Daniel Michelbrink
Stochastic Calculus with Applications to Stochastic Portfolio Optimisation
ISBN: 978-3-8366-1287-6
Druck Diplomica® Verlag GmbH, Hamburg, 2008
Zugl. University of Wales Swansea, Swansea, Großbritannien, Magisterarbeit, 2007
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Contents
Introduction
4
1
Stochastic Processes
6
1.1
Construction of a Stochastic Process . . . . . . . . . . . . . . . . . .
6
1.1.1
Construction of a Canonical Process . . . . . . . . . . . . . .
6
1.1.2
Markovian Semigroups and Projective Families . . . . . . . .
9
1.1.3
Continuous Modification of a Stochastic Process . . . . . . .
12
1.2
The Multi-Variate Normal Distribution
. . . . . . . . . . . . . . . .
13
1.3
The Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4
Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.4.1
Conditional Expectation . . . . . . . . . . . . . . . . . . . . .
18
1.4.2
Martingales and Filtrations . . . . . . . . . . . . . . . . . . .
20
1.4.3
Stopping Times and Local Martingales . . . . . . . . . . . . .
21
2
Stochastic Calculus
24
2.1
Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.1.1
The Quadratic Variation Process . . . . . . . . . . . . . . . .
25
2.1.2
The Stochastic Integral w.r.t. Martingales and the It^
o Integral
27
2.2
Change of Variable Formula . . . . . . . . . . . . . . . . . . . . . . .
30
2.2.1
Semimartingales and It^
o's Formula . . . . . . . . . . . . . . .
30
2.2.2
It^
o Processes and It^
o's Formula . . . . . . . . . . . . . . . . .
31
2.3
Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . .
33
2.3.1
Solutions of Stochastic Differential Equations . . . . . . . . .
33
2.3.2
Solving Stochastic Differential Equations
. . . . . . . . . . .
35
2.3.3
Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4
Stochastic Control Theory . . . . . . . . . . . . . . . . . . . . . . . .
37
3
Stochastic Portfolio Theory
40
3.1
The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.1.1
A Stock Market Model . . . . . . . . . . . . . . . . . . . . . .
40
3.1.2
Portfolios and Value Processes . . . . . . . . . . . . . . . . .
42
3.1.3
The Riskless Asset and the State Price Density . . . . . . . .
45
3.1.4
Consumption and Income Processes . . . . . . . . . . . . . .
50
3.2
Relative Return Processes and Portfolio Generating Functions . . . .
52
3.2.1
Relative Return Processes . . . . . . . . . . . . . . . . . . . .
53
3.2.2
The Market Portfolio and Portfolio Generating Functions . .
56
2
CONTENTS
CONTENTS
3.3
The Long-Term Behaviour of Portfolios and Markets . . . . . . . . .
58
3.3.1
Long-Term Behaviour of Portfolios . . . . . . . . . . . . . . .
59
3.3.2
Market Long-Term Behaviour and Diversity . . . . . . . . . .
63
4
Stochastic Portfolio Optimisation
67
4.1
Decisions under Risk . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.2
Some Optimisation Problems . . . . . . . . . . . . . . . . . . . . . .
69
4.2.1
Markowitz Portfolio Optimisation Problem . . . . . . . . . .
70
4.2.2
Mean-Variance Problems corresponding to Relative Returns .
71
4.2.3
Maximise the Probability of Reaching a Goal . . . . . . . . .
72
4.3
Consumption and Investment with Labour Earnings . . . . . . . . .
73
4.3.1
Maximising Expected Utility from Consumption . . . . . . .
74
4.3.2
Maximising Expected Utility from Terminal Wealth . . . . .
78
4.3.3
Utility from Consumption and Terminal Wealth
. . . . . . .
83
4.4
Solving Investment-Consumption Problems via Stochastic Control
.
87
4.5
Maximising Utility from Relative Return Processes . . . . . . . . . .
88
Bibliography
92
3
Introduction
The present paper is about continuous time stochastic calculus and its application
to stochastic portfolio selection problems. The paper is divided into two parts:
The first part provides the mathematical framework and consists of Chapters 1 and
2, where it gives an insight into the theory of stochastic process and the theory of
stochastic calculus.
The second part, consisting of Chapters 3 and 4, applies the first part to problems
in stochastic portfolio theory and stochastic portfolio optimisation.
Chapter 1, "Stochastic Processes", starts with the construction of stochastic process.
The significance of Markovian kernels is discussed and some examples of process and
semigroups will be given. The simple normal-distribution will be extended to the
multi-variate normal distribution, which is needed for introducing the Brownian mo-
tion process. Finally, another class of stochastic process is introduced which plays
a central role in mathematical finance: the martingale.
Chapter 2, "Stochastic Calculus", begins with the introduction of the stochastic
integral. This integral is different to the Lebesgue-Stieltjes integral because of the
randomness of the integrand and integrator. This is followed by the probably most
important theorem in stochastic calculus: It^
o's formula. It^
o's formula is of central
importance and most of the proofs of Chapters 3 and 4 are not possible without it.
We continue with the notion of a stochastic differential equations. We introduce
strong and weak solutions and a way to solve stochastic differential equations by
removing the drift. The last section of Chapter 2 applies stochastic calculus to
stochastic control. We will need stochastic control to solve some portfolio problems
in Chapter 4.
