class="bi x1 y1 w3 h2"
Giulia Di Nunno · Bernt Øksendal
Frank Proske
Malliavin Calculus
for L
´
evy Processes
with Applications
to Finance
ABC
Giulia Di Nunno
Bernt Øksendal
Frank Proske
Department of Mathematics
University of Oslo
0316 Oslo
Blindern
Norway



ISBN 978-3-540-78571-2 e-ISBN 978-3-540-78572-9
Library of Congress Control Number: 2008933368
Mathematics Subject Classification (2000): 60H05, 60H07, 60H40, 91B28, 93E20, 60G51, 60G57
c
 2009 Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-
casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of
this publication or parts thereof is permitted only under the provisions of the German Copyright Law

we present a general Malliavin calculus for L´evy processes, covering both the
Brownian motion case and the pure jump martingale case via Poisson random
measures, and also some combination of the two. We also present many of the
recent applications to finance, including the following:
• The Clark–Ocone theorem and hedging formulae
• Minimal variance hedging in incomplete markets
• Sensitivity analysis results and efficient computation of the “greeks”
• Optimal portfolio with partial information
• Optimal portfolio in an anticipating environment
• Optimal consumption in a general information setting
• Insider trading
To be able to handle these applications, we develop a general theory of
anticipative stochastic calculus for L´evy processes involving the Malliavin
derivative, the Skorohod integral, the forward integral, which were originally
introduced for the Brownian setting only. We dedicate some chapters to the
generalization of our results to the white noise framework, which often turns
out to be a suitable setting for the theory. Moreover, this enables us to prove
VII
VIII Preface
results that are general enough for the financial applications, for example, the
generalized Clark–Ocone theorem.
This book is based on a series of courses that we have given in different
years and to different audiences. The first one was given at the Norwegian
School of Economics and Business Administration (NHH) in Bergen in 1996,
at that time about Brownian motion only. Other courses were held later, every
time including more updated material. In particular, we mention the courses
given at the Department of Mathematics and at the Center of Mathematics for
Applications (CMA) at the University of Oslo and also the intensive or com-
pact courses presented at the University of Ulm in July 2006, at the University
of Cape Town in December 2006, at the Indian Institute of Science (IIS) in

2.4 Exercises 25
3 Malliavin Derivative via Chaos Expansion 27
3.1 The Malliavin Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Computation and Properties of the Malliavin Derivative . . . . . . 29
3.2.1 Chain Rules for Malliavin Derivative . . . . . . . . . . . . . . . . . 29
3.2.2 Malliavin Derivative and Conditional Expectation . . . . . 30
3.3 Malliavin Derivative and Skorohod Integral . . . . . . . . . . . . . . . . . 34
3.3.1 Skorohod Integral as Adjoint Operator to the
Malliavin Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.2 An Integration by Parts Formula and Closability
oftheSkorohodIntegral 36
3.3.3 A Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . 37
3.4 Exercises 40
4 Integral Representations and the Clark–Ocone Formula 43
4.1 The Clark–Ocone Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 The Clark–Ocone Formula under Change of Measure . . . . . . . . . 45
IX
X Contents
4.3 Application to Finance: Portfolio Selection . . . . . . . . . . . . . . . . . . 48
4.4 Application to Sensitivity Analysis and Computation
ofthe“Greeks”inFinance 54
4.5 Exercises 59
5 White Noise, the Wick Product, and Stochastic
Integration 63
5.1 White Noise Probability Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 The Wiener–ItˆoChaosExpansionRevisited 65
5.3 The Wick Product and the Hermite Transform . . . . . . . . . . . . . . 70
5.3.1 Some Basic Properties of the Wick Product . . . . . . . . . . . 72
5.3.2 Hermite Transform and Characterization
Theorem for (S)

