DSpace at VNU: The First Attempt on the Stochastic Calculus on Time Scale

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ISSN 0736-2994 print/1532-9356 online

DOI: 10.1080/07362994.2011.610169

The First Attempt on the Stochastic

Calculus on Time Scale

1Faculty of Mathematics, Mechanics, and Informatics,

University of Science-VNU, Hanoi, Vietnam

2Department of Mathematics, Vinh University, Nghe An, Vietnam

The aim of this article is to study the Doob–Meyer decomposition theorem,

-stochastic integration and Ito’s formula for stochastic processes defined on time scale The obtained results can be considered as a first attempt on the stochastic calculus on time scale.

Keywords Doob–Mayer decomposition; Ito’s formula; Martingale; Natural

increasing process; Stochastic integration; Time scale

1991 Mathematics Subject Classification 60H10; 60J60; 34A40; 39A13.

Introduction

The stochastic calculus for discrete and continuous time had been studying for longtimes It is used to describe the mathematical stochastic models in economy, physics,biology, medicine and social sciences   Some of basic problems show concern

for studying are stochastic integration, Doob–Meyer decomposition theorem, stochastic

differential equation, Ito’s formulawhich have been studied carefully for both discreteand continuous time (see for example [9, 11–13])

Moreover, in recent years, the theory of time scale, which was introduced byHilger in his PhD thesis [6], has been born in order to unify continuous and discreteanalysis Since then, this topic has received much attention from many researchgroups [2, 6, 7] However, almost works for this topic focus only on the deterministicanalysis For stochastic analysis, there are not too much in mathematical literature.Here, we mention one of the first attempts on this direction, the article of Bhamidiand his research group [1] in which the authors developed the theory of Brownianmotion on time scale After, Suman, in his Ph.D dissertation [14], tried to defined

“stochastic integral on time scales” but he just deals with discrete time scale.

Received June 25, 2010; Accepted May 10, 2011

This work was done under the support of the Grand NAFOSTED, no 101.02.63.09.Address correspondence to Nguyen Huu Du, Faculty of Mathematics, Mechanics, andInformatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam;E-mail: dunh@vnu.edu.vn

1057

The aim of this article is to develop the theory of stochastic calculus on the time

scale which leads us to a way to unify the presentation of classical problems ofstochastic calculus in the discrete and continuous time The first attempt on thistopic is to consider the Doob–Meyer decomposition theorem, stochastic integration,Ito’s formula for stochastic processes indexed by a time scale

When constructing a stochastic integration on a time scale, the difficulty we arefaced here is the forward jump operator t since it can make an adapted processbecome non-adapted One can avoid this disadvantage by using the -integration

on the semi-open intervals of the form ti ti+1 Meanwhile, by the predictablerequirement of the integrand, we need to use semi-open intervals ti ti+1 Thedifference between them makes a wide gap on the stochastic calculus with the

-integration In using
-integration for stochastic calculus on time scale, we canovercome this difficulty Although it makes some inconveniences when we try

to define a stochastic dynamic equations on time scale because the -dynamicequations are more popular in references,
-dynamic equations are also interesting

in both theory and practice

The organization of this article is as follows In Section 1 we survey some basicnotions and properties of the analysis on time scale Section 2 presents Doob–Meyerdecomposition theorem for a submartingale indexed by a time scale In Section 3,

we give a definition of the
-stochastic integration, and deals with some basicproperties

1 Preliminaries on Time Scale

This section surveys some basic notions on the theory of the analysis on time scalewhich was introduced by Hilger [6] A time scale is a nonempty closed subset of thereal numbers
, and we usually denote it by the symbol  We assume throughoutthat a time scale  is endowed with the topology inherited from the real numbers

with the standard topology We define the forward jump operator and the backward

sup

(respectively backward) graininessis given by t= t − t (respectively t = t −) A point t∈  is said to be right-dense if t = t, right-scattered if t > t,

left-dense

left-scattered For every a b

The set k is defined to be  if  does not have a right-scattered minimum;otherwise it is  without this right-scattered minimum The set k is defined to

be  if  does not have a left-scattered maximum; otherwise it is  without thisleft-scattered maximum Let f be a function defined on , valued in
m We say

that f is
-differentiable (or simply: differentiable) at t∈k provided there exists

a vector f
t∈
m, called the derivative of f , such that for all  > 0 there is a

all s∈ V If f is differentiable for every t ∈k, then f is said to be differentiable on

 If  =
then delta derivative is ft from continuous calculus; if = , thedelta derivative is the backward difference,
f , from discrete calculus A function

f defined on  is ld-continuous if it is continuous at every left-dense point and if

the right-sided limit exists at every right-dense point The set of all ld-continuousfunction from  to a Banach space X is denoted by Cld X It is similar tonotation of rd-continuous If f  →
is a function, then we write f  →

for the function f t k Denote lims↑tfsbyft−or ft−if this limit exists It is easy to see that t is left-scatted then ft−= ft Let

that1is semi-ring of subsets of  Let m1 be the set function defined on1by

m1a b = Ab− Aa (1.1)