Chapter 3, "Stochastic Portfolio Theory", deals mainly with the problem of intro-
ducing an appropriate model for stock prices and portfolios. These models will be
needed in Chapter 4. The first section of Chapter 3 introduces a stock market model,
portfolios, the risk-less asset, consumption and labour income processes. The sec-
ond section, Section 3.2, introduces the notion of relative return as well as portfolio
generating functions. Relative return finds application in Chapter 4 where we deal
with benchmark optimisation. Benchmark optimisation is optimising a portfolio
with respect to a given benchmark portfolio. The final section of Chapter 3 contains
some considerations about the long-term behaviour of portfolios and markets. It
finishes with the surprising result that under some apparently irrelevant assumption
the market is not diverse.
Chapter 4, "Stochastic Portfolio Optimisation", is dedicated to apply all prior
groundwork to the problem of finding optimum stock portfolios. Section 4.1 intro-
4
Introduction
5
duces two ways to make decisions under risk. The first is the mean-variance approach
used by Markowitz and the second is by using an utility function. In Section 4.2 we
solve various optimisation problems mainly in the mean-variance sense. Section 4.3
uses utility functions to maximise an investors terminal wealth, consumption, and
a combination of both. It generalises the typical investment-consumption problem
by introducing labour earnings to the portfolio value process. Section 4.4 solves the
problem from Section 4.3 by using stochastic control theory. We finish the paper
by solving the problem of maximising expected utility from relative return, which
is the problem of maximising the expected utility from a portfolio with respect to a
given benchmark portfolio.
Acknowledgments
I wish to thank my supervisor Dr. Jiang-Lun Wu for his help and guidance. I also
would like to thank my parents and family who supported me during my time of
study. Special thanks goes also to my brother Markus who inspired me to start
a degree in mathematics and has encouraged me to come to Swansea University.
Furthermore, I would like to thank the Amt f¨
ur Ausbildungsf¨
orderung that has
made it financially possible to study in Swansea.
Chapter 1
Stochastic Processes
In stochastic calculus and its application to finance, the notion of a stochastic process
plays a central role. We therefore wish to give some background about stochastic
processes.
The whole Chapter 1 is strongly influenced by the monograph of H. Bauer [4]. All
missing proofs in Chapter 1 can be found in this monograph. Some other references
for the first chapter are to be found in N. Jacob [19] and L.C.G. Rogers & D. Williams
[35].
Throughout the paper the space (,
A, P ) is a probability space. This means that
is an arbitrary non-empty set,
A is a -algebra on and P is a probability measure
on the measurable space (,
A).
1.1
Construction of a Stochastic Process
In the following we briefly describe the construction of a canonical process.
It
follows an overview of Markovian kernels and semigroups where we also give some
examples of common semigroups. At the end of the section we consider continuous
modifications of stochastic processes.
1.1.1
Construction of a Canonical Process
To construct the so-called canonical process we first need to know what a stochastic
process is. Let (E,
B) be a Polish space equipped with its Borel--algebra B and let
I be a time parameter set, e.g. I = [0,
).
Definition 1.1.1. A stochastic process with state space (E,
B) and time parameter
set I is a quadruple (,
A, P, (X
t
)
t0
) where (,
A, P ) is a probability space and
X
t
:
E is for each t I a random variable.
We will often refer to a stochastic process as, simply, a process. Sometimes we will
only write (X
t
)
t0
for a stochastic process (,
A, P, (X
t
)
t0
), and sometimes just X.
For a fixed
we call the mapping t X
t
() a path of the process (X
t
)
tI
.
The process is called continuous, right-continuous or left-continuous if almost surely
every path is continuous, right-continuous and left-continuous, respectively.
In order to construct a stochastic process we need the notion of an infinite probability
6
1.1 Construction of a Stochastic Process
7
space. Given a family of probability spaces (
j
,
A
j
, P
j
)
jI
there exists the product
probability space
I
,
A
I
, P
I
:=
jI
j
,
jI
A
j
,
jI
P
j
(1.1)
where
jI
j
is the Cartesian product constructed by
I
:=
jI
j
:= : I
jI
j
:
j
j
and the product -algebra
jI
A
j
is constructed by the projections
j
from
jI
j
onto
j
, namely
A
I
:=
jI
A
j
:= (
j
; j
I).
The probability measure P =
jI
P
j
is a probability measure on the measure space
(
jI
j
,
jI
A
j
) such that for any projection
J
from
I
onto a finite subspace
J
I we have that
J
(P ) = P
J
holds. Thereby,
J
(P ) denotes the image measure of the mapping
J
. The right-
hand side is the finite probability measure on the finite measure space
(
jJ
j
,
jJ
A
j
).
Such a probability measure
jI
P
j
does indeed exist and is unique.
Theorem 1.1.2. Let (
j
,
A
j
, P
j
)
jI
be a family of probability spaces with I =
.
Then there exists exactly one probability measure P on
jJ
A
j
such that for all
finite subsets J
I
J
(P ) = P
J
.
The construction of the measurable space (
I
,
A
I
) with the probability measure P
leads us to the (infinite) probability space given in equation (1.1).
In the same manner as above, we can define for a Polish space E and its -algebra
B and a nonempty subset J I
E
J
:=
jJ
E
j
with
E
j
:= E
and
B
J
:=
jJ
B
j
with
B
j
:=
B.