toHedging 109
7.2 The Donsker Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.3 The Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.4 Exercises 127
8 The Forward Integral and Applications 129
8.1 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 TheForwardIntegral 132
8.3 ItˆoFormula forForward Integrals 135
Contents XI
8.4 Relation Between the Forward Integral
andtheSkorohodIntegral 138
8.5 ItˆoFormula forSkorohodIntegrals 140
8.6 Application to Insider Trading Modeling . . . . . . . . . . . . . . . . . . . 142
8.6.1 Markets with No Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.6.2 Markets with Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.7 Exercises 154
Part II The Discontinuous Case: Pure Jump L´evy Processes
9 A Short Introduction to L´evy Processes 159
9.1 Basics on L´evyProcesses 159
9.2 The ItˆoFormula 163
9.3 The Itˆo Representation Theorem for Pure Jump
L´evyProcesses 166
9.4 Application to Finance: Replicability . . . . . . . . . . . . . . . . . . . . . . . 169
9.5 Exercises 171
10 The Wiener–Itˆo Chaos Expansion 175
10.1 Iterated Itˆo Integrals 175
10.2 The Wiener–Itˆo ChaosExpansion 176
10.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11 Skorohod Integrals 181
11.1 The Skorohod Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

∗
227
13.5 The Malliavin Derivative on G
∗
228
13.6 A Generalization of the Clark–Ocone Theorem . . . . . . . . . . . . . . 230
13.7 A Combination of Gaussian and Pure Jump L´evy Noises
in the White Noise Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
13.8 Generalized Chain Rules for the Malliavin Derivative . . . . . . . . 237
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
14 The Donsker Delta Function of a L´evy Process
and Applications 241
14.1 The Donsker Delta Function of a Pure Jump L´evy Process . . . . 242
14.2 An Explicit Formula for the Donsker Delta Function . . . . . . . . . 242
14.3 Chaos Expansion of Local Time for L´evyProcesses 247
14.4 Application to Hedging in Incomplete Markets . . . . . . . . . . . . . . 253
14.5 A Sensitivity Result for Jump Diffusions . . . . . . . . . . . . . . . . . . . 256
14.5.1 A Representation Theorem for Functions
of a Class of Jump Diffusions . . . . . . . . . . . . . . . . . . . . . . . 256
14.5.2 Application: Computation of the “Greeks” . . . . . . . . . . . . 261
14.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
15 The Forward Integral 265
15.1 Definition of Forward Integral and its Relation
with the Skorohod Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
15.2 Itˆo Formula for Forward and Skorohod Integrals . . . . . . . . . . . . . 268
15.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
16 Applications to Stochastic Control: Partial
and Inside Information 273
16.1 The Importance of Information in Portfolio Optimization . . . . . 273
16.2 Optimal Portfolio Problem under Partial Information . . . . . . . . 274

17.1 The Pure Jump Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
17.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
17.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
18 Absolute Continuity of Probability Laws 341
18.1 Existence of Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
18.2 Smooth Densities of Solutions to SDE’s Driven
by L´evyProcesses 345
18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Appendix A: Malliavin Calculus on the Wiener Space 349
A.1 Preliminary Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
A.2 Wiener Space, Cameron–Martin Space,
and Stochastic Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
A.3 Malliavin Derivative via Chaos Expansions . . . . . . . . . . . . . . . . . 359
Solutions 363
References 395
Notation and Symbols 407
Index 411
Introduction
The mathematical theory now known as Malliavin calculus was first intro-
duced by Paul Malliavin in [157] as an infinite-dimensional integration by
parts technique. The purpose of this calculus was to prove the results about
the smoothness of densities of solutions of stochastic differential equations
driven by Brownian motion. For several years this was the only known appli-
cation. Therefore, since this theory was considered quite complicated by many,
Malliavin calculus remained a relatively unknown theory also among mathe-
maticians for some time. Many mathematicians simply considered the theory
as too difficult when compared with the results it produced. Moreover, to a
large extent, these results could also be obtained by using H¨ormander’s earlier
theory on hypoelliptic operators. See also, for example, [20, 113, 224, 229].
This was the situation until 1984, when Ocone in [172] obtained an explicit

