∈T; (2) maxi⁡(ρ(si)-si-1)≤1/2n for i=1,2,…,n. Notice that by definition ρ(si)=sup⁡{s<si:s∈T}, so that we can always find si-1∈T so that ρ(si)-si-1 is sufficiently small.
Let the setsC and I be defined as before. Let us fix a partition π(n), and consider a classification of its corresponding intervals si-1,si,i=1,2,…,n. We will classify all intervals (si-1,si) such that for all Ik∈I we have Ik∩(si-1,si)=∅ as class (a); and we classify all intervals (si-1,si) such that there exist some Ik∈I with (si-1,si)∩Ik≠∅ as class (b). For an interval (si-1,si) in class (a), since for all Ik∈I we have Ik∩(si-1,si)=∅, we see that ρ(si)=si, because otherwise (ρ(si),si) will be one of the Ik’s. Thus in this case we have si-si-1<1/2n. For an interval (si-1,si) in class (b), since both si-1 and si are in T, we see that we have in fact Ik⊆(si-1,si). In this case either Ik=(si-1,si), or Ik≠(si-1,si). If the latter happens, then (ρ(si),si)∈I is one of the Ik’s and ρ(si)-si-1<1/2n. We also see from the above analysis that all Ik’s are contained in intervals (si-1,si) that belong to class (b). On the other hand, either each interval (si-1,si) is entirely one of the Ik’s, or it contains an interval (ρ(si),si) that is one of the Ik’s. For the latter case, that is, when si-1<ρ(si)<si, the set of intervals of the form (si-1,ρ(si)) are disjoint open intervals such that(21)∑si-1,si∈b,si-1<ρsi<siρsi-si-1<n2n.
Now we have(22)ft2,Wt2-ft1,Wt1=∑i=1nfsi,Wsi-fsi-1,Wsi-1=∑afsi,Wsi-fsi-1,Wsi-1+∑bfsi,Wsi-fsi-1,Wsi-1=I+II.
We apply (19) term by term in part (I) of (22), and we get(23)I=∑afsi,Wsi-fsi-1,Wsi-1=∑afΔsi-1,Wsi-1si-si-1+∂f∂xsi-1,Wsi-1Wsi-Wsi-1+12∂2f∂x2si-1,Wsi-1Wsi-Wsi-12+Rsi-1,si;Wsi-1,Wsi=∑i=1nfΔsi-1,Wsi-1si-si-1+∂f∂xsi-1,Wsi-1Wsi-Wsi-1+∑a12∂2f∂x2si-1,Wsi-1Wsi-Wsi-12+∑b12∂2f∂x2si-1,Wsi-1si-si-1-∑bfΔsi-1,Wsi-1si-si-1+∂f∂xsi-1,Wsi-1Wsi-Wsi-1+12∂2f∂x2si-1,Wsi-1si-si-1+∑aRsi-1,si;Wsi-1,Wsi=III1+III2+IV+V.
We have the following four convergence results.
Convergence Result
1.1. By Lemma 6 ((35) and (36)) established below we have(24)PIII1⟶∫t1t2fΔs,WsΔs+∫t1t2∂f∂xs,WsΔWsasn⟶∞=1.
Convergence Result 1.2. By Lemma 7, (43), and Lemma 6, (35), established below we have(25)PIII2⟶∫t1t212∂2f∂x2s,WsΔsasn⟶∞=1.
Convergence Result 2. We have, with probability one, that(26)V=∑aRsi-1,si;Wsi-1,Wsi⟶0as n→∞.
In fact, by the Kolmogorov–Čentsov theorem proved in Theorem 3.1 of [5] we know that for almost all trajectories of Wt on T, for each fixed trajectory Wt(ω), there exists an n0=n0(ω) such that for all n≥n0, for a partition π(n) with a classification of its intervals (si-1,si) into classes (a) and (b) as above, sup(a)⁡Wsi-Wsi-1≤δ/2γn/5 for some fixed δ>0 and γ>0. From here we can estimate (27)E∑aRsi-1,si;Wsi-1,Wsi≤E∑arsi-si-1si-si-1+rWsi-Wsi-1Wsi-Wsi-12≤r12nt2-t1+rδ2γn/5t2-t1;that is,(28)Plimn→∞∑aRsi-1,si;Wsi-1,Wsi=0=1.