(1.2)
Let us take the stochastic process as defined above (,
A, P, (X
t
)
tI
) with state space
(E,
B). We define
X
J
:=
jJ
X
j
and
1.1 Construction of a Stochastic Process
8
P
J
:= X
J
(P ).
(1.3)
Clearly, P
J
is the joint distribution of the random variables (X
j
)
jJ
. This means
taking a finite subset J =
{t
1
, ..., t
m
} I and m events B
j
B, j J, the finite
dimensional distribution of the process (,
A, P, (X
t
)
tI
) is given by
P
J
(B
1
× ... × B
m
) = P (X
-1
J
(B
1
× ... × B
m
))
= P (X
t
1
B
1
, ..., X
t
m
B
m
).
(1.4)
The above equation (1.4) gives us the probability that the process hits the set B
j
for each j
J at time t
j
.
Now, let
K
J
denote for J
K I the projection from
K
onto
J
, which is
simply the restriction of
K
to the space
J
, i.e.
K
J
(
K
) =
K
|
J
. We denote with
H = H(I) the set of all finite subsets of I.
Definition 1.1.3. Let (P
J
)
JH
be the family of the finite dimensional distributions
of a stochastic process (,
A, P, (X
t
)
t0
) with state space (E,
B). We call (P
J
)
JH
a projective family of probability measures over (E,
B) if for all J, K H with
= J K we have
J
(P
K
) = P
J
.
It is possible to extend such a projective family of probability measures to the whole
measure space (E
I
,
B
I
). This extension is called the projective limit and it allows
us to construct the canonical process.
Theorem 1.1.4 (A. N. Kolmogorov). Let E be a Polish space,
B its Borel--algebra
and let I =
. For every projective family (P
J
)
JH
of probability measures over
(E,
B) there exists a unique probability measure P
I
on (E
I
,
B
I
) such that
J
(P
I
) = P
J
holds.
The unique probability measure P
I
determined as above is called the projective limit
of the projective family (P
J
)
JH
. It is often also denoted by
lim
-
P
J
:= P
I
.
Let us now construct the canonical process. Let (P
J
)
JH
be a projective family
of probability measures and let P
I
be its projective limit. The canonical process
corresponding to the projective family (P
J
)
JH
is defined as the process
(,
A, P, (X
t
)
t0
) := (E
I
,
B
I
, P
I
, (
t
)
t0
)
Thus, the canonical process is simply a projection. We have
X
t
:
E
X
t
() := (t) :=
t
().
(1.5)
1.1 Construction of a Stochastic Process
9
The definition of the canonical process does make sense since (E
I
,
B
I
, P
I
) is, as seen
earlier, a probability space and the projections are measurable.
Taking J
H and B B
J
we obtain
P (X
J
B) = P (
-1
J
(B)) =
J
(P
I
(B)) = P
J
(B).
Thus, the probability that X
J
hits B is given by P
J
(B), which coincides with our
earlier considerations.
However, this easy construction raises several questions. It is not yet clear how we
are going to obtain a projective family. H. Bauer shows in his monograph [4] that
a projective family characterises the stochastic process completely (up to the initial
distribution, which we are going to consider later). Therefore, it is important to find
a way to construct such a projective family.
A different problem is that the paths of the canonical process are in principle all
kinds of mappings from [0,
) into E, i.e. : [0, ) E. This is quite a large set
of possible paths and it would be nice if we could restrict the paths somehow, for
instance to only continuous paths.
In the following subsection, we are going to face these two problems. We start with
finding a projective family (P
J
)
JH
.
1.1.2
Markovian Semigroups and Projective Families
We want to introduce the notion of a kernel. Kernels give us a key tool to construct
a projective family.
Definition 1.1.5. Let (,
A) and ( , A ) be two measurable spaces. Then a kernel
from (,
A) to ( , A ) is a function
K :
× A [0, +]
such that
K(, A ) is A-measurable for every A A
A
K(, A ) is a measure on A for every .
A kernel is Markovian (or sub-Markovian) if
K(, ) = 1,
1 respectively, for all .
For a kernel K from (,
A) to itself we simply say K is a kernel on (, A).
Denote by
M(, A) all numerical, A-measurable functions and by M
+
(,
A) all
non-negative,
A-measurable functions. Then, given a kernel K from (, A) to
( ,
A ) we can introduce an operator from M
+
(,
A) into M(, A) by
(K
op
u)() :=
u( )K(, d ).
(1.6)
It can be shown that K
op
is positivity preserving and K
op
is actually an operator from
M
+
(,
A) into itself. Moreover, K
op
is linear and for every sequence of functions
(u
n
)
nN
M
+
(,
A) that converges to u M
+
(,
A) we have K
op
u
n
K
op
u, see
1.1 Construction of a Stochastic Process
10
[19] for details. We now introduce the composition of two kernel operators in order to
introduce a semigroup of kernels. Let us take three measurable spaces (
i
,
A
i
)
i=1,2,3
and two kernels K
i
from (
i
,
A
i
) to (
i+1
,
A
i+1
), for i = 1, 2, and let K
i
op
be the
corresponding kernel operators for i = 1, 2. Then we obtain the composition K
12
op
of
K
1
op
and K
2
op
from
M
+
(
3
,
A
3
) to
M
+
(
1
,
A
1
) by
(K
1
op
K
2
op
u)(
1
)
=
2
K
1
(
1
, d
2
)
3
u(
3
)K
2
(
2
, d
3
)
=
3
u(
3
)
2
K
2
(
2
, d
3
)K
1
(
1
, d
2
)
=:
3
u(
3
)
2
K
12
(
1
, d
3
)
(1.7)
for a function u
M
+
(
3
,
A
3
). Thus, for the composition K
12
op
we get a new kernel
K
12
(
1
, A
3
) =
2
K
2
(
2
, A
3
)K
1
(
1
, d
2
),
which is a kernel from (
1
,
A
1
) to (
3
,
A
3
). In particular, K
12
is Markovian (or
sub-Markovian) if both kernels, K
1
and K
2
, are Markovian (or sub-Markovian).