Other extension of Malliavin calculus within quantum probability have also
appeared, see, for example, [83, 84].
One way of interpreting the Malliavin derivative of a given random vari-
able F = F(ω), ω ∈ Ω, on the given probability space (Ω, F,P) is to regard
it as a derivative with respect to the random parameter ω. For this to make
sense, one needs some mathematical structure on the space Ω. In the original
approach used by Malliavin, for the Brownian motion case, Ω is represented
as the Wiener space C
0
([0,T]) of continuous functions ω :[0,T] −→ R with
ω(0) = 0, equipped with the uniform topology. In this book we prefer to use
the representation of Hida [98], namely to represent Ω as the space S

of tem-
pered distributions ω : S−→R, where S is the Schwartz space of rapidly
decreasing smooth functions on R (see Chap. 5). The corresponding probabil-
ity measure P is constructed by means of the Bochner–Minlos theorem. This
is a classical setting of white noise theory. This approach has the advantage
that the Malliavin derivative D
t
F of a random variable F : S

−→ R can
simply be regarded as a stochastic gradient.
In fact, if γ is deterministic and in L
2
(R) (note that L
2
(R) ⊂S


F (ω):=Ψ (ω, t),ω∈S

as the Malliavin–(Hida) derivative (or stochastic gradient)ofF at t.
This gives a simple and intuitive interpretation of the Malliavin deriva-
tive in the Brownian motion case. Moreover, some of the basic properties of
calculus such as chain rule follow easily from this definition. See Chap. 6.
Alternatively, the Malliavin derivative can also be introduced by means of
the Wiener–Itˆo chaos expansion [119]:
F =
∞

n=0
I
n
(f
n
)
Introduction 3
of the random variable F as a series of iterated Itˆo integrals of symmetric
functions f
n
∈ L
2
(R
n
) with respect to Brownian motion. In this setting, the
Malliavin derivative gets the form
D
t
F =

It is the purpose of this book to give an introductory presentation of the theory
of Malliavin calculus and its applications, mainly to finance. For pedagogical
reasons, and also to make the reading easier and the use more flexible, the
book is divided into two parts:
Part I. The Continuous Case: Brownian Motion
Part II. The Discontinuous Case: Pure Jump L´evy Processes
In both parts the emphasis is on the topics that are most central for the
applications to finance. The results are illustrated throughout with examples.
In addition, each chapter ends with exercises. Solutions to some selection of
exercises, with varying level of detail, can be found at the back of the book.
We hope the book will be useful as a graduate text book and as a source
for students and researchers in mathematics and finance. There are several
possible ways of selecting topics when using this book, for example, in a
graduate course:
Alternative 1. If there is enough time, all eighteen chapters could be included
in the program.
4 Introduction
Alternative 2. If the interest is only in the continuous case, then the whole
Part I gives a progressive overview of the theory, including the white noise
approach, and gives a good taste of the applications.
Alternative 3. Similarly, if the readers are already familiar with the continuous
case, then Part II is self-contained and provides a good text choice to cover
both theory and applications.
Alternative 4. If the interest is in an introductory overview on both the con-
tinuous and the discontinuous case, then a good selection could be the reading
from Chaps. 1 to 4 and then from Chaps. 9 to 12. This can be possibly sup-
plemented by the reading of the chapters specifically devoted to applications,
so according to interest one could choose among Chaps. 8, 15, 16, and also
Chaps. 17 and 18.
1


s<t
F
s

,
respectively,
F
t
= lim
ut
F
u
:=

u>t
F
u
.
See, for example, [128] or [206].
G.D. Nunno et al., Malliavin Calculus for L´evy Processes with Applications 7
to Finance,
c
 Springer-Verlag Berlin Heidelberg 2009
8 1 The Wiener–Itˆo Chaos Expansion
Definition 1.1. Arealfunctiong :[0,T]
n
→ R is called symmetric if
g(t
σ


[0,T ]
n
g
2
(t
1
, ,t
n
)dt
1
···dt
n
< ∞. (1.3)
Let

L
2
([0,T]
n
) ⊂ L
2
([0,T]
n
) be the space of symmetric square integrable
Borel real functions on [0,T]
n
. Let us consider the set
S
n

∈ L
2
(S
n
)and
g
2
L
2
([0,T ]
n
)
= n!

S
n
g
2
(t
1
, ,t
n
)dt
1
dt
n
= n!g
2
L
2

, ,t
n
)=
1
n!