Convergence Result
3. Let(29)II+IV=An=Anω=∑bfsi,Wsi-fsi-1,Wsi-1-fΔsi-1,Wsi-1si-si-1-∂f∂xsi-1,Wsi-1Wsi-Wsi-1-12∂2f∂x2si-1,Wsi-1si-si-1,Bn=Bnω=∑Ik∈I,Ik⊂t1,t2fsIk+,WsIk+-fsIk-,WsIk--fΔsIk-,WsIk-sIk+-sIk--∂f∂xsIk-,WsIk-WsIk+-WsIk--12∂2f∂x2sIk-,WsIk-sIk+-sIk-.
We claim that we have(30)PAnω-Bnω⟶0asn⟶∞=1. In fact, from the analysis that leads to estimate (17) we see that we can write An-Bn as(31)An-Bn=∑si-1,si∈b,si-1<ρsi<sifsi,Wsi-fsi-1,Wsi-1-fΔsi-1,Wsi-1si-si-1-∂f∂xsi-1,Wsi-1Wsi-Wsi-1-12∂2f∂x2si-1,Wsi-1si-si-1-∑si-1,si∈b,si-1<ρsi<sifsi,Wsi-fρsi,Wρsi-fΔρsi,Wρsisi-ρsi-∂f∂xρsi,WρsiWsi-Wρsi-12∂2f∂x2ρsi,Wρsisi-ρsi-∑Ik∈I,Ik⊂si-1,ρsifor  somesi-1,si∈bfsIk+,WsIk+-fsIk-,WsIk--fΔsIk-,WsIk-sIk+-sIk--∂f∂xsIk-,WsIk-WsIk+-WsIk--12∂2f∂x2sIk-,WsIk-sIk+-sIk-=VI1+VI2+VI3+VI4-VII.
Here(32)VI1=∑si-1,si∈b,si-1<ρsi<sifρsi,Wρsi-fsi-1,Wsi-1,VI2=∑si-1,si∈b,si-1<ρsi<sifΔρsi,Wρsisi-ρsi-fΔsi-1,Wsi-1si-si-1=∑si-1,si∈b,si-1<ρsi<sifΔρsi,Wρsi-fΔsi-1,Wsi-1si-si-1-fΔρsi,Wρsiρsi-si-1,VI3=∑si-1,si∈b,si-1<ρsi<sifΔρsi,WρsiWsi-Wρsi-fΔsi-1,Wsi-1Wsi-Wsi-1=∑si-1,si∈b,si-1<ρsi<sifΔρsi,Wρsi-fΔsi-1,Wsi-1Wsi-Wsi-1-fΔρsi,WρsiWρsi-Wsi-1,VI4=∑si-1,si∈b,si-1<ρsi<sifΔρsi,Wρsisi-ρsi-fΔsi-1,Wsi-1si-si-1=∑si-1,si∈b,si-1<ρsi<sifΔsi-1,Wsi-1si-1-ρsi+fΔρsi,Wρsi-fΔsi-1,Wsi-1si-ρsi,VII=∑Ik∈I,Ik⊂si-1,ρsifor  somesi-1,si∈bfsIk+,WsIk+-fsIk-,WsIk--fΔsIk-,WsIk-sIk+-sIk--∂f∂xsIk-,WsIk-WsIk+-WsIk--12∂2f∂x2sIk-,WsIk-sIk+-sIk-.
From (21), the Kolmogorov–Čentsov theorem proved in Theorem 3.1 of [5], as well as the assumptions about function f, we see that(33)PVI1+VI2+VI3+VI4+VII⟶0asn⟶∞=1.From here we immediately see the claim (30).
