Stochastic Analysis,
Stochastic Systems, and
Applications to Finance
8197.9789814355704-tp.indd 1 5/19/11 12:05 PM
NEW JERSEY
•
LONDON
•
SINGAPORE
•
BEIJING
•
SHANGHAI
•
HONG KONG
•
TAIPEI
•
CHENNAI
World Scientic
Allanus Tsoi
University of Missouri, Columbia, USA

David Nualart
University of Kansas, USA

George Yin
Wayne State University, Michigan, USA

Edited by
Stochastic Analysis,

Contributors and Addresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Part I. Stochastic Analysis and Systems
1. Multidimensional Wick-Itˆo Formula for Gaussian Processes . . . . . . . . 3
D. Nualart and S. Ortiz-Latorre
2. Fractional White Noise Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 27
A. H. Tsoi
3. Invariance Principle o f Regime-Switching Diffusions . . . . . . . . . . . 43
C. Zhu and G. Yin
Part II. Finance and Stochastics
4. Real Options and Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
A. Bensoussan, J. D. Diltz, and S. R. Hoe
5. Finding Exp e ctations of Monotone Functions of Binary Random
Var iables by Simulation, with Applications to Reliability,
Finance, and Round Robin Tournaments . . . . . . . . . . . . . . . . . . . . . 101
M. Brown, E. A. Pek¨oz, and S. M. Ross
6. Filtering with Counting Process Observations and Other
Facto rs: Applications to Bond Price Tick Data . . . . . . . . . . . . . . . . . 115
X. Hu, D. R. Kuipers, and Y. Zeng
May 13, 2011 11:8 WSPC - Proceedings Trim Size: 9in x 6in cnts
vi Contents
7. Jump Bond Markets Some Steps towards General Models
in Applications to Hedging and Utility Problems . . . . . . . . . . . . . . . 145
M. Kohlmann and D. Xiong
8. Recombining Tree for Regime-Switching Model: Algorithm
and Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193
R. H. Liu
9. Optimal Re insurance under a Jump Diffusion Model . . . . . . . . . . . 215
S. Luo
10. Applications of Counting Processes and Martingales in
Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

April 27, 2011 9:46 WSPC - Proceedings Trim Size: 9in x 6in 01-preface
viii Preface
finding expectations of monotone functions of binary random variables by
simulation, with applications to reliability, finance, and round robin tour-
naments, jump bond markets with general models in applications to hedg-
ing and utility proble ms , algorithm and weak convergence for reco mbining
tree in a regime-switching model, applications of counting processes and
martingales in survival analysis, extended filtering micro-movement model
with counting process observations a nd applicatio ns to bond price tick data,
optimal reinsurance for a jump diffusion mo del, recursive algorithms and
numerical studies for mean-reverting asset trading.
Without the encouragement and assistance of many colleag ue s, this vol-
ume would have never come into being. We thank all the authors of this
volume, and all of the speakers of the conference for their contributions. The
financial support provided by the University of Mis souri for this conference
is also greatly acknowledged.
Allanus Tsoi
Columbia, Missouri
David Nualart
Lawrence, Kansas
George Yin
Detroit, Michigan
April 21, 2011 16:30 WSPC - Proceedings Trim Size: 9in x 6in names
ix
Contributors and Addresses
• Alain Bensoussan, School of Management, University of Texas at
Dallas, Richardson, TX 75083-068 8, USA. & The Ho ng Kong Poly-
technic University, Hong Kong. Email: alain.bensoussan@utdallas.
edu
• Mark Brown, Department of Mathematics, City Colle ge, CUNY,

• Sheldon M. Ross, Department of Industrial and Systems Engineer-
ing, University of Souther n California, Los Angeles, CA 90089,
USA. Email: du
• Jianguo Sun, Department of Statistics, University of Mis souri,
USA. Email:
• Allanus Hak-Man Tsoi, Department of Mathematics, University o f
Missouri, Columbia, MO 65211, USA. Email:
• Dewen Xiong, Department of Mathematics, Shanghai Jiaotong
University, Shanghai 200240, People’s Republic of China. Email:

• George Yin, Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA. Email:
• Yong Zeng, Department of Mathematics and Statistics, University
of Missouri at Kansas City, Kansas City, MO 64110, USA. Email:

• Qing Zhang, Department of Mathematics, University of Georgia,
Athens, GA 30602, USA. Email:
• Chao Zhu, Department of Mathematical Sciences, University
of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA. Email:

• Chao Zhuang, Marshall School of Business, University of Southern
California, Los Angeles, CA 90089, USA. Email:
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
3
Multidimensional Wick-Itˆo Formula for Gaussian Processes
D. Nualart
∗
Department of Mathematics, University of Kansas
Lawrence, KS 66045, USA
E-mail:

and was further developed by Carmona
∗
Supported by the NSF Grant DMS-0604207
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
4 D. Nualart and S. Ortiz-Latorre
and Coutin
2
and Duncan, Hu and Pasik-Duncan
4
(see also Hu
5
and Nu-
alart
9
for a general survey papers on the stochastic calculus for the fBm).
The divergence integral can be approximated by Riemma n sums defined
using the Wick pr oduct, and it has the important property of having zero
expectation.
Nualart and Taqqu
11,12
have proved a Wick-Itˆo formula for general
Gaussian processes. In 11 they have considered Gaussian processes with
finite qua dratic variation, which includes the fBm with Hurst parameter
H > 1/2. The paper 12 deals with the change-of-variable formula for Gaus-
sian proces ses with infinite quadratic variation, in particular the fBm with
Hurst parameter H ∈ (1/4, 1/2). The lower bound for H is a natural one,
see Al`os, Mazet and Nua lart.
1
The aim of this pa per is to generalize the results of Nualart and Taqqu
12

for i, j = 1, . . . , d. For s = t, we have the covariance matrix V
t
= R(t, t).
We denote by H be the space obtained as the completion of the set of
step functions in A = [0, T ] × {1, . . . , d} with respect the sc alar product
1
i
[0,s]
, 1
j
[0,t]

H
= R
i,j
(s, t) , 0 ≤ s, t ≤ T, 1 ≤ i, j ≤ d,
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 5
where
1
i
[0,s]
= 1
[0,s]×{i}
(x, k) , (x, k) ∈ A.
The mapping 1
i
[0,t]
→ X
i

i

H
,
where h
1
, . . . , h
m
, g
1
, . . . , g
m
∈ H. The subspace of mth symmetric tensors
will be denoted by H

m
. In H

m
we introduce the modified scalar prod-
uct given by ·, ·
H

m
= m! ·, ·
H
⊗
m
. In this way, the multiple stochastic
integral I

n
∈ H, n ≥ 1, and f ∈ C
∞
b
(R
n
) (f and all its partial de riva-
tives are bounded). In S one can define the derivative operator D as
DF =
n

i=1
∂
i
f (X (h
1
) , . . . , X (h
n
)) h
i
,
which is an element o f L
2
(Ω; H). By iteration one obtains
D
m
F =
n

i

Definition 2.1. For m ≥ 1, the spac e D
m,2
is the completion of S with
respect to the norm F 
m,2
defined by
F 
2
m,2
= E[F
2
] +
m

i=1
E[


D
i
F


2
H
⊗
i
].
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
6 D. Nualart and S. Ortiz-Latorre

inf
≤ D,
where |π| = max
0≤i≤n−1
(t
i+1
− t
i
) , |π|
inf
= min
0≤i≤n−1
(t
i+1
− t
i
) , and
D is a positive constant.
Definition 2.3. Let u = {u
t
, t ∈ [0, T ]} be a d-dimensional stochastic pro-
cess such that u
i
t
∈ D
1,2
for all t ∈ [0, T ] and i = 1, . . . , d. The Wick
integral

T

 (X
j
t
i+1
− X
j
t
i
)
as |π| tends to zero, where π runs over all the partitions of the interval [0, T ]
in the class D.
May 27, 2011 13:32 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 7
3. Main Result
We will make use of the following assumptions.
Assumptions.
(A1) For all j, k ∈ {1, . . . , d} the function t → V
j,k
t
has bounded varia-
tion on [0, T ].
(A2) For all k, l ∈ {1, . . . , d}
n−1

i,j=0


E[∆
i
X

where ∆
i
X
j
= X
j
t
i+1
− X
j
t
i
, and π runs over all partitions of [0, T ]
in the class D.
Our purpose is to de rive a change-of-variable formula for the process
f(X
t
), where f : R
d
→ R if a function satisfying the following co nditio n.
(A4) For every multi-index α = (α
1
, , α
d
) ∈ N
d
with |α| := α
1
+ ··· +
α