A family of kernels (P
t
)
t0
on the measurable space (E,
B) forms a semigroup of
kernels if for all t, s
0
P
s+t
= P
s
P
t
.
Such a semigroup is called (sub-)Markovian if all kernels P
t
are (sub-)Markovian.
Important to mention are the Chapman-Kolmogorov-equations which hold for all
semigroups of kernels, that is
P
s+t
(x, B) =
P
s
(x, dy)P
t
(y, B)
for any (x, B)
E × B and s, t 0.
Having a semigroup of Markovian kernels (P
t
)
t0
then for (x, B)
E × B the
probability P
t
(x, B) can be interpreted as the transition probability that a particle
starting at x hits the set B at time t. With this in mind we can construct a projective
family of probability measures on (E,
B) using the transition probabilities.
Theorem 1.1.6. On a measurable space (E,
B) let (P
t
)
t0
be a Markovian semi-
group and let be a probability measure. Set for every set J =
{t
1
, ..., t
m
} H with
t
1
< ... < t
m
and every B
B
J
P
J
(B) :=
...
1
B
(x
1
, ..., x
m
)P
t
n
-t
n-1
(x
n-1
, dx
n
)...P
t
1
(x
0
, dx
1
)(dx
0
).
(1.8)
Then (P
J
)
JH
is a projective family of probability measures on (E,
B).
The probability measure is the distribution of the starting point of the process.
Therefore, it is called the initial distribution of the process.
Having, additionally, a convolution semigroup of probability measures (
t
)
t0
on
1.1 Construction of a Stochastic Process
11
(
R
d
,
B
d
) we can construct a translation invariant Markovian semigroup (P
t
)
t0
on
(
R
d
,
B
d
) (and so a projective family) by
P
t
(x, B) :=
t
(B
- x)
(1.9)
for any (x, B)
(R
d
,
B
d
). On the other hand, it is possible to obtain a convo-
lution semigroup of probability measures given a translation invariant Markovian
semigroup by setting
t
(B) := P
t
(0, B).
It can be shown that, constructing a projective family in the way described above,
the corresponding canonical process has stationary and independent increments.
Moreover, the canonical process is in this case a Markov process. In the following,
we are going to give some examples for convolution semigroups.
Example 1.1.7 (Brownian semigroup). The Brownian semigroup is the convolution
semigroup given by
t
:=
0,t
...
0,t
(d factors)
(1.10)
and
0
=
0
(the Dirac-measure). This convolution semigroup is of particular im-
portance and we will study it in more detail in the following sections.
Example 1.1.8 (Poisson semigroup). The Poisson convolution semigroup on (
R, B)
is defined by
t
:=
t
,
:=
k=0
e
-t
(t)
k
k!
k
where ,
0 and
0
=
,
0
=
0
.
Example 1.1.9 (Cauchy semigroup). The Cauchy convolution semigroup on (
R, B)
defined by
t
:= c
t
1
where c
t
is the density given by c
t
(x) :=
t
(t
2
+ x
2
)
-1
, and
0
=
0
=
0
.
Example 1.1.10 (Ornstein-Uhlenbeck process). The Ornstein-Uhlenbeck Marko-
vian semigroup,
P
t
(x, B) :=
B
p
t
,c
(x, y)dy
p
,c
t
(x,
·) being the density of the one-dimensional normal distribution N(
x,
2
)
with
:= e
-t
and
2
:= c
·
1
- e
-2t
2
,
is not translation invariant and therefore does not form a convolution semigroup in
the way described above.
We have seen, that we can obtain a projective family of probability measures by a
Markovian semigroup of kernels. A translation invariant Markovian semigroup in
turn can be constructed by a convolution semigroup of probability measures.
1.1 Construction of a Stochastic Process
12
1.1.3
Continuous Modification of a Stochastic Process
Now we want to pay our attention to the question of how to restrict the set of all
possible paths to only the set of paths which are continuous. Since a stochastic pro-
cess is completely characterised by its corresponding projective family, the following
definition makes sense.
Definition 1.1.11. We call two stochastic processes (,
A, P, (X
t
)
t0
) and
( ,
A , P , (X
t
)
t0
) with the same parameter set I and the same state space (E,
B)
equivalent if they lead to the same projective family (P
J
)
JH
.
Similarly, two processes are equivalent if they have the same finite distributions P
J
.
Definition 1.1.12. Let E be a Polish space and (P
J
)
JH
a projective family of prob-
ability measures over E. A set
E
I
is called essential with respect to (P
J
)
JH
whenever a stochastic process exists with state space E, time-parameter set I, finite-
dimensional distributions P
J
, and path-set .