It is easy to show that m1is a countably additive measure on 1We write A

the Caratheodory extension of the set function m1 associated with the family 1

and call it the Stieltjes–Lebesgue
-measure associated with A on

Let E be a A

-measurable subset of k and f  →
 be an A

-measurablefunction The integral of f associated with the measures A

2 Doob–Meyer Decomposition

For a∈ k, denote a t t ∈ a be a probabilityspace with filtration t t∈ asatisfying the usual conditions (see [10]) The notions ofcontinuous process, rd-continuous process, ld-continuous process, cadlag process,martingale, submartingale, semimartingale, stopping time   for a stochastic process

X= Xt t∈ a on probability space   t t ∈ a are defined as usually

A right continuous process A= At t ∈ a is said to be increasing if it is tadapted, Aa= 0 and the sample paths of A are increasing functions on afor almostsure ∈  The increasing process A = At t∈ a is called integrable ifAt < forall t∈ a

-By convention, we write fa = fa for all function defined ona

Proposition 2.1 If M is a bounded martingale, A is increasing, integrable, then for



at

M
A = 

lim

Definition 2.2 An increasing process A= Att∈a is said to be natural if thereholds

MtAt= 

at

M

for every bounded martingale M= Mtt∈a

Proposition 2.3 Let Att∈a be an increasing process These statements hold

1) If At is rd-continuous and At is t −-measurable for t∈  ∩ a then At is natural Wheret − stands for

s<ts 2) If At is natural, then At ist −-measurable for t∈ a.

Proof 1) Since Atis rd-continuous, A

= 0 for any t ∈ a\ Further, Mtis amartingale then the set of points t where Mt−= Mt is at most countable These factsimply



at \M− M −
A= 0 as

2) Let A= At be a natural increasing process We need to show that At is

t −-measurable for t∈ a Consider a cadlag martingale Mt defined ona For any

M=  At  if  < t

It is easy to see that Mis-martingale Therefore, by (2.2) we have

Mt− Mt −At− At t − = At− At t − 2

= 0

Hence, At− At t − = 0 a.s The proof is complete

Example 2.4 Atis increasing process on time scale

i) In the case  = , Atis natural iff it is previsible, i.e., At ist−1-measurablefor t= 1 2   

ii)  =
then every continuous increasing process Atis natural

In the following, to simplify notations, we omit the upper case n if there is noconfusion

Let tt∈

a be an increasing process Fix a T ∈ a For any  > 0 we consider

a partition of a T inductively by letting t0= a and for i = 1 2    set:

in the sense of convergence in 1  

Proof First, we assume2

T< and show that lim →0S= T in 2   It

T < Further, the continuity

of ton the compact set a T implies that supii ti+1− ti

with probability 1 Hence, by using Lebesgue’s dominated convergence theorem,

ii)  =
the above theorem is the content of Theorem 21 in [5]

iii) The conclusion of Theorem 2.5 is still valid if in the expression(2.4), a= t0<

t1<· · · < tk= T is an arbitrary partition of 0 T , provided maxi i+1− ti

≤  for i = 0     k − 1

We recall the Dunford–Pettis theorem in [10].

Theorem 2.7 (Dunford–Pettis [10]) If Ynn∈ is uniformly integrable sequence of random variables, there exists an integrable random variable Y and a subsequence

Lemma 2.8 If Ynn∈ is integrable sequence of random variables on probability space

 1to an integrable random variable Y , then for each -field ⊂  the sequence  Yn  converges weakly to Y in 1

Proof For an arbitrary bounded random variable  on    we have

lim

n→ Yn =  Y  

Definition 2.9 A process X is said belong to class DL if for any t∈ a the set

X is a stopping time satisfying a

is uniformly integrable

Theorem 2.10 (Doob–Meyer Decomposition) Let X be a right continuous submartingale of class DL Then, there exist a right continuous martingale and a right continuous increasing process A such that

Xt= Mt+ At ∀t ∈ a a.s

If A is natural then M and A are uniquely determined up to indistinguishability.

Proof We firstly prove the uniqueness Suppose there exist two right continuousmartingales M, Mand two right continuous natural increasing processes A, Asuchthat

X = M + A = M+ A ∀t ∈  as

This relation implies that

Bt= At− A

t = M

t− Mt

is right continuous martingale

For each partition n a= t0< t1<· · · < tkn = t such that maxi i +1− ti

≤ 2−nof a t set

B n

kn−1 i=0

we may assume that Xa= 0 Consider a sequence of partitions n a= tn

i=1Xti− Xt i−1 t i−1 is an tj

knj=0–previsible increasing process and

It is easy to see that if X belong to class DL then Anb n

uniformly integrable Therefore, by Dunford–Pettis theorem, there is a subsequence