Condition (3) holds if det V
t
> 0 for all t ∈ (0, T ], and the partial
derivatives ∂
α
f satisfy the exponential growth condition
|∂
α
f (x)| ≤ C
T
e
c
T
|x|
2
, (4)
for all t ∈ [0, T ] , x ∈ R
d
, where C
T
> 0 and c
T
are such that
0 < c
T
<
1
4
inf
0<t≤T

m
i
1
, ,i
m
f = ∂
i
m

∂
i
m−1
(···∂
i
2
(∂
i
1
f)

, i
k
∈ {1, . . . , d}, k = 1, . . . , m.
The next theorem is the main result of the paper.
Theorem 3.1. Suppose that the Gaussian process X and the function f
satisfy the preceding assumptions (A1) to (A4). Then the forward int egrals
(see Definition 2.3)

t
0

d

j,k=1

t
0
∂
2
j,k
f (X
s
) dV
j,k
s
.
Proof. Using the Taylor expansion of f up to fourth order in two consec-
utive points of a partition π = {0 = t
0
< t
1
< ··· < t
n
= t} in the class D
we obtain
f

X
t
i+1


X
j
∆
i
X
k
+
1
3!
T
π
3
(i) +
1
4!
T
π
4
(i) ,
where
T
π
3
(i) =
d

j,k,l=1
∂
3
j,k,l

X
j
∆
i
X
k
∆
i
X
l
∆
i
X
m
,
and
X
i
= λX
t
i
+ (1 − λ) X
t
i+1
, 0 ≤ λ ≤ 1.
By the definition of the Wick product, see Definition 2.2, one has
∂
j
f (X
t

= (t
i
, t
i+1
]. Taking into account that
D (∂
j
f (X
t
i
)) =
d

k=1
∂
2
j,k
f (X
t
i
) 1
k
[0,t
i
]
,
one gets
d

j=1

t
i
) 1
k
[0,t
i
]
, 1
j
δ
i

H
.
Using the definition of ·, ·
H
and adding and subtracting
1
2
E

∆
i
X
j
∆
i
X
k


−
1
2
E

∆
i
X
j
∆
i
X
k

,
where
ϕ
j,k
i
= E

X
j
t
i+1
− X
j
t
i


i
X
j
+
1
2
d

j,k=1
∂
2
j,k
f (X
t
i
)

∆
i
X
j
∆
i
X
k
− E

∆
i
X

t
) = f (X
0
) +
n−1

i=0

f

X
t
i+1

− f (X
t
i
)

= f (X
0
) +
n−1

i=0
d

j=1
∂
j

2
+
1
3!
R
π
3
+
1
4!
R
π
4
,
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
10 D. Nualart and S. Ortiz-Latorre
where
R
π
2
=
n−1

i=0
d

j,k=1
∂
2
j,k

i=0
T
π
3
(i) =
n−1

i=0
d

j,k,l=1
∂
3
j,k,l
f (X
t
i
) ∆
i
X
j
∆
i
X
k
∆
i
X
l
,

∆
i
X
l
∆
i
X
m
.
Note that
1
2
d

j,k=1
∂
2
j,k
f (X
t
i
) ϕ
j,k
i
=
1
2
d

j=1

2
d

j=1
∂
2
j,j
f (X
t
i
) (V
j,j
t
i+1
− V
j,j
t
i
)
+
d

k>j=1
∂
2
j,k
f (X
t
i
) (V

lim
|π|→0
1
2
n−1

i=0
d

j,k=1
∂
2
j,k
f (X
t
i
) (V
j,k
t
i+1
− V
j,k
t
i
) =
1
2
d

j,k=1

d

j=1
∂
j
f (X
t
i
)  ∆
i
X
j
=
d

j=1

t
0
∂
j
f (X
s
)  dX
j
s
,
and the result follows.
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 11

n
D
m
F, h
1
 ···h
m

H
⊗
m
, g
1
 ···g
n

H
⊗
n
=

D
m+n
F, h
1
 ···h
m
 g
1
 ···g

E [F ξ] = E[

D
2
F, h  g

H
⊗
2
].
Proof. It is an immediate consequence of the preceding lemma.
Lemma 4.4. Let F ∈ D
4,2
, h
1
, h
2
, g
1
, g
2
∈ H, ξ
1
= X (h
1
) X (g
1
) −
h
1

1
 g
1

H
⊗
4
] + E[

D
2
F, h
2
 g
1

H
⊗
2
] h
1
, g
2

H
+ E[

D
2
F, g

H
+ E[

D
2
F, h
1
 g
2

H
⊗
2
] h
2
, g
1

H
+ 2E [F ] h
1
 g
1
, h
2
 g
2

H
⊗

(F ξ
1
) =

D
2
F

ξ
1
+ 2DF Dξ
1
+ F D
2
ξ
1
,
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
12 D. Nualart and S. Ortiz-Latorre
where
Dξ
1
= h
1
X (g
1
) + X (h
1
) g
1