Besides the notion of equivalence for stochastic processes we have also some stronger
notations.
Definition 1.1.13. Let (X
t
)
t0
and (Y
t
)
t0
be two stochastic processes over the
same probability space (,
A, P ) with the same state space (E, B). We call each
process a modification of the other one if for each t
0
X
t
= Y
t
,
a.s.,
and we call these two processes indistinguishable if almost surely we have that
X
t
= Y
t
,
for all t
0.
Obviously, if the two processes are indistinguishable, then they are a modification
of each other; and if they are a modification of each other, then they are equivalent.
The reverse does not hold in general.
The process (X
t
)
t0
is called a continuous modification of (X
t
)
t0
if its paths are con-
tinuous and (X
t
)
t0
is a modification of (X
t
)
t0
. If such a continuous modification
exists for the process (X
t
)
t0
then the set of all continuous paths C := C([0,
), E)
is essential, and the process (X
t
)
t0
is equivalent to the corresponding C-canonical
process (for a proof see [4]). That means, if a continuous modification exists we
can restrict our observation to the continuous paths. Thus, our goal is to prove the
existence of a continuous modification for a given process (X
t
)
t0
, because then we
know that we can find a continuous canonical process that is equivalent to (X
t
)
t0
.
The following theorem and its corollary gives us a condition for the existence of a
continuous modification.
Theorem 1.1.14 (Kolmogorov's continuity theorem). The following condition is
sufficient for the existence of a continuous modification of a process (,
A, P, (X
t
)
t0
)
with state space
R
d
: For some real numbers > 0, > 0, and > 0 the inequality
E(
|X
s
- X
t
|
)
|t - s|
1+
(1.11)
holds for all s, t
[0, +).
1.2 The Multi-Variate Normal Distribution
13
The next corollary gives us even more information.
Corollary 1.1.15 (Theorem of Kolmogorov and Chentov). Suppose the stochastic
process (X
t
)
t0
with state space
R
d
satisfies condition (1.11) with positive real con-
stants , , . Then the process has a modification (X
t
)
t0
of which every path is
locally H¨
older-continuous of order
(0, /).
Of course, we can also be interested in a different modification of a stochastic process.
Another important modification is a c`
adl`
ag modification which is a modification
that is right-continuous with left limits. However, such a modification will not play
a role here. It becomes important when dealing with jump processes like the Poisson
process.
1.2
The Multi-Variate Normal Distribution
The normal distribution is one of the essential distributions in probability theory.
The aim in this section is to present this distribution and highlight its most im-
portant properties. The multi-variate normal distribution is important for the un-
derstanding of the Brownian motion process, which we will introduce in the next
section.
For
R and > 0, we define the probability density function
g
,
2
(x) := (2
2
)
-1/2
e
(x-)2
22
.
(1.12)
The normal distribution is the probability measure on (
R, B
1
) that has density g
,
2
with respect to the Lebesgue measure
1
, i.e.
N (,
2
) :=
,
2
:= g
,
2
1
.
The expectation of a normally distributed random variable is and its variance
is given by
2
. If = 0 and = 1 then N (0, 1) is called the standard normal
distribution. For the case = 0 we have
=
,0
and call it the degenerated
normal distribution. Instead of normal distribution we often also refer to
,
2
as
the Gauß-measure. For a random variable X that is normally distributed with mean
and variance
2
, we write
X
N(,
2
).
From now on let
N
1
denote the set of all one-dimensional normal distributions
together with the degenerated normal distributions, i.e.
N
1
:=
{
,
2
:
R, 0}
Using the linear transformation
x
T (x) := x +
(1.13)
it is possible to transform every normal distribution to the standardised normal
distribution by
T (
0,1
) =
,
2
.
1.2 The Multi-Variate Normal Distribution
14
The n-th moment of a normally distributed random variable X is defined as
M
n
:= E(X
n
) =
x
n
g
0,1
(x)dx
for n
N.
Using the transformation (1.13) we can also rewrite the moments as
E(X
n
) =
n
k=0
n
k
k
n-k
M
k
,
since
E(X
n
) =
x
n
,
2
(dx) =
x
n
T (
0,1
)(dx) =
(x + )
n
0,1
(dx).
We can characterise the normal distribution in the following way presented by S.N.
Bernstein.
Theorem 1.2.1. For two independent random variables X and Y we have the equiv-
alence of the following:
(i) X and Y have a distribution in
N
1
with the same variance.
(ii) The random variables X + Y and X
- Y are independent.
Let us now construct the multi-dimensional normal distributions on (
R
d
,
B
d
).
Definition 1.2.2. A probability measure on (
R
d
,
B
d
) is a d-dimensional normal
distribution if for every linear mapping h :
R
d
R the image measure of h is in
N
1
:
h()
N
1
From now on we denote the set of d-dimensional normally distributions with
N
d
.
A random vector X = (X
1
, ..., X
d
) :
R
d
is called normal distributed if its
distribution is in
N
d
. Clearly, this is the case if and only if for any choice of real
numbers
1
, ...,
d
we have that the distribution of
Y :=
1
X
1
+ ... +
d
X
d
is in
N
1
. To see this we only have to take Y = h(X) with the linear mapping
h(x) = h(x
1
, ..., x
d
) :=
1
x
1
+ ... +
d
x
d
. Thus, all components of the vector X are
one-dimensional normally distributed.