Therefore, by Lemma 2.8,

weak− lim

k→Mn k 

b   = Mb 

for any sub -field of 

Let =n∈ nand a≤ s ≤ t ≤ b with s t ∈  be fixed It is easy to see that



≥ 0 as

Since  is countable and dense with respect to a b and A is right continuous,

At ≥ Asa.s for all t > s in a b This means that A is increasing

Next, we check A to be natural Let tbe any bounded martingale Put

Proposition 2.11 For any T∈ a, let ti be defined by (2.3) Suppose that the

characteristic of a martingale M is continuous Then,

Proof By assumption,Mt is a natural increasing continuous process Therefore,

by virtue of Theorem 2.5, we have

3 The Ito–Stieltjes Integrals

a×  with the left continuous paths on a and -adapted

processes in

Definition 3.1 Every set in the -algebra

to be predictable if it is measurable with respect to

It is easily to see that

s t∈ a s < t F ∈ s We note that in general a left continuous process is notnecessarily predictable

Remark 3.2.

i) If = then the process t is predictable if t ist−1-measurable

ii) If  =
then the process t is predictable if it is understood by stochasticprocess, measurable with respect to -algebra generated by adapted leftcontinuous processes

Let M ∈ 2 Denote by2Mthe space of all real-valued, predictable processes

= t t ∈ a satisfying

2

at 2
M<for any t > a Fix a b > a and let2a b  Mbe the restriction of2Mon a b

We endow2a b  Mwith the norm

2

ab 2
Mand identify  and  in2a b  Mif − bM= 0

A process  defined on a b is called simple if there exist a partition  of a b

a= t0< t1<· · · < tn = b and bounded random variables fi such that fi is t i−1measurable for all i= 1 n and

-t=n

i=1

fit

i−1 ti t t∈ a b  (3.1)Denote the set of the simple processes by0

Lemma 3.3. 0 is dense in2a b  M with respect to the metric

d 2=  − 2

ab − 2
M

Proof Clearly, 0⊂ 2a b  M Take ∈ 2a b  M and set Kt =

t −KK t Then K∈ 2a b  Mand − KbM→ 0 as K → +Thus, it is enough to show that for any bounded ∈ 2a b  M we can find

n∈ 0 n= 1 2     such that  − nbM→ 0 as n →  Let

 =  ∈ 2a b  M  is bounded and there exist

where ti is a partition of a b such that maxi i+1− ti≤ 2−n It is clear that

n∈ 0 and n− bM→ 0 as n → by the bounded convergence theorem.From [9, Proposition I-5.1, p 21] we can conclude that  contains all boundedpredictable processes Thus,  = 2a b  MThe proof is complete

Definition 3.4 For a simple process  with the form (3.1) in0, define

It is easy to prove that the
-stochastic integral

ab 
M is b-measurableand the following relations hold

Lemma 3.5 For any ∈ 0, there hold

2   to a random variables  with 2< This limit does not depend

on the choice of the sequence n This leads to the following definition

Definition 3.6 The
-stochastic integration of a process ∈ 2a b  M withrespect to the square integrable martingale M on a b , denoted by

ab 
M, isdefined by

The stochastic integration has the following usual properties.

Proposition 3.8 Let ∈ 2a b  M and let   be two real numbers Then, the

following relations hold

ab + 
M = 
ab 
M+ 
ab 
M a.s.

(vi) If  is a real a-measurable bounded random variable, then ∈ 2a b  M

Proof Those above properties are true for  in 0 For  in 2a b  M we

Theorem 3.9 Let M ∈ 2and ∈ 2a b  M Then

Proof The formula (3.7) follows directly from the definition of
-stochasticintegral and (3.6) Further,

Definition 3.10 Let ∈ 2a b  M For each t∈ a b we define

at 
M for It

Theorem 3.11 For any ∈ 2a b  M the indefinite
-stochastic integral

t∈ab is an t



sup

t∈ abe an adapted, cadlag process satisfying
ab t2
Mt< and

n be a certain partition of a b We define the process nt sampled at nto be

Consider the case when t is a cadlag process but the assumption


ab t2
Mt< does not satisfy For any m ∈ , let m

Since, mn converges to m in 2, it converges to m in probability Therefore, forany  > 0,

Remark 3.12 To simplify the presentation, we define the integration only for a

martingale M ∈ 2 However, it is easy to deal with the definition of stochasticintegration for a semimartingale which can be decomposed as the sum of a boundedvariation process and a square integrable martingale (square semimartingale forshort)

We come to the definition of the quadratic co-variation Let X Y be twostochastic processes defined on a For each t∈ a we consider the partition n

Definition 3.13 The quadratic co-variation of X and Y on the interval a t is the

limit (as n→ ) in the sense of convergence in probability of the sums Bnt,provided this limit exists

The quadratic co-variation of the process Xtand Yton the interval a t isdenoted by X Y t If X= Y then X X t  X t is called the quadratic variation of

ii)  =
the above theorem is the content of Theorem 21 in [5]

iii) The conclusion of Theorem 2.5 is still valid if in the expression(2.4), a=...

We come to the definition of the quadratic co-variation Let X Y be twostochastic processes defined on a For each t∈ a we consider the partition n...

Definition 3.13 The quadratic co-variation of X and Y on the interval a t is the< /b>

limit (as n→ ) in the sense of convergence in probability of the sums Bnt,provided