Introduction
[8]
D
[10]
[5]
stochastic calculus on the time scale
1. Preliminaries on time scales
[5]
R T
T
forward jump operator
backward jump operator σ, ρ : T → T σ(t) = inf{s ∈ T : s > t}
inf ∅ = sup T ρ(t) = sup{s ∈ T : s < t} sup ∅ = inf T
graininess µ : T → R
+
∪ {0} µ(t) = σ(t) − t t ∈ T right-
dense σ(t) = t right-scattered σ(t) > t left-dense ρ(t) = t left-scattered ρ(t) < t
1
isolated t a, b ∈ T [a, b]
{t ∈ T : a  t  b} T
k
T T
T
f T R
m
f delta differentiable
differentiable t ∈ T
k
f
∆
(t) ∈ R

1.1. Proposition ([7]). The set I of all right-scattered points of T is at most countable.
A rd− T F
1
T
F
1
= {[a; b) : a, b ∈ T}.
F
1
T m
1
F
1
m
1
([a, b)) = A(b) − A(a).
m
1
F
1
. µ
A
∆
m
1
, F
1
∆− A T
E A
∆

− measurable, and
µ
A
∆
({t
0
}) = A(σ(t
0
)) − A(t
0
) (1.1)
(2) If a, b ∈ T and a  b, then
µ
A
∆
((a, b)) = A(b) − A(σ(a))
µ
A
∆
((a, b]) = A(σ(b)) − A(σ(a))
µ
A
∆
([a, b]) = A(σ(b)) − A(a)
2. Doob - Meyer decomposition
a ∈ T
k
T
a
= {x ∈ T : x  a} (Ω, F, {F

a
= 0
and the almost sure sample paths of A are increasing on T
a
.
2.2. Proposition. If M is a right continuous bounded martingale, A is increasing then for
any t ∈ T
a
, then
EM
t
A
t
= E

[a,t)
M
σ
s
∆A
s
. (2.1)
Proof. Fix a t ∈ T
a
. For any n ∈ N, consider a partion π
(n)
= {a = t
(n)
0
< t



M
σ(a)
if s = a
M
σ(t
(n)
i+1
)
if s ∈ (t
(n)
i
, t
(n)
i+1
] ∀i = 0, , k
n
− 2
M
t
if s ∈ (t
(n)
k
n
−1
, t).
Since M is right continuous, M
σ
s

N
π
(n)
s
∆A(s)

= lim
δ
π
(n)
→0
E


[a,t)
N
π
(n)
s
∆A(s)

= lim
δ
π
(n)
→0
E

M
σ(a)

k
n
−1
)
)

= lim
δ
π
(n)
→0
E

M
t
A
t
+
k
n
−1

i=1
A
σ(t
(n)
i−1
)
(M
σ(t

.
The proof is complete. 
f : T
a
→ R f
t−
f
t−
= lim
s↑t
f(s)
t ∈ T \ {min T} f
a−
= f(a)
2.3. Definition. An increasing process A = (A
t
)
t∈T
a
is said to be natural if for every
bounded cadlag martingale M = (M
t
)
t∈T
a
we have
EM
t
A
t

. Let M
t
be a
F
t
− cadlag martingale and t ∈ T
a
arbitrary. For any n ∈ N, we consider a partition
π
(n)
= {a = t
(n)
0
< t
(n)
1
< · · · < t
(n)
k
n
= t} of [a, t] such that δ
π
(n)
= max |t
(n)
i+1
−σ(t
(n)
i
)|  2

(n)
n
− 2.
M
σ(t
(n)
k
n
−1
)
if s ∈ (t
(n)
k
n
−1
; t)
Since M is a cadlag process,
M
s−
= lim
δ
π
(n)
→0
M
π
(n)
s
∀s ∈ [a, t).
Therefore, by the b ounded convergence theorem we have