Note that for any intervalIk=(sIk-,sIk+) we have fΔ(sIk-,WsIk-)=f(sIk+,WsIk-)-f(sIk-,WsIk-)/sIk+-sIk-; therefore we see that(34)Bn=Bnω=∑Ik∈I,Ik⊂t1,t2fsIk+,WsIk+-fsIk-,WsIk--fsIk+,WsIk--fsIk-,WsIk--∂f∂xsIk-,WsIk-WsIk+-WsIk--12∂2f∂x2sIk-,WsIk-sIk+-sIk-=∑Ik∈I,Ik⊂t1,t2fsIk+,WsIk+-fsIk+,WsIk--∂f∂xsIk-,WsIk-WsIk+-WsIk--12∂2f∂x2sIk-,WsIk-sIk+-sIk-.
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 Itô’s formula, but they are also of independent interest.Lemma 6 (convergence ofΔ-deterministic and stochastic integrals).
Given a time scaleT and t1,t2∈T, t1<t2; a probability space (Ω,F,P); a Brownian motion {Wt}t∈T on the time scale T, for any progressively measurable random function f that is continuous on [t1,t2]∩T, viewed as a L2([t1,t2]T)-progressively measurable random function f(t,ω) on T, and the families of partitions π(n):t1=s0<s1<⋯<sn=t2, s0,s1,…,sn∈T, maxi=1,2,…,n⁡(ρ(si)-si-1)<1/2n, one has (35)Plimn→∞∑i=1nfsi-1,ωsi-si-1=∫t1t2fs,ωΔs=1,(36)Plimn→∞∑i=1nfsi-1,ωWsi-Wsi-1=∫t1t2fs,ωΔWs=1.Proof.
As we have seen in the proof of Itô’s formula, for a given partitionπ(n):t1=s0<s1<⋯<sn=t2, such that si∈T for i=0,1,…,n, and maxi=1,2,…,n⁡ρsi-si-1<1/2n, we can classify all intervals of the form (si-1,si) into two classes (a) and (b): class (a) is those open intervals (si-1,si) such that it does not contain any open intervals Ik∈I; class (b) is those open intervals (si-1,si) such that it contains at least one open interval Ik∈I, the latter of which has endpoints that are left- or right-scattered.
Let us form a family of partitionsσ(n):t1=r0<r1<⋯<rm=t2, so that the partition σ(n) is the partition π(n) together with all points in T that are of the form rj=ρ(si) for some si in the partition π(n). Note that under this construction we have r0,r1,…,rm∈T. In fact, for any interval (si-1,si) in (a), there is an identical interval (rj-1,rj) in the partition σ(n) corresponding to it; for any interval (si-1,si) in (b), there are two intervals (rj-2,rj-1) and (rj-1,rj) corresponding to it, so that rj-1=ρ(si). And by (21) we know that(37)∑si-1,si∈b,rj-2,rj-1is  corresponding  to  itrj-1-rj-2<n2n.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 stochasticΔ-integrals as defined in Section 2. Let f~ be the extension of f that we have in (8): for any t∈T,(38)f~t,ω=fsup⁡0,tT,ω.Note that if t∈T is such that ρ(t)=t, then f~(t,ω)=f(t,ω); otherwise if t∈T is such that ρ(t)<t, then f~(t,ω)=f(ρ(t),ω). Thus we see that (39)Plimn→∞∑j=1mfrj-1,ωrj-rj-1=∫t1t2f~s,ωds=1,Plimn→∞∑j=1mfrj-1,ωWrj-Wrj-1=∫t1t2f~s,ωdWs=1.
So it suffices to prove that(40)Plimn→∞∑i=1nfsi-1,ωsi-si-1-∑j=1mfrj-1,ωrj-rj-1=0=1,Plimn→∞∑i=1nfsi-1,ωWsi-Wsi-1-∑j=1mfrj-1,ωWrj-Wrj-1=0=1.