We define the expectation of a random vector X = (X
1
, ..., X
d
) as
m := E(X) := (E(X
1
), ..., E(X
d
)).
The covariance of X
j
and X
k
is
Cov(X
j
, X
k
) = E((X
j
- E(X
j
))(X
k
- E(X
k
)))
and the covariance matrix C of X is defined as
C := Cov(X) := (Cov(X
j
, X
k
))
j,k=1,...,d
.
1.3 The Brownian Motion
15
It can be shown that the covariance matrix C is positive definite. We write analo-
gously to the 1-dimensional case N
d
(m, C) for a d-dimensional normal distribution
with expectation m and covariance matrix C.
Now consider the probability distribution given by
:=
d
k=1
m
k
,
2
k
=
m
1
,
2
1
...
m
d
,
2
d
where m
k
R and
k
0, k = 1, ..., d. This is the general form of the distribution
already given earlier by the Brownian convolution semigroup (1.10). Let us show
that this distribution is a d-dimensional normal distribution.
Take d independent (
R, B
1
)-random variables X
1
, ..., X
d
with X
k
N(m
k
,
2
k
). Then
the random vector X := (X
1
, ..., X
d
) follows the distribution . Define
Y :=
1
X
1
+ ... +
d
X
d
,
then the summands are normally distributed with
k
X
k
N(
k
m
k
, (
|
k
|)
2
) and
since the random variables are independent we have
Y
N(
d
k=1
k
m
k
,
d
k=1
(
|
k
|
k
)
2
)
N
1
.
Thus, the probability measure is indeed a d-dimensional normal distribution.
To demonstrate the parallels to the one-dimensional density function (1.12) we finish
this section with an important theorem.
Theorem 1.2.3. Every normal-distribution
m,C
= N
d
(m, C), with m
R
d
and C
being a symmetric positive definite matrix, has a density with respect to the Lebesgue
measure
d
. We have
m,C
= g
m,C
d
with
g
m,C
(x) := (2)
-d/2
|C|
-1/2
e
-
1
2
(x-m)
t
C
-1
(x-m)
where
|C| > 0 is the determinant of C.
1.3
The Brownian Motion
We have already met the Brownian convolution semigroup in (1.10). This semi-
group leads to the stochastic process of Brownian motion. The Brownian motion
has its name from the Scottish botanist Robert Brown (17731858) who observed
the random motion of pollen grains in water. Since then the phenomenon has been
named after its discoverer. A rather mathematical examination of the phenomenon
is due to Albert Einstein (18791955), Marian Smoluchowski (18721917) and Nor-
bert Wiener (18941912). N. Wiener, in particular, made major contributions to
the theory of Brownian motion. That is why the process is often also called the
Wiener process.
An interesting approach to economics was due to the French mathematician Louis
1.3 The Brownian Motion
16
Bachelier (18701946) who was one of the pioneers in the study of financial math-
ematics. In his thesis `Th´
eorie de la sp´
eculation' in 1900, he used the Brownian
motion to model stock volatilities on the Paris stock exchange.
Definition 1.3.1. A stochastic process (,
A, P, (B
t
)
t0
) with state space (
R
d
,
B
d
)
is a d-dimensional Brownian motion if the following two properties are fulfilled:
(i) The process (B
t
)
t0
has stationary and independent increments and the incre-
ments are normally distributed such that for 0
s < t
B
t
- B
s
0,t-s
...
0,t-s
(d factors).
(ii) The paths t
B
t
() are a.s. continuous.
If moreover, B
0
= 0 a.s., then the process is called Brownian motion with starting
point 0, or normalised Brownian motion. Every Brownian motion is a centred Gauß-
process in the sense that by taking any finite points t
1
< ... < t
n
, the random variable
B
t
1
... B
t
n
has Gaussian distribution (i.e. normal distribution).
Taking the Brownian convolution semigroup in (1.10)
t
:=
0,t
...
0,t
with d factors
and using (1.9) and Theorem 1.1.6 we obtain a projective family. In [4] it has
been shown that the canonical process is equivalent to a continuous C([0,
) × R
d
)-
canonical process. Thus, we can construct a Brownian motion process with the
Brownian semigroup (1.10). In particular, if we choose in (1.8) the Dirac measure
0
as the initial distribution, then we get a standard or normalised Brownian motion.
The following theorem shows us some other properties of the Brownian motion.
Theorem 1.3.2. 1. (Homogeneity) If (B
t
)
t0
is a Brownian motion in (
R
d
,
B
d
)
and s
0, then (B
s+t
- B
s
)
t0
is a normalised Brownian motion.
2. (Temporal translation) Along with (B
t
)
t0
, the process (B
s+t
)
t0
is also a Brow-
nian motion in (
R
d
,
B
d
) for every s
0.
3. (Symmetry) Along with (B
t
)
t0
, the process (
-B
t
)
t0
is also a normalised Brow-
nian motion in (
R
d
,
B
d
).
4. (Scale changing) Along with (X
t
)
t0
the process ( X
t/
2
)
t0
is also a Brownian
motion in (
R
d
,
B
d
) for every = 0.
5. (Coordinate process) If (B
t
)
t0
is a Brownian motion in (
R
d
,
B
d
) and B
1
t
, ..., B
d
t
denote the coordinates of B
t
, then each of the coordinate processes (B
j
t
)
t0
, j =
1, ..., d is a Brownian motion in (
R
1
,
B
1
). In the normalised case the random vari-
ables B
1
t
, ..., B
d
t
are independent for each t
0.