1
)
+2F (h
1
 g
1
) = A
1
+ 2A
2
+ 2A
3
+ 2A
4
.
Then,
E [A
1
, h
2
 g
2

H
⊗
2
] = E[ξ
1

D

2
]
= E[

D
4
F, h
2
 g
2
 h
1
 g
1

H
⊗
4
],
where we have applied Lemmas 4.3 and 4.1 in the second and third equalities
respectively. For the term B, we have
E [A
2
, h
2
 g
2

H
⊗

+
1
2
[X (g
1
) DF, g
2

H
] h
1
, h
2

H
=
1
2
E[

D
2
F, h
2
 g
1

H
⊗
2

3
we obtain
E [A
3
, h
2
 g
2

H
⊗
2
]
=
1
2
E[

D
2
F, h
1
 h
2

H
⊗
2
] g
1

2
 g
2

H
⊗
2
] = E [F ] h
1
 g
1
, h
2
 g
2

H
⊗
2
.
Adding up a ll the terms the result follows.
Lemma 4.5. The exponential growth condition (4) implies (3).
Proof. The exponential growth assumption (4) implies
E[|∂
α
f (X
t
)|
2
] ≤ C

2c
T
|X
t
|
2
] =
1
(2π)
d/2
|V
t
|
1/2

R
d
e
−x,Ax
dx =
1
2
d/2
|V
t
|
1/2
|A|
1/2
,

|
2
] = |I
d
− 4c
T
V
t
|
−1/2
,
provided A is symmetric and positive definite. This matrix is positive defi-
nite if and only if for all x ∈ R
d
with |x| > 0
x
T

1
4c
T
V
−1
t
− I
d

x =
1
4c

|
−1/2
=: a
T
,
which is finite by condition (5).
5. Convergence Results
From now on, C will denote a finite positive constant that may change from
line to line.
Proposition 5.1. Let
R
π
2
=
n−1

i=0
d

j,k=1
∂
2
j,k
f (X
t
i
)

∆
i

j,k
i
= ∂
2
j,k
f(X
t
i
) and
ϕ
j,k
i
= ∆
i
X
j
∆
i
X
k
− E

∆
i
X
j
∆
i
X
k

1
,i
2
=0
d

j
1
,j
2
,k
1
,k
2
=1
E[F
j
1
,k
1
i
1
F
j
2
,k
2
i
2
ϕ

j
1
,k
1
i
1
ϕ
j
2
,k
2
i
2
] = B
1
+ B
2
+ B
3
+ B
4
+ B
5
+ B
6
,
where
B
1
= E[D

δ
i
1
 1
k
2
δ
i
2

H
⊗
4
,
B
2
= E[D
2
(F
j
1
,k
1
i
1
F
j
2
,k
2


H
,
B
3
= E[D
2
(F
j
1
,k
1
i
1
F
j
2
,k
2
i
2
)], 1
k
1
δ
i
1
 1
k
2

1
i
1
F
j
2
,k
2
i
2
)], 1
j
1
δ
i
1
 1
j
2
δ
i
2

H
⊗
2
1
k
1
δ

j
1
δ
i
1
 1
k
2
δ
i
2

H
⊗
2
1
j
2
δ
i
2
, 1
k
1
δ
i
1

H
,

2
δ
i
2
 1
k
2
δ
i
2

H
⊗
2
.
Notice that the terms B
h
, h = 1, . . . , 6, depend on the indices i
1
, i
2
, j
1
, j
2
,
k
1
, and k
2

] =

6
h=1
B
h
. We have that
B
1
=
4

p=0

4
p

E[D
p
(F
j
1
,k
1
i
1
) D
4−p
(F
j

H
⊗
4
.
On the other hand
D
p
(F
j,k
i
) =
p

u
1
, ,u
d
=0
u
1
+···+u
d
=p
p!
u
1
! ···u
d
!
∂

1
=
4

p=0

4
p

p

u
1
, ,u
d
=0
u
1
+···+u
d
=p
4−p

v
1
, ,v
d
=0
v
1

v
(∂
2
j
2
,k
2
f(X
t
i
2
)

×(1
1
[0,t
i
1
]
)

u
1
 ···(1
d
[0,t
i
1
]
)