In fact, for any interval(si-1,si) in class (a), there exist an interval (rj-1,rj) identical to the interval (si-1,si), so that(41)fsi-1,ωsi-si-1-frj-1,ωrj-rj-1=0,fsi-1,ωWsi-Wsi-1-frj-1,ωWrj-Wrj-1=0.For any open interval (si-1,si) in class (b), there are two corresponding intervals (rj-2,rj-1) and (rj-1,rj) such that rj-2=si-1, rj-1=ρ(si), and rj=si. In this case(42)fsi-1,ωsi-si-1-frj-1,ωrj-rj-1-frj-2,ωrj-1-rj-2=fsi-1,ωsi-si-1-fρsi,ωsi-ρsi-fsi-1,ωρsi-si-1=fsi-1,ω-fρsi,ωsi-ρsi,fsi-1,ωWsi-Wsi-1-frj-1,ωWrj-Wrj-1-frj-2,ωWrj-1-Wrj-2=fsi-1,ωWsi-Wsi-1-fρsi,ωWsi-Wρsi-fsi-1,ωWρsi-Wsi-1=fsi-1,ω-fρsi,ωWsi-Wρsi.
From the above calculations and the fact that we have (21) and that f is continuous on [t1,t2]∩T, together with the fact that sj-1,ρ(sj)∈T, 0≤ρ(sj)-sj-1≤1/2n, we see the claim as follows.Lemma 7 (convergence of quadratic variation of Brownian motion on time scale).
Given a time scaleT and t1,t2∈T, t1<t2; a probability space (Ω,F,P); a Brownian motion {Wt}t∈T on the time scale T, let any L2([t1,t2]T)-progressively measurable random function f(t,ω) on T be defined such that Ef2(t,ω) is uniformly bounded on [t1,t2]. Consider the families of partitions π(n):t1=s0<s1<⋯<sn=t2, s0,s1,…,sn∈T, maxi=1,2,…,n⁡(ρ(si)-si-1)<1/2n. One classifies all the intervals (si-1,si),i=1,2,…,n into two classes (a) and (b) as before. Then one has (43)Plimn→∞∑afsi-1,ωWsi-Wsi-12-∑afsi-1,ωsi-si-1=0=1.Proof.
We notice that for all intervals(si-1,si)∈(a) we have ρ(si)=si-1 and thus si-si-1<1/2n. Let us denote that (44)Vn=∑afsi-1,ωWsi-Wsi-12-∑afsi-1,ωsi-si-1.
Sincef(t,ω) is progressively measurable, we see that f(si-1,ω) is independent of Wsi-Wsi-1. Thus(45)EVn=E∑afsi-1,ωWsi-Wsi-12-∑afsi-1,ωsi-si-1=∑aEfsi-1,ωsi-si-1-∑aEfsi-1,ωsi-si-1=0.
Furthermore(46)var⁡Vn=E∑afsi-1,ωWsi-Wsi-12-∑afsi-1,ωsi-si-12=E∑afsi-1,ωfsj-1,ωWsi-Wsi-12-si-si-1·Wsj-Wsj-12-sj-sj-1=∑aEfsi-1,ωfsj-1,ωWsi-Wsi-12-si-si-1·Wsj-Wsj-12-sj-sj-1.
Ifi<j, then f(si-1,ω)f(sj-1,ω)[(Wsi-Wsi-1)2-(si-si-1)] is independent of [(Wsj-Wsj-1)2-(sj-sj-1)], so we have Ef(si-1,ω)f(sj-1,ω)[(Wsi-Wsi-1)2-(si-si-1)]·[(Wsj-Wsj-1)2-(sj-sj-1)]=0. Similarly, for i>j we also have Ef(si-1,ω)f(sj-1,ω)[(Wsi-Wsi-1)2-(si-si-1)]·[(Wsj-Wsj-1)2-(sj-sj-1)]=0. This implies that(47)var⁡Vn=∑aEf2si-1,ωWsi-Wsi-12-si-si-12=∑aEf2si-1,ωEWsi-Wsi-12-si-si-12=∑aEf2si-1,ωEWsi-Wsi-14-2Wsi-Wsi-12si-si-1+si-si-12=∑aEf2si-1,ω3si-si-12-2si-si-12+si-si-12=2∑aEf2si-1,ωsi-si-12≤12n-1maxs∈t1,t2Ef2s,ω∑asi-si-1⟶0as n→∞. This together with the fact that EVn=0 for any n implies claim (43) of the lemma.The argument above leads us to an Itô formula forf(t,Wt). Making use of the same methods, one can derive a more general Itô formula for the solution Xt to the Δ-stochastic differential equation (11). We will not repeat the proof, but we will claim the following theorem.Theorem 8 (general Itô’s formula).