6. (Covariance matrix) If (B
t
)
t0
is a normalised real Brownian motion then for ev-
ery choice of finitely many instants 0
t
1
< ... < t
n
, the random vector B
t
1
...B
t
n
is an n-dimensional Gaussian random vector having the zero vector as expected value
and the covariance matrix
Cov(B
t
1
... B
t
n
) = (t
i
t
j
)
i,j=1,...,n
.
Although the paths of a Brownian motion process are highly fluctuating and ex-
tremely irregular it is possible to give some constrains for the fluctuation of the
paths.
1.4 Martingales
17
Theorem 1.3.3 (Law of the Iterated Logarithm). For every standard Brownian
motion (B
t
)
t0
we have for almost every
lim sup
t0
B
t
()
2t log log(1/t)
= 1,
lim inf
t0
B
t
()
2t log log(1/t)
= 1,
lim sup
t+
B
t
()
2t log log(t)
= 1,
lim inf
t+
B
t
()
2t log log(t)
= 1.
(1.14)
Another important property of the Brownian motion process is that the Brownian
motion is local H¨
older continuous of order for all
(0, 1/2). Recall that a
function B is H¨
older continuous if there exists non-negative constants C and such
for all s, t
R
|B(t) - B(s)| C|t - s|
.
Recalling Theorem 1.1.14 we know that there must be a continuous modification
of the process. Thus, every Brownian motion is equivalent to a C([0,
) × R
d
)
canonical process. On the other hand it can be shown that a Brownian motion is
nowhere H¨
older continuous for a parameter
1/2. This also tells us that the
paths of the Brownian motion process are not differentiable for almost all
.
Let us consider a Brownian motion with a filtration. A filtration
F = (F
t
)
t0
is a
collection of sub--algebras of
A such that for every t > s 0 we have
F
s
F
t
.
(1.15)
A process (X
t
)
t0
is adapted to a filtration (
F
t
)
t0
if X
t
is
F
t
-measurable for every
t
0. The canonical filtration of the random variable (X
t
)
t0
is defined by
F
0
t
:=
(X
s
; s
t). Trivially, every process (X
t
)
t0
is adapted to its canonical filtration.
To emphasise that a process (X
t
)
t0
is adapted to a filtration (
F
t
)
t0
we often write
(X
t
,
F
t
)
t0
.
Definition 1.3.4. The stochastic process (B
t
,
F
t
)
t0
:= (,
A, P, B
t
,
F
t
)
t0
, adapted
to the filtration (
F
t
)
t0
, and with state space (
R
d
,
B
d
), is called a d
-dimensional
Brownian motion with filtration if the following conditions are fulfilled:
(i) For each 0
s < t, the process B
t
- B
s
is independent of
F
s
and is normally
distributed, i.e.
B
t
- B
s
0,t-s
...
0,t-s
(d factors).
(ii) Almost every path t
B
t
() is continuous.
Again, we say (B
t
,
F
t
)
t0
is normalised if it starts at 0 almost surely, i.e. B
0
= 0
a.s.
1.4
Martingales
Martingales were first introduced by Joseph L. Doob. They are a powerful tool in
stochastic calculus and mathematical finance.
To understand martingales we need to introduce the notion of conditional expecta-
tion. Thereafter, we establish the notion of a martingale, a submartingale, and a
supermartingale. At the end of the section we consider the more general case of a
local martingale.
1.4 Martingales
18
1.4.1
Conditional Expectation
Let X be a integrable random variable and let us interpret every sub--algebra of
A
as amount of information available. Consider the expectation
E(X) of the random
variable X. The expectation of X is a.s. constant and provides us with relatively
little information. Every sub--algebra is measurable w.r.t. a constant function and
therefore the information given by
E(X) is contained in every sub--algebra of A.
In contrast to this, the -algebra (X), which is generated by X, provides us with
a lot of information about X. Actually, if we know (X) then we have complete
information about X. However, all sub--algebras of (X) give us less information
about X and we need some tool to approximate X w.r.t. the information available.
Such a tool can be found in the conditional expectation of a random variable.
Let I be a countable index set and let (B
j
; j
I) be a partition of such that
P (B
j
)
0, for every j I. Taking one of the B
j
, recall the definition of the
conditional probability measure P
B
j
under the hypothesis B
j
:
P
B
j
(A) := P (A
|B
j
) :=
P (A
B
j
)
P (B
j
)
=
1
P (B
j
)
(1
B
j
P )(A)
for all A
A.
or
P
B
j
=
1
P (B
j
)
(1
B
j
P ).
The conditional expectation of X under the hypothesis B
j
is the expectation of X
w.r.t. the probability measure P
B
j
, i.e.
E
B
j
(X) :=
XdP
B
j
=
1
P (B
j
)
B
j
XdP.
(1.16)
We use the conditional expectation under B
j
(1.16) to introduce a new random
variable defined by
X
0
:=
jI
E
B
j
(X)1
B
j
.
X
0
assigns to each
the conditional expectation under B
j
, and is measurable
with respect to the -algebra
C := (B
j
; j
I). Moreover, we have the characteris-
ing property that for every C
C, we have that
C
X
0
dP =
C
XdP.
