Tài liệu Stochastic Analysis, Stochastic Systems, and Applications to Finance docx

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,

Stochastic Systems, and
Applications to Finance
8197.9789814355704-tp.indd 2 5/19/11 12:05 PM
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STOCHASTIC ANALYSIS, STOCHASTIC SYSTEMS, AND APPLICATIONS
TO FINANCE
He Yue - Stochastic Analysis.pmd 5/11/2011, 3:59 PM1
May 13, 2011 11:8 WSPC - Proceedings Trim Size: 9in x 6in cnts
v
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

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

J. Sun
11. Stochastic Algorithms and Numerics for Mean-Reverting
Asset Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Q. Zhang, C. Zhuang, and G. Yin
April 27, 2011 9:46 WSPC - Proceedings Trim Size: 9in x 6in 01-preface
vii
Preface
This volume contains 11 chapters. It is an expanded version of the papers
presented at the first K ansas–Missouri Winter School of Applied Probabil-
ity, which was organized by Allanus Tsoi and was held at the University of
Missouri, February 14 and 15, 2008. It brought together resea rchers from
different parts of the country to review and to update the recent advances,
and to identify future directions in the areas of applied proba bility, stochas-
tic pro cesses, and their applications.
After the successful conference was over, there was a strong support
of publishing the paper s delivered in the conference as an archival volume.
Based on the support, we began the preparation on this project. In addition
to paper s reported at the conference, we have invited a number of collea gues
to contribute additional papers.
As an archive, this volume presents some of the highlights of the con-
ference, as well as some of most recent developments in stochastic sys tems
and applications . This book is naturally divided into two parts. The first
part contains some recent results in stochastic analysis, stochastic processes
and related fields. It explores the Itˆo formula for multidimensional Gaussian
processes using the Wick integral, introduces the notion of fractional white
noise multiplication, and discusses the LaSalle type of invariance principles
for hybrid sw itching diffusions. The second part of the book is devoted to fi-
nancial mathematics, insurance models, and applications. Included here are
optimal investment policies for irreversible capital investment projects un-
der uncertainty in monopoly and Stackelberg le ader-follower environments,

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,

New York, NY, USA. Email: cy om
• J. David Diltz, Department of Finance and Real Estate, The Uni-
versity of Texas at Arlington, Arlington, TX 76019, USA. Email:

• SingRu Hoe, School of Management, University of Texas at Dallas,
Richardson, TX 75083-0688, USA. Email:
• Xing Hu, Department of Economics, Princeton University, Prince-
ton, 08544, USA. Email:
• Michael Kohlmann, Department of Mathematics and Statistics,
University of Konstanz, D-78457, Konstanz, Germany. Email:

• David R. Kuipers, Department of Finance, Henry W. B loch
School of Business and Public Administration, University of Mis-
souri at Kansas City, Kansas City, MO 64 110, USA. Email:
kuip e
• Ruihua Liu, Department of Mathematics, University of Dayton,
300 College Park, Dayton, OH 45469-23 16, USA. Email: rui-

April 21, 2011 16:30 WSPC - Proceedings Trim Size: 9in x 6in names
x Contributors and Addresses
• Shangzhen Luo, Department of Mathematics, University of North-
ern Iowa, Cedar Falls, Iowa, 5 0614-0 506, USA. Email: luos @uni.edu
• David Nualart, Department of Mathematics, University of Kansas,
Lawrence, KS 66045, USA. Email:
• Salvador Ortiz-Latorre, Departament de Probabilitat, L`ogica i Es-
tad´ıstica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona,
Spain. Email:
• Erol A. Pekoz, School of Management, Boston University,
595 Commonwealth Avenue, Boston, MA 02215, USA. Email:

• 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:

S. Ortiz-Latorre
Departament de Probabilitat, L`ogica i Estad´ıstica, Universitat de Barcelona
Gran Via 585, 08007 Barcelona, Spain
Email:
An Itˆo formula for multidimensional Gaussian processes using the Wick inte-
gral is obtained. The conditions allow us to consider processes with infinite
quadratic variation. As an example we consider a correlated heterogenous frac-
tional Brownian motion. We als o use this Itˆo formula to compute the price of
an exchange option in a Wick-fractional Black-Scholes model.
Keywords: Wick-Itˆo formula; Gaussian processes; Malliavin calculus.
1. Introduction
The classical stochastic calculus and Itˆo’s formula c an be extended to semi-
martingales. There has been a recent interest in developing a stochastic
calculus for Gaussia n processes which are not semimartingales such as the
fractional Brownian motion (fBm for short). These developments are mo-
tivated by the fact that fBm a nd other related processes are suitable input
noises in practical problems arising in a variety of fields including finance,
telecommunications and hydrology (see, for instance, Mandelbrot and Van
Ness
7
and Sottinen
13
).
A possible definition of the stochastic integra l with respect to the fBm
is based on the divergence operator app earing in the stochastic calculus
of variations. This approach to define stochastic integrals star ted from the
work by Decreusefond and
¨
Ust¨unel
3

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

to the multidimensional case. We introduce the multidimensional Wick-Itˆo
integral as a limit of forward Riemann sums and prove a Wick-Itˆo formula
under conditions simila r to those in Nualart and Taqqu,
12
allowing infinite
quadratic variation processes.
The paper is organized as follows. In Section 2, we introduce the con-
ditions that the multidimensional Gaussian process must satify and state
the Itˆo formula. Section 3 co ntains some definitions in order to introduce
the Wick integral. In Section 4 we prove some technical lemmas using ex-
tensively the integration by parts formula for the derivative operator. The
convergence r e sults used in the proof o f the main theorem are proved in
Section 5. Section 6 is devoted to study two examples related to the multi-
dimensional fBm with parameter H > 1/4. Finally, in section 7 we use o ur
Itˆo formula to compute the price o f an exchange option on a Wick-fractional
Black-Scholes market.
2. Preliminaries
Let X = {X
t
, t ∈ [0, T ]} be a d-dimensional centered Gaussian proces s with
continuous covariance function ma trix R(s, t), that is,
R
i,j
(s, t) = E[X
i
s
X
j
t
],

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

t
can be extended to a linear isometry between the
space H and the Gaussian Hilbert space generated by the process X. We
denote by let I
1
: h → X (h) , h ∈ H this isometry.
Let H
⊗
m
denote the mth tensor power of H, equipped with the following
scalar pro duct
h
1
⊗ ···⊗h
m
, g
1
⊗ ···⊗g
m

H
⊗
m
=
m

i=1
h
i
, g

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

m
is a n isometry between H

m
and the mth Wiener chaos (see
Nualart
10
and also Janson
6
for a more detailed discussion of tensor prod-
ucts of Hilbert spaces). We denote by h
1
··· h
m
the symmetr iz ation of
the tensor pro duct h
1
⊗ ···⊗h
m
.
Now consider the set of smooth random variables S. A random variable
F ∈ S has the form
F = f (X (h
1
) , . . . , X (h
n
)) , (1)
with h
1
, . . . , h

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

1
, ,i
m
=1
∂
m
f
∂x
i
1
···∂x
i
m
(X (h
1
) , . . . , X (h
n
)) h
i
1
⊗ ···⊗h
i
m
,
which is an element o f L
2
(Ω; H

m
).

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

The Wick product F X (h) between a random var iable F ∈ D
1,2
and
the Gaussian random variable X (h) is defined as follows.
Definition 2.2. Let F ∈ D
1,2
and h ∈ H. Then the Wick product F X (h)
is defined by
F X (h) = F X (h) − DF, h
H
.
Actually, the Wick product coincides with the divergence (or the Sko-
rohod integral) of F h, and by the properties of the divergence operator we
can write
E [F X (h)] = E [DF, h
H
] . (2)
The Wick integral of a stochastic process u with respect to X is defined
as the limit of Riemann sums constructed using the Wick product. For this
we need some nota tion. Denote by D the set of all partitions of [0, T ]
π = {0 = t
0
< t
1
< ··· < t
n
= T }
such that
|π|
|π|

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

0
u
t
 dX
t
=
d

j=1

T
0
u
j
t
dX
j
t
is defined as the limit in probability, if it exis ts , of the forward Riemann
sums
d

j=1
n−1

i=0
u
j
t
i

 (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

k
∆
j
X
l
]


2
→ 0, as |π| → 0.
(A3) For all j, k ∈ {1, . . . , d}
n−1

i=0
sup
0≤s≤t


E[X
j
s
∆
i
X
k
]