LetXt be the solution to the Δ-stochastic differential equation (11). Let any function f:T×R→R be such that fΔ(t,x), fΔ2(t,x), ∂f/∂x(t,x), ∂2f/∂x2(t,x), ∂fΔ/∂x(t,x), and ∂2fΔ/∂x2(t,x) are continuous on T×R. For any t1≤t2, t1,t2∈[0,∞)T one has(48)ft2,Xt2-ft1,Xt1=∫t1t2bs,XsfΔs,XsΔs+∫t1t2σs,Xs∂f∂xs,XsΔWs+12∫t1t2σ2s,Xs∂2f∂x2s,XsΔs+∑Ik∈I,Ik⊂t1,t2fsIk+,WsIk+-fsIk+,WsIk--σsIk-,WsIk-∂f∂xsIk-,WsIk-WsIk+-WsIk--12σ2sIk-,WsIk-∂2f∂x2sIk-,WsIk-sIk+-sIk-.


## 5. The Stochastic Exponential on Time Scales

Our target in this section is to establish a closed-form formula for thestochastic exponential in the case of general time scales T.Definition 9.
One says an adapted stochastic processA(t) defined on the filtered probability space (Ω,Ft,P) is stochastic regressive with respect to the Brownian motion Wt on the time scale T if and only if for any right-scattered point t∈T one has (49)1+AtWσt-Wt≠0,a.s.∀t∈T.The set of stochastic regressive processes will be denoted by RW.The following definition of a stochastic exponential was also introduced in [3].Definition 10 (stochastic exponential).
Lett0∈T and A∈RW; then the unique solution of the Δ-stochastic differential equation(50)ΔXt=AtXtΔWt,Xt0=1,t∈Tis called the stochastic exponential and is denoted by (51)X∙=EA∙,t0.We note thatEA(t,t0) as a solution to (50) can be written into an integral equation(52)EAt,t0=1+∫t0tAsEAs,t0ΔWs,∀t∈T.We will be making use of the set-up we have in Section 4 about Itô’s formula. Let t0<t and t0,t∈T. Let the sets C and I be defined as in Section 4 corresponding to the interval [t1,t2]=[t0,t]. Let Ik∈I and Ik=(sIk-,sIk+). We note that sIk-=ρ(sIk+), sIk+=σ(sIk-). Let(53)Dt,t0=∑Ik∈I,Ik⊂t0,tAsIk-WsIk+-WsIk--12∑Ik∈I,Ik⊂t0,tA2sIk-sIk+-sIk-.We define(54)Ut,t0=∏Ik∈I,Ik⊂t0,t1+AsIk-WsIk+-WsIk-,Vt,t0=exp∫t0tAsΔWs-12∫t0tA2sΔs-Dt,t0.Theorem 11 (stochastic exponential on time scales).
The stochastic exponential has the closed-form expression(55)EAt,t0=Ut,t0Vt,t0.Proof.
Consider the process(56)Yt=∫t0tAsΔWs-12∫t0tA2sΔs-Dt,t0.
Let us introduce another functionα(t) such that (57)αt=0,whent=sIk-ort=sIk+,At,otherwise.
We see now that the processYt is a solution to the Δ-stochastic differential equation (58)ΔYt=αtΔWs-12α2tΔs,Yt0=0.
Notice thatYsIk-=YsIk+ for any Ik=(sIk-,sIk+)∈I. Taking this into account, as well as the fact that α(s)=0 whenever t=sIk-ort=sIk+, we can apply the general Itô formula (48) to the function V(t,t0)=exp(Yt) and we will get(59)exp⁡Yt-1=∫t0tαsexp⁡YsΔWs-12∫t0tα2sexp⁡YsΔs+12∫t0tα2sexp⁡YsΔs=∫t0tαsexp⁡YsΔWs.
Thus(60)Vt,t0=1+∫t0tαsVs,t0ΔWs,or in other words(61)ΔVt,t0=αtVt,t0ΔWt.