)∆A
s
= E lim
δ
π
(n)
→0

[a,t)
(N
π
(n)
s
− M
π
(n)
s
)∆A
s
= lim
δ
π
(n)
→0
E

(M
σ(a)
− M
a

i
)
) + (M
t
− M
σ(t
(n)
k
n
−1
)
)(A
t
− A
σ(t
(n)
k
n
−1
)
)

.
Because σ(t
(n)
i
)  t
(n)
i+1
 σ(t

− measurable for s ∈ I ∩ [a, t) then
E [(M
σ
s
− M
s
)(A
σ
s
− A
s
)] = E [E(M
σ
s
− M
s
)(A
σ
s
− A
s
)|F
s
]
= E [(A
σ
s
− A
s
)E(M

[a,t)
M
s−
∆A
s
= EM
t
tA
t
,
i.e., (A
t
) is natural increasing processes
Necessary condition. Let A = (A
t
) be a natural increasing process. We need drive that A
σ
t
is F
t
−measurable for t ∈ I∩ T
a
. Let t ∈ I∩T
a
. It is easy to see the process

A
s
= A
s

σ(t)
− M
t
)(A
σ(t)
− A
t
) = 0 =⇒ E(M
σ(t)
− M
t
)A
σ(t)
= 0.
Since EM
t
(A
σ(t)
− E[A
σ(t)
| F
t
]) = 0,
E(M
σ(t)
− M
t
)(A
σ(t)
− E[A

])
= E(A
σ(t)
− E[A
σ(t)
| F
t
])(A
σ(t)
− E[A
σ(t)
| F
t
]) = 0,
which implies that A
σ(t)
− E[A
σ(t)
| F
t
] = 0 a.s. The proof is complete. 
2.5. Corollary. (A
t
) is increasing process on time scale T
i) T = N then (A
t
) is natural iff it is previsible.
ii) T = R then evry increasing process (A
t
) is natural if it is continuous.

k∈N
such that
weak-lim
k→∞
Y
n
k
= Y , i.e., for all bounded random variables ξ we have
lim
k→∞
E(ξY
n
k
) = E(ξY )
X
- (D)
{X
τ
: τ is a stopping time satisfying a  τ < ∞}
- (DL) t ∈ T
a
{X
τ
: τ is a stopping time satisfying a  τ  t}
2.7. Theorem (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
X
t
= M

B
t
= A
t
− A

t
= M

t
− M
t
is right continuous martingale.
Let ξ
t
be an arbitrary right continuous bounded martingale. For each partition π
(n)
: a =
t
(n)
0
< t
(n)
1
< · · · < t
(n)
n
= t of [a, t], we set
ξ
π

if s ∈ (t
(n)
n−1
; t)
.
we have
ξ
s−
= lim
δ
π
(n)
→0
ξ
π
(n)
s
∀s ∈ [a, t)
By the bounded convergence theorem we have
E

[a,t)
ξ
s−
∆A(s) = E lim
δ
π
(n)
→0


− A
σ(t
i
)
) + ξ
σ(t
n−1
)
(A
t
− A
σ(t
n−1
)
)

Therefore,
Eξ
t
(A
t
− A

t
) = E

[a,t)
ξ
s−
∆A(s) − E

)
− B
σ(t
i
)
)
+ ξ
σ(t
n−1
)
(B
t
− B
σ(t
n−1
)
)

= lim
δ
π
(n)
→0
E

ξ
0
(B
σ(0)
− B

Thus
Eξ
t
(A
t
− A

t
) = 0.
Now let X be an arbitrary bounded random variable and let us define the bounded mar-
tingale ξ by taking a right continuous version of E(X|F
t
)
t∈T
a
. From the above,
E(X(A
t
− A

t
)) = E

E(X(A
t
− A

t
)|F
t


are
indistinguishable. Hence, M = X − A and M

= X − A

are indistinguishable as well.
Next, we prove the existence of the decomposition. By uniqueness, it suffices to prove
the existence of the processes M and A on the interval [a; b] for fixed b ∈ T
a
. Without loss
of generality we may assume that X
a
= 0. Let π
(n)
: a = t
(n)
0
< t
(n)
1
< · · · < t
(n)
N
= b be
a partition of [a, b] such that δ
π
(n)
= max
t