2
→ 0, as |π| → 0,

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

d
≤ 7, the iterated derivatives
∂
α
f (x) =
∂
|α|
f
∂x
α
1
1
···∂x
α
d
d
(x)
exist, are c ontinuous, and satisfy
sup
t∈[0,T ]
E

|∂
α
f (X
t
)|
2

< ∞. (3)

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

x∈R
d
,|x|>0
x
T
V
−1
t
x
|x|
2
< ∞ (5)
(see Le mma 4.5).
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
8 D. Nualart and S. Ortiz-Latorre
Besides the multi-index notation for the derivatives, we will also us e the
following notation for iterated derivatives. Let f(x
1
, . . . , x
d
) be a sufficiently
smooth function, then
∂
i
f =
∂f
∂x
i
, i ∈ {1, . . . , d}
∂

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

∂
j
f (X
s
)  dX
j
s
, 0 ≤ t ≤ T, j = 1, . . . , d
exist and the following Wick-Itˆo formula holds:
f (X
t
) = f (X
0
) +
d

j=1

t
0
∂
j
f (X
s
)  dX
j
s
+
1
2

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


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

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

f (X
t
i
) ∆
i
X
j
∆
i
X
k
∆
i
X
l
,
T
π
4
(i) =
d

j,k,l,m=1
∂
4
j,k,l,m
f(X
i
)∆
i

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

i
) ∆
i
X
j
= ∂
j
f (X
t
i
)  ∆
i
X
j
+ D (∂
j
f (X
t
i
)) , 1
j
δ
i

H
,
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 9
where δ
i

= (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

∂
j
f (X
t
i
) ∆
i
X
j
=
d

j=1
∂
j
f (X
t
i
) ∆
i
X
j
+
d

j,k=1
∂
2
j,k
f (X

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


we have
1
k
[0,t
i
]
, 1
j
δ
i

H
= E[X
k
t
i
(X
j
t
i+1
− X
j
t
i
)] =
1
2
ϕ
j,k
i

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

X
k
t
i+1
+ X
k
t
i

.
This gives
f

X
t
i+1

= f (X
t
i
) +
d

j=1
∂
j
f (X
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

j
∆
i
X
k

+
1
2
d

j,k=1
∂
2
j,k
f (X
t
i
) ϕ
j,k
i
+ T
π
3
(i) + T
π
4
(i) .
Hence,
f (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

f (X
t
i
)  ∆
i
X
j
+
1
2
n−1

i=0
d

j,k=1
∂
2
j,k
f (X
t
i
) ϕ
j,k
i
+
1
2
R
π

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

f (X
t
i
)

∆
i
X
j
∆
i
X
k
− E

∆
i
X
j
∆
i
X
k

R
π
3
=
n−1


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
,

R
π
4
=
n−1

i=0
T
π
4
(i) =
n−1

i=0
d

j,k,l,m=1
∂
4
j,k,l,m
f(X
i
)∆
i
X
j
∆
i
X
k

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

j,k
t
i+1
− V
j,k
t
i
)
=
1
2
d

j,k=1
∂
2
j,k
f (X
t
i
) (V
j,k
t
i+1
− V
j,k
t
i
).
Using Assumption (A1) it is easy to show the almost sure convergence

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

∂
2
j,k
f (X
s
) dV
j,k
s
as |π| → 0. The convergences of R
π
2
and R
π
3
to zero in L
2
(Ω) as |π| → 0 are
proved in Propositions 5.1 and 5.2. The convergence of R
π
4
to zero in L
1
(Ω)
as |π| → 0 is proved in Proposition 5.3. This clearly implies the convergence
in probability
lim
|π|→0
n−1

i=0

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

Remark 3 .1. We can also consider a function f(t, x) depending on time
such that the par tial derivative
∂f
∂t
(t, x) exists and is continuous. In this
case we obtain the additional term

t
0
∂f
∂t
(s, X
s
)ds.
In order to prove Propositions 5.1, 5.2 and 5.3 we need to introduce
some technical c oncepts a nd prove a number of lemmas.
4. Technical Lemmas
In this section we establish some preliminary lemmas. The first one is trivial.
Lemma 4.1. Let F ∈ D
m+n,2
and h
1
, . . . , h
m
, g
1
, . . . , g
n
∈ H. Then
D

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

n

H
⊗
m+n
.
The next le mmas are based on the integration by parts formula.
Lemma 4.2. Let F ∈ D
2,2
and h, g ∈ H. Then
E [F X (h) X (g)] = E[