(1.17)
The random variable X
0
is a.s. uniquely determined by (1.17).
Theorem 1.4.1. Let X be a non-negative (integrable), real random variable on
(,
A, P ). Then there exists to every sub--algebra C A an equivalent non-negative
(integrable), real random variable X
0
on which is
C-measurable and fulfils
C
X
0
dP =
C
XdP
for all C
C.
(1.18)
Now it makes sense to give the following definition of the conditional expectation.
1.4 Martingales
19
Definition 1.4.2. Assume that the circumstances of the last theorem are given and
let X
0
be the P -a.s. determined random variable such that (1.18) is fulfilled. Then
we call X
0
the conditional expectation under the hypothesis
C and define
E(X
|C) := X
0
.
We call a random variable which is either non-negative or integrable an admissible
random variable, because for these random variables the condition expectation is
defined.
Let us summarise some properties of the conditional expectation. In the following,
let X and Y be two admissible random variables and let ,
R
+
or
R (depending
on whether X and Y are non-negative or integrable). We have
1.
E(E(X|C)) = E(X).
2. If X is
C-measurable, then E(X|C) = X, a.s.
3. If X = Y , a.s., then
E(X|C) = E(Y |C), a.s.
4. If X = is almost surely constant, then
E(X|C) = , a.s.
5.
E(X + Y |C) = E(X|C) + E(Y |C), a.s.
6. If X
Y , a.s., then E(X|C) E(Y |C), a.s.
7. If X is
C-measurable, then E(XY |C) = XE(Y |C), a.s.
8. If
C
1
is a sub--algebra of
C
2
, then
E(E(X|C
1
)
|C
2
) =
E(E(X|C
2
)
|C
1
) =
E(X|C
1
),
a.s.
Also the Jensen inequality holds for the conditional expectation.
Theorem 1.4.3. Let X be an integrable random variable with values in an open
interval I
R, and let q be a real-valued convex function on I. Then for every
sub--algebra
C A
E(X|C) I a.s.
and, in case q
X is admissible,
q(
E(X|C) E(q X|C) a.s.
(1.19)
We finish our considerations about the conditional expectation with two ways of
interpreting it. First, we can consider the conditional expectation as the mean-square
approximation of the random variable X. The other way is to take a geometric
approach and interpret the conditional expectation as a projection.
Let us assume that X is square integrable, i.e. X
L
2
(P ). Then it can be shown
that the conditional expectation X
0
is the best approximation in the Gauß mean-
square sense. The conditional expectation X
0
of X is the random variable that
minimises
E((X - X
0
)
2
) =
E(X
2
)
- E(X
2
0
).
1.4 Martingales
20
A rather geometric idea is to see the conditional expectation as a projection. Let
L
2
(P ) denote the space of all equivalence classes of
L
2
(P ), and let two random
variables be equivalent if they are almost surely equal. Then we can obtain L
2
(P )
as a direct sum of orthogonal spaces, i.e.
L
2
(P ) = L
2
C
(P )
L
2
C
(P )
.
Here, L
2
C
(P ) denotes the subspace of L
2
(P ) that contains all the
C-measurable func-
tions. The conditional expectation for a square integrable random variable is then
simply interpreted as the projection of the random variable X from L
2
(P ) onto
L
2
C
(P ).
1.4.2
Martingales and Filtrations
With the conditional expectation we now have a tool to come to the notion of a
martingale. As mentioned in the last section we interpret sub--algebras of
A as the
amount of information available. The increasing property (1.15) in Section 1.3 means
that we accumulate information over the time. This amount of information does not
decrease since we assume that we do not forget any information once obtained.
In the following we are going to extend the notion of the filtration to prepare our
further steps. Let us define for a given filtration (
F
t
)
t0
the -algebra
F
:= (
t0
F
t
).
We define the -algebra of events strictly prior to t as
F
t-
:= (
s<t
F
s
),
t
0,
where we set
F
0-
:= 0, and the -algebra of events immediately after t as
F
t+
:= (
s>t
F
s
),
t
0.
Furthermore, we call the filtration (
F
t
)
t0
right-continuous if (
F
t
)
t0
= (
F
t+
)
t0
and
left-continuous if (
F
t
)
t0
= (
F
t-
)
t0
. If (X
t
)
t0
is adapted to the left-continuous
filtration (
F
t
)
t0
then we can obtain X
t
by knowing all the prior variables X
s
,
s
[0, t). Similar, right-continuity means that if X
s
has been observed for 0
s < t,
then nothing more can be learned by peaking infinitesimally far into the future.
In addition, a filtration is said to fulfil the usual conditions if every
F
t
is right-
continuous and
F
0
contains all null sets.
Definition 1.4.4. A stochastic process (,
A, P, (X
t
)
t0
) that is adapted to the fil-
tration (
F
t
)
t0
and for which X
t
L
1
(P ), t
0, is called a martingale if
E(X
t
|F
s
) = X
s
,
for t
s 0, a.s.,
(1.20)
a supermartingale if
E(X
t
|F
s
)
X
s
,
for t
s 0, a.s.,
(1.21)
Ende der Leseprobe aus 96 Seiten
- nach oben
Details
- Arbeit zitieren
- , 2007, Stochastic Calculus with Applications to Stochastic Portfolio Optimisation, Hamburg, Bedey Media GmbH, https://www.diplom.de/document/225773
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