Let us now consider the functionEA(t,t0)=U(t,t0)V(t,t0). We claim that(62)Ut,t0Vt,t0-1=∫t0tAsUs,t0Vs,t0ΔWs.
Notice that(63)UsIk+,t0=1+AsIk-WsIk+-WsIk-UsIk-,t0=UsIk-,t0+AsIk-UsIk-,t0WsIk+-WsIk-;that is,(64)UsIk+,t0-UsIk-,t0=AsIk-UsIk-,t0WsIk+-WsIk-.Using this fact, the above claimed identity (62) can be written as(65)Ut,t0Vt,t0-1=∫t0tαsUs,t0Vs,t0ΔWs+∑Ik∈I,Ik⊂t0,tAsIk-UsIk-,t0VsIk-,t0WsIk+-WsIk-=∫t0tαsUs,t0Vs,t0ΔWs+∑Ik∈I,Ik⊂t0,tVsIk-,t0UsIk+,t0-UsIk-,t0.
In fact, with respect to the partitionπ(n):t0=s0<s1<⋯<sn-1<sn=t that we have been using, we have(66)Ut,t0Vt,t0-1=∑i=1nUsi,t0Vsi,t0-Usi-1,t0Vsi-1,t0=∑i=1nUsi,t0-Usi-1,t0Vsi,t0-Vsi-1,t0+Usi-1,t0Vsi,t0-Vsi-1,t0+Vsi-1,t0Usi,t0-Usi-1,t0=∑i=1nUsi,t0-Usi-1,t0Vsi,t0-Vsi-1,t0+∑i=1nUsi-1,t0Vsi,t0-Vsi-1,t0+∑i=1nVsi-1,t0Usi,t0-Usi-1,t0=I+II+III.
Here(67)I=∑i=1nUsi,t0-Usi-1,t0Vsi,t0-Vsi-1,t0,II=∑i=1nUsi-1,t0Vsi,t0-Vsi-1,t0,III=∑i=1nVsi-1,t0Usi,t0-Usi-1,t0.
We can apply the previous arguments and classify the intervals(si-1,si) into classes (a) and (b). Notice that, on each interval (sIk-,sIk+), the function V(t,t0) remains constant and the function U(t,t0) has a jump, and on each interval (si-1,si) in class (a) the function U(t,t0) 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→∞, we will have (68)I⟶0,II⟶∫t0tαsUs,t0Vs,t0ΔWs,III⟶∑Ik∈I,Ik⊂t0,tVsIk-,t0UsIk+,t0-UsIk-,t0.
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 Section3 (originally from [2]).Let us consider two processes: the standard Brownian motion{Wt}t∈T on (Ω,Ft,P) on the time scale T and the process (69)Bt=Wt-∫0tAsΔs,on the time scale t∈T.Let us consider an extension of the (probably random) functionA(s) as in (8). Let us define the so obtained extension function to be A~(s). Recall that (8) implies that (70)A~s,ω=Asup⁡0,sT,ω.Let W~t be a standard Brownian motion on [0,∞). If we define (71)B~t=W~t-∫0tA~sds,then the process B~t agrees with Bt for any time point t∈T.For anyt,t0∈T, t>t0, let(72)GAt,t0=exp∫t0tA~sdWs-12∫t0tA~2sds=exp∫t0tA~sdWs-12∫t0tA2~sds=exp∫t0tAsΔWs-12∫t0tA2sΔs.It is easy to see that the functionGA(t,t0) 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 Bt and Wt on the time scale T, we conclude with the following two Theorems.Theorem 12 (Novikov’s condition on time scales).
If for everyt≥0 one has(73)Eexp⁡∫0tA2sΔs<∞,then for every t≥0 one has(74)EGAt,t0=1.Let (73) be satisfied. Let T>0 and pick T>t0, t0,T∈T. Consider a new measure PB on (Ω,Ft), defining by it Radon–Nikodym derivative with respect to PW, as (75)dPBdPW=GAT,t0.Theorem 13 (Girsanov’s change of measure on time scales).