t
(n)
j
+ A
(n)
t
(n)
j
Thus, M
(n)
= {M
(n)
t
(n)
j
}
t
(n)
j
∈π
(n)
is a martingale satisfying M
(n)
a
= X
a
and A
(n)
= {A
(n)

j
)
2.8. Lemma. {A
(n)
b
: n = 1, 2, · · · } is uniformly integrable.
Proof. Let λ > 0 be fix and define the random variable T
(n)
λ
by
T
(n)
λ
=





min{t
(n)
j−1
: j = 1, 2, , N and A
(n)
t
(n)
j
> λ}
b if {t
(n)

(n)
λ
is a
stopping time. By noting that
{T
(n)
λ
< b} = {A
(n)
b
> λ}
and that A
(n)
T
(n)
λ
 λ on this set, we obtain
0 
1
2

A
(n)
b
>2λ
A
(n)
b
dP 


T
(n)
λ
)dP
=

Ω
(X
b
− X
T
(n)
λ
)dP =

{A
(n)
b
>λ}
(X
b
− X
T
(n)
λ
)dP.
By using Chebyshev inequality we have:
P{A
(n)
b

(n)
>2λ
A
(n)
b
dP → 0 where λ → ∞,
i.e., {A
(n)
b
} is uniform integrability. 
Now we return to the proof of Theorem 2.8. By the Dunford - Pettis theorem, there
is a subsequence (A
(n
k
)
b
)
k∈N
converging weakly to an integrable random variable A
b
. We
claim that for any sub σ− algebra G of F,
weakly- lim
k→∞
E(A
(n
k
)
b
|G) = E(A

)
b
E(η|G)|G))
= lim
k→∞
E(A
(n
k
)
b
E(η|G)) = E(A
b
E(η|G)) = E(ηE(A
b
|G)).
We now define the processes M and A by
M
t
= E(X
b
− A
b
|F
t
); A
t
= X
t
− M
t

(n
k
)
b
|F
a
)
= X
a
− weak- lim
k→∞
E(M
(n
k
)
b
|F
a
) = X
a
− weak- lim
k→∞
M
(n
k
)
a
= weak- lim
k→∞
A

s
)
= X
t
− X
s
− weak- lim
k→∞

E(X
b
− A
(n
k
)
b
|F
t
) − E(X
b
− A
(n
k
)
b
|F
s
)

= X

− EA
(n
k
)
s

 0.
Since Π is countable and A is right continuous, it follows that there is a version of A
t
such
that A
t
 A
s
for all t > s in [a; b] almost surely. It follows that A is increasing. Next
we check that A is natural. Let ξ be any right continuous bounded martingale. By the
predictability of A
(n)
then
E

ξ
b
A
(n)
b

= Eξ
b
(A

(n)
σ(a)
− A
(n)
a
) +

t
(n)
k
∈π
(n)
Eξ
σ(t
(n)
k−1
)
(A
(n)
σ(t
(n)
k
)
− A
(n)
σ(t
(n)
k−1
)
).

Letting n → ∞ we obtain
Eξ
σ(a)
(A
σ(a)
− A
a
) +

t
(n)
k
∈π
(n)
Eξ
σ(t
(n)
k−1
)
(A
σ(t
(n)
k
)
− A
σ(t
(n)
k−1
)
) → E

Replacing ξ = (ξ
s
) by ξ = ξ
t∧s
for each t ∈ [a, b] it easy to conclude that
E [ξ
t
A
t
] = E

[a,t)
ξ
s−
∆A(s).
Thus A = (A
t
) is natural.
Finally, if X is of class (D), then X is uniformly integrable and the limit X
∞
=
lim
t→∞
X
t
exists almost surely and this limit belongs to L
1
. The Doob - Meyer decom-
positions of the discrete submartingales X
(n)

∆
∆