σ
f(t
σ
1
, ,t
σ
n
), (1.5)
where the sum is taken over all permutations σ of (1, ,n). Note that

f = f
if and only if f is symmetric.
Example 1.2. The symmetrization of the function
f(t
1
,t
2
)=t
2
1
+ t
2
sin t
1

2

, (t
1
,t
2
) ∈ [0,T]
2
.
Definition 1.3. Let f be a deterministic function defined on S
n
(n ≥ 1) such
that
f
2
L
2
(S
n
)
:=

S
n
f
2
(t
1
, ,t
n

1
)dW (t
2
) ···dW (t
n−1
)dW (t
n
).
(1.6)
1.1 Iterated ItˆoIntegrals 9
Note that at each iteration i =1, , n the corresponding Itˆo integral with
respect to dW (t
i
) is well-defined, being the integrand

t
i
0
···

t
2
0
f(t
1
, , t
n
)
dW (t
1


1/2
=


Ω
X
2
(ω)P (dω)

1/2
.
Applying the Itˆo isometry iteratively, if g ∈ L
2
(S
m
)andh ∈ L
2
(S
n
), with
m<n, we can see that
E

J
m
(g)J
n
(h)


m

0
···
t
2

0
h(t
1
, ,t
n−m
,s
1
, ,s
m
)dW (t
1
) ···dW (t
n−m
)dW (s
1
) ···dW (s
m
)

=
T

0

t
2

0
h(t
1
, ,s
m−1
,s
m
)dW (t
1
) ···dW (s
m−1
)

ds
m
=
=
T

0
s
m

0
···
s
2

) ···dW (t
n−m
)

ds
1
···ds
m
=0
(1.7)
because the expected value of an Itˆo integral is zero. On the contrary, if both
g and h belong to L
2
(S
n
), then
E

J
n
(g)J
n
(h)

=
T

0
E


, ,s
n
)dW (s
1
) ···dW (s
n−1
)

ds
n
=
=
T

0
···
s
2

0
g(s
1
, ,s
n
)h(s
1
, ,s
n
)ds
1

2
(S
n
)
:=

S
n
g(t
1
, ,t
n
)h(t
1
, ,t
n
)dt
1
···dt
n
is the inner product of L
2
(S
n
). In particular, we have
J
n
(h)
L
2

(af + bg)=aJ
n
(f)+bJ
n
(g), for f,g ∈ L
2
(S
n
)
and a, b ∈ R.
Definition 1.7. If g ∈

L
2
([0,T]
n
) we define
I
n
(g):=

[0,T ]
n
g(t
1
, ,t
n
)dW (t
1
) dW(t

(S
n
)
= n!g
2
L
2
([0,T ]
n
)
(1.12)
for all g ∈

L
2
([0,T]
n
). Moreover, if g ∈

L
2
([0,T]
m
)andh ∈

L
2
([0,T]
n
), we

polynomials h
n
(x), x ∈ R, n =0, 1, 2, are defined by
h
n
(x)=(−1)
n
e
1
2
x
2
d
n
dx
n
(e
−
1
2
x
2
),n=0, 1, 2, , (1.13)
1.2 The Wiener–Itˆo Chaos Expansion 11
Thus, the first Hermite polynomials are
h
0
(x)=1,h
1
(x)=x, h

2
2
dx (see, e.g., [214]).
Proposition 1.8. If ξ
1
,ξ
2
, are orthonormal functions in L
2
([0,T]),we
have that
I
n

ξ
⊗α
1
1
ˆ
⊗···
ˆ
⊗ξ
⊗α
m
m

=
m

k=1

and the symmetrized tensor product f
ˆ
⊗g is the symmetrization of f ⊗ g.In
particular, from (1.14), we have
n!
T

