Stochastic Calculus with Applications to Stochastic Portfolio Optimisation

Michelbrink, Daniel

Stochastic Calculus with Applications to Stochastic Portfolio Optimisation

Zusammenfassung

Inhaltsangabe: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 emigroups will be given. The simple normal-distribution will be extended to the multi-variate normal distribution, which is needed for introducing the Brownian motion 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: Itos formula. Itos 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 introducing 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 second 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 […]

Leseprobe

Inhaltsverzeichnis

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

Dieses Werk ist urheberrechtlich geschützt. Die dadurch begründeten Rechte,

insbesondere die der Übersetzung, des Nachdrucks, des Vortrags, der Entnahme von

Abbildungen und Tabellen, der Funksendung, der Mikroverfilmung oder der

Vervielfältigung auf anderen Wegen und der Speicherung in Datenverarbeitungsanlagen,

bleiben, auch bei nur auszugsweiser Verwertung, vorbehalten. Eine Vervielfältigung

dieses Werkes oder von Teilen dieses Werkes ist auch im Einzelfall nur in den Grenzen

der gesetzlichen Bestimmungen des Urheberrechtsgesetzes der Bundesrepublik

Deutschland in der jeweils geltenden Fassung zulässig. Sie ist grundsätzlich

vergütungspflichtig. Zuwiderhandlungen unterliegen den Strafbestimmungen des

Urheberrechtes.

Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in

diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme,

dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei

zu betrachten wären und daher von jedermann benutzt werden dürften.

Die Informationen in diesem Werk wurden mit