D
2
F, h  g

H
⊗
2
] + E [F ] h, g
H
.
Proof. See Nualart a nd Taqqu
12
, Lemma 6.
Lemma 4.3. Let F ∈ D
2,2
, h, g ∈ H, ξ = X (h) X (g) − h, g
H
. Then

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

, g
1

H
and ξ
2
= X (h
2
) X (g
2
) − h
2
, g
2

H
. Then
E [F ξ
1
ξ
2
] = E[

D
4
F, h
2
 g
2
 h

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

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

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
⊗

2
.
Proof. Applying the last lemma with F replaced by Fξ
1
and ξ by ξ
2
, we
get
E [F ξ
1
ξ
2
] = E[

D
2
(F ξ
1
) , h
2
 g
2

H
⊗
2
].
Now, by the Leibniz rule for the derivative operator,
D
2

(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

,
D
2
ξ
1
= 2 (h
1
 g
1
) ,
and thus
D
2
(F ξ
1
) =

D
2
F

ξ
1
+ 2X (g
1
) (DF h
1
) + 2X (h
1
) (DF  g

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

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
⊗

2
]
= E [X (g
1
) DF h
1
, h
2
 g
2

H
⊗
2
]
=
1
2
[X (g
1
) DF, h
2

H
] h
1
, 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

] h
1
, g
2

H
+
1
2
E[

D
2
F, g
1
 g
2

H
⊗
2
] h
1
, h
2

H
.
Where we have used the integration by parts formula (2) and L emma 4.1.
Analogously, for A

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

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

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

2
T
sup
0≤t≤T
E[e
2c
T
|X
t
|
2
]. (6)
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 13
For any symmetric and positive definite matrix A we have

R
d
e
−x,Ax
dx =

π
d
|A|

1/2
,
where |A| = det(A). As a consequence,
E[e

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
,

with
A =
1
2
V
−1
t
− 2c
T
I
d
= 2c
T

1
4c
T
V
−1
t
− I
d

,
and this gives
E[e
2c
T
|X
t

|
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

T
x
T
V
−1
t
x − |x|
2
> 0,
which is implied by (5). Therefore,
E

|∂
α
f (X
t
)|
2

≤ C
2
T
sup
0≤t≤T
|I
d
− 4c
T
V
t

|
−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

X
j
∆
i
X
k
− E

∆
i
X
j
∆
i
X
k

.
Then
lim
|π|→0
E[(R
π
2
)
2
] = 0.
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
14 D. Nualart and S. Ortiz-Latorre
Proof. Set F

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


= X(1
j
δ
i
)X(1
k
δ
i
) − 1
j
δ
i
, 1
k
δ
i

H
.
Then
E[(R
π
2
)
2
] =
n−1

i

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
],
and by Lemma 4.4 we get the decomposition
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

4
(F
j
1
,k
1
i
1
F
j
2
,k
2
i
2
)], 1
j
1
δ
i
1
 1
j
2
δ
i
2
 1
k
1

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

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

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

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

i
1
, 1
k
2
δ
i
2

H
,
B
5
= E[D
2
(F
j
1
,k
1
i
1
F
j
2
,k
2
i
2
)], 1

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
,

B
6
= 2E[F
j
1
,k
1
i
1
F
j
2
,k
2
i
2
]1
j
1
δ
i
1
 1
k
1
δ
i
1
, 1
j

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

. We omit this dependence to simplify the notation and we set
B
h
=
n−1

i
1
,i
2
=0
d

j
1
,j
2
,k
1
,k
2
=1
B
h
,
so E[(R
π
2
)
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

2
,k
2
i
2
)], 1
j
1
δ
i
1
1
j
2
δ
i
2
1
k
1
δ
i
1
1
k
2
δ
i
2


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
!
∂

u
(∂
2
j,k
f(X
t
i
))(1
1
[0,t
i
]
)

u
1
···(1
d
[0,t
i
]
)

u
d
.
May 11, 2011 10:49 WSPC - Proceedings Trim Size: 9in x 6in 01-nu
Multidimensional Wick-Itˆo Formula for Gaussian Processes 15
Hence,
B

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
d
=4−p
p!
u
1
! ···u
d
!
(4 − p)!
v
1
! ···v
d
!
E