0
t
n

0
···
t
2

0
g(t
1
)g(t
2
) ···g(t
n
)dW (t
1
) ···dW (t
n
)=g
n
h

0
1 dW (t
1
)dW (t
2
)dW (t
3
)=T
3/2
h
3

W (T )
T
1/2

= W
3
(T ) −3TW(T ).
1.2 The Wiener–Itˆo Chaos Expansion
Theorem 1.10. The Wiener–Itˆo chaos expansion. Let ξ be an F
T
-
measurable random variable in L
2
(P ). Then there exists a unique sequence
{f
n
}
∞


n=0
n!f
n

2
L
2
([0,T ]
n
)
. (1.17)
Proof By the Itˆo representation theorem there exists an F-adapted process
ϕ
1
(s
1
), 0 ≤ s
1
≤ T, such that
E

T

0
ϕ
2
1
(s
1

1
)
to conclude that there exists an F-adapted process ϕ
2
(s
2
,s
1
), 0 ≤ s
2
≤ s
1
,
such that
E

s
1

0
ϕ
2
2
(s
2
,s
1
)ds
2


ξ = g
0
+
T

0
g
1
(s
1
)dW (s
1
)+
T

0
s
1

0
ϕ
2
(s
2
,s
1
)dW (s
2
)dW (s
1

1
)

2

=
T

0
s
1

0
E[ϕ
2
2
(s
2
,s
1
)]ds
2
ds
1
≤ E[ξ
2
].
Similarly, for almost all s
2
≤ s

2
3
(s
3
,s
2
,s
1
)ds
3

≤ E[ϕ
2
2
(s
2
,s
1
)] < ∞ (1.23)
and
ϕ
2
(s
2
,s
1
)=E[ϕ
2
(s
2

1
)+
T

0
s
1

0
g
2
(s
2
,s
1
)dW (s
2
)dW (s
1
)
+
T

0
s
1

0
s
2

)], 0 ≤ s
2
≤ s
1
≤ T.
By (1.18), (1.20), (1.23), and the Itˆo isometry we have
E

T

0
s
1

0
s
2

0
ϕ
3
(s
3
,s
2
,s
1
)dW (s
3
)dW (s

0
,g
1
, ,g
n
, with g
0
constant and g
k
defined on S
k
for 1 ≤ k ≤ n, such that
ξ =
n

k=0
J
k
(g
k
)+

S
n+1
ϕ
n+1
dW
⊗(n+1)
,
where

)
is the (n + 1)-fold iterated integral of ϕ
n+1
.Moreover,
E


S
n+1
ϕ
n+1
dW
⊗(n+1)

2

≤ E

ξ
2

.
14 1 The Wiener–Itˆo Chaos Expansion
In particular, the family
ψ
n+1
:=

S
n+1

(P )
=
n

k=0
J
k
(g
k
)
2
L
2
(P )
+ ψ
n+1

2
L
2
(P )
.
In particular,
n

k=0
J
k
(g
k

L
2
(P )
=0
for all k and for all f
k
∈ L
2
([0,T]
k
). In particular, by (1.15) this implies that
E

h
k

θ
g

· ψ

=0
for all g ∈ L
2
([0,T]) and for all k ≥ 0, where θ =
T

0
g(t)dW (t). But then, from
the definition of the Hermite polynomials,

) (1.26)
and
ξ
2
L
2
(P )
=
∞

k=0
J
k
(g
k
)
2
L
2
(P )
. (1.27)
Finally, to obtain (1.16)–(1.17) we proceed as follows. The function g
n
is
defined only on S
n
, but we can extend g
n
to [0,T]
n

n
)=n!J
n
(g
n
)=J
n
(g
n
)
and (1.16) and (1.17) follow from (1.26) and (1.27), respectively. 
Example 1.11. What is the Wiener–Itˆo expansion of ξ = W
2
(T )? From (1.15)
we get
2
T

0
t
2

0
1 dW (t
1
)dW (t
2
)=Th
2


2
)dW (t
1
)dW (t
2
)=

T
t
W (t)dW (t
2
)=W (t)

W (T ) − W (t)

.
Hence, if we put
ξ = W (t)(W (T ) − W(t)),g(t
1
,t
2
)=χ
{t
1
<t<t
2
}
we can see that
ξ = J
2

2
<t<t
1
}

.
Here and in the sequel we denote the indicator function by
χ = χ
A
(x)=χ
{x∈A}
:=

1,x∈ A,
0,x/∈ A.