Under the measurePB the process Bt, t∈[0,T]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(76)qZ≔qk:k∈Z,qZ¯≔qZ∪0.The quantum time scale (q-time scale) is defined by T=qZ¯. Given the quantum time scale T, one can then construct a Brownian motion Wt on T according to Definition 3.We have(77)σt=inf⁡qn:n∈m+1,∞=qm+1=qqm=qtif t=qm∈T and obviously σ(0)=0. So we obtain (78)σt=qt,ρt=tq,∀t∈Tand consequently (79)μt=σt-t=q-1t∀t∈T.Here0 is a right-dense minimum and every other point in T is isolated. For a function f:T→R we have (80)fΔt=fσt-ftμt=fqt-ftq-1t∀t∈T∖0,fΔ0=lims→0f0-fs0-s=lims→0fs-f0sprovided the limit exists.The open intervalsIk that we have constructed in Section 4 have the form Ik=(qk,qk+1) where k∈Z. For any two points t1<t2, t1,t2∈T, if t1,t2≠0, then t1=qk1 and t2=qk2 for two integers k1<k2. In this case we can apply (14) and we get(81)fqk2,Wqk2-fqk1,Wqk1=∫qk1qk2fΔs,WsΔs+∫qk1qk2∂f∂xs,WsΔWs+12∫qk1qk2∂2f∂x2s,WsΔs+∑k=k1k2-1fqk+1,Wqk+1-fqk+1,Wqk-∂f∂xqk,WqkWqk+1-Wqk-12∂2f∂x2qk,Wqkqk+1-qk.SinceT∖{0} is a discrete time scale, we have (82)∫qk1qk2∂f∂xs,WsΔWs=∑k=k1k2-1∂f∂xqk,WqkWqk+1-Wqk,12∫qk1qk2∂2f∂x2s,WsΔs=∑k=k1k2-112∂2f∂x2qk,Wqkqk+1-qk.Moreover, we have(83)∫qk1qk2fΔs,WsΔs=∑k=k1k2-1fqk+1,Wqk-fqk,Wqkqk+1-qkqk+1-qk=∑k=k1k2-1fqk+1,Wqk-fqk,Wqk.Therefore (81) becomes(84)fqk2,Wqk2-fqk1,Wqk1=∑k=k1k2-1fqk+1,Wqk-fqk,Wqk+fqk+1,Wqk+1-fqk+1,Wqk=∑k=k1k2-1fqk+1,Wqk+1-fqk,Wqk,which is a trivial telescoping identity. This justifies (14) in the case away from 0.Let us consider now the case whent1=0 and t2=qk>0 for some k∈Z. In this case we have, according to (14), that(85)fqk,Wqk-f0,0=∫0qkfΔs,WsΔs+∫0qk∂f∂xs,WsΔWs+12∫0qk∂2f∂x2s,WsΔs+∑j=-∞k-1fqj+1,Wqj+1-fqj+1,Wqj-∂f∂xqj,WqjWqj+1-Wqj-12∂2f∂x2qj,Wqjqj+1-qj.One can justify that in this case we have(86)∫0qk∂f∂xs,WsΔWs=∑j=-∞k-1∂f∂xqj,WqjWqj+1-Wqj,12∫0qk∂2f∂x2s,WsΔs=∑j=-∞k-112∂2f∂x2qj,Wqjqj+1-qj.Moreover, we have(87)∫0qkfΔs,WsΔs=∑j=-∞k-1fqj+1,Wqj-fqj,Wqjqj+1-qjqj+1-qj=∑j=-∞k-1fqj+1,Wqj-fqj,Wqj.Therefore (81) becomes(88)fqk,Wqk-f0,0=∑j=-∞k-1fqj+1,Wqj-fqj,Wqj+fqj+1,Wqj+1-fqj+1,Wqj=∑j=-∞k-1fqj+1,Wqj+1-fqj,Wqj,which is also a telescoping identity. This justifies (14) in the case including 0.Making use of Theorem11, it is easy to write down the stochastic exponential for the quantum time scale: (89)EA0,qk=∏j=-∞k-11+AqjWqj+1-Wqj.



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