∂
u
(∂
2
j
1
,k
1
f(X
t
i
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
]
)

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

)

u
1
 ···  (1
d
[0,t
i
1
]
)

u
d
 (1
1
[0,t
i
2
]
)

v
1
 ···(1
d
[0,t
i
2
]

)

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

k
∈ {1, . . . , d}, s
k
∈ {t
i
1
, t
i
2
}, k = 1, . . . , 4. But for any 0 ≤ s
k
≤
t, w
k
∈ {1, . . . , d}, k = 1, . . . , 4,
|1
w
1
[0,s
1
]
 1
w
2
[0,s
2
]
 1
w
3

[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

δ
i
2

H
⊗
4
|
≤
1
4!

σ∈Σ
4
|1
w
σ(1)
[
0,s
σ(1)
]
, 1
j
1
δ
i
1

H
1

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
σ(4)
[
0,s
σ(4)
]
, 1
k
2
δ
i
2

H
|
≤ sup
0≤s≤t
1≤w ≤d
|1
w
[0,s]
, 1
j
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


H
|
= sup
0≤s≤t
1≤w ≤d


E[X
w
s
∆
i
1
X
j
1
]




E[X
w
s
∆
i
2
X
j

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



.
Furthermore, by Assumption (A4), we have
E[


∂
u
(∂
2
j
1
,k
1
f(X
t
i
))∂
v
(∂
2
j
2
,k
2
f(X
t
i
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

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


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=0

2
p

p

u
1
, ,u
d
=0
u
1
+···+u
d
=p
2−p

v
1
, ,v
d
=0
v
1
+···+v
d

=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

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

[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

1

H
⊗
2
1
j
1
δ
i
1
, 1
k
2
δ
i
2

H
≤ Ca
T
sup
0≤s≤t
1≤w ≤d


E[X
w
s
∆

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

∆
i
2
X
k
2
]


.
Therefore, by Cauchy-Schwartz inequality
B
2
≤ Ca
T



n−1

i=0
d

j=1
sup
0≤s≤t
1≤w ≤d


E[X

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

1
X
j
∆
i
2
X
k
]


2


1/2
which tends to zero as |π| → 0 by Assumptions (A2) and (A3). The proof
for the terms B
3
, B
4
and B
5
is almost the same as for the term B
2
. Finally,
B
6
= E[F
j
1

,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

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

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

1
∆
i
2
X
j
2
]




E[∆
i
1
X
k
1
∆
i
2
X
k
2
]


+ a
T


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

]


.
Hence,
B
6
≤ Ca
T
n−1

i
1
,i
2
=0


d

j,k=1
E[∆
i
1
X
j
∆
i
2
X

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



2
,
which tends to zero as |π| → 0 by Assumption (A2).
Proposition 5.2. If
R
π
3
=
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
,
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
j
∆
i
X
k
− E

∆
i
X
j
∆
i
X
k

∆
i
X
l
+E

∆
i
X
j

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

3
j,k,l
f (X
t
i
) ∆
i
X
l

∆
i
X
j
∆
i
X
k
− E

∆
i
X
j
∆
i
X
k



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

i
X
j
∆
i
X
k



2



= 2C
1
+ 2C
2
.
To prove the convergence of C
1
to ze ro, observe that
C
1
≤ C
d

l=1
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


∆
i
X
j
∆
i
X
k



2


.
So it s uffice s to fix l and apply Proposition 5.1 with the term ∂
2
j,k
f (X
t
i
)
replaced by ∂
3
j,k,l
f (X
t
i
) ∆
i

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

,l
1
,j
2
,k
2
,l
2
=1
E[∂
3
j
1
,k
1
,l
1
f(X
t
i
1
)∂
3
j
2
,k
2
,l
2
f(X

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


E

∆
i
2
X
j
2
∆
i
2
X
k
2

= E
1
+ E
2
,
where E
h
=

n−1
i
1
,i
2

=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

1
f(X
t
i
1
)∂
3
j
2
,k
2
,l
2
f(X
t
i
2
)), 1
l
1
δ
i
1
 1
l
2
δi
2

H

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

k
2

,
E
2
= E[∂
3
j
1
,k
1
,l
1
f(X
t
i
1
)∂
3
j
2
,k
2
,l
2
f(X
t
i
2

)]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


∆
i
2
X
j
2
∆
i
2
X
k
2

.