 1
j
2
δ
i
2
 1
k
1
δ
i
1
 1
k
2
δ
i
2

H
⊗
4
.
Notice that
(1
1
[0,t
i
1
]

)

v
d
= 1
w
1
[0,s
1
]
 1
w
2
[0,s
2
]
 1
w
3
[0,s
3
]
 1
w
4
[0,s
4
]
,
where w

[0,s
3
]
 1
w
4
[0,s
4
]
, 1
j
1
δ
i
1
 1
j
2
δ
i
2
 1
k
1
δ
i
1
 1
k
2

w
σ(2)
[
0,s
σ(2)
]
, 1
j
2
δ
i
2

H
1
w
σ(3)
[
0,s
σ(3)
]
, 1
k
1
δ
i
1

H
1

w
[0,s]
, 1
j
2
δ
i
2

H
||1
w
[0,s]
, 1
k
1
δ
i
1

H
||1
w
[0,s]
, 1
k
2
δ
i
2

2
]




E[X
w
s
∆
i
1
X
k
1
]




E[X
w
s
∆
i
2
X
k
2
]

)


] ≤ a
T
< ∞.
Hence, using Cauchy-Schwartz inequality,
B
1
≤ Ca
T



n−1

i=0
d

j,k=1
sup
0≤s≤t
1≤w ≤d


E[X
w
s
∆
i


i=0
sup
0≤s≤t
1≤w ≤d


E[X
w
s
∆
i
X
j
]


2



2
.
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
16 D. Nualart and S. Ortiz-Latorre
The last expression tends to zero as |π| → 0 by Assumption (A3). Analo-
gously
B
2
=

=2−p
p!
u
1
! ···u
d
!
(2 − p)!
v
1
! ···v
d
!
E[∂
u
(∂
2
j
1
,k
1
f(X
t
i
1
))∂
v
(∂
2
j

[0,t
i
2
]
)

v
1
 ···  (1
d
[0,t
i
2
]
)

v
d
,
1
j
2
δ
i
2
 1
k
1
δ
i

i
2
X
j
2
]




E[X
w
s
∆
i
1
X
k
1
]




E[∆
i
1
X
j
1

w
s
∆
i
X
j
]


2



×


n−1

i
1
,i
2
=0
d

j,k=1


E[∆
i

,k
1
i
1
F
j
2
,k
2
i
2
]1
j
1
δ
i
1
, 1
j
2
δ
i
2

H
1
k
1
δ
i

k
2
δ
i
2

H
1
k
1
δ
i
1
, 1
j
2
δ
i
2

H
≤ a
T


E[∆
i
1
X
j


E[∆
i
1
X
j
1
∆
i
2
X
k
2
]




E[∆
i
1
X
k
1
∆
i
2
X
j
2

k
]


2
≤ Ca
T
d

j,k=1
n−1

i
1
,i
2
=0


E[∆
i
1
X
j
∆
i
2
X
k
]

∆
i
X
l
,
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 17
then
lim
|π|→0
E[(R
π
3
)
2
] = 0.
Proof. Setting
∆
i
X
j
∆
i
X
k
∆
i
X
l
=

∆
i
X
k

∆
i
X
l
,
one gets
E[(R
π
3
)
2
]
≤ 2E





n−1

i=0
d

j,k,l=1
∂


2



+2E





n−1

i=0
d

j,k,l=1
∂
3
j,k,l
f (X
t
i
) ∆
i
X
l
E

∆




n−1

i=0
d

j,k=1
∂
3
j,k,l
f (X
t
i
) ∆
i
X
l

∆
i
X
j
∆
i
X
k
− E


X
l
=: g

X
t
i
, X
t
i+1

whose exact form does not
matter because it satisfies the exponential condition (4). Using Lemma 4.2,
we obtain tha t
C
2
=
n−1

i
1
,i
2
=0
d

j
1
,k
1

t
i
2
)∆
i
1
X
l
1
∆
i
2
X
l
2
]
×E

∆
i
1
X
j
1
∆
i
1
X
k
1

=0

d
j
1
,k
1
,l
1
,j
2
,k
2
,l
2
=1
E
h
, for h = 1, 2, and
E
1
= E[D
2
(∂
3
j
1
,k
1
,l


2
]
×E

∆
i
1
X
j
1
∆
i
1
X
k
1

E

∆
i
2
X
j
2
∆
i
2
X

)]1
l
1
δ
i
1
, 1
l
2
δ
i
2

H
×E

∆
i
1
X
j
1
∆
i
1
X
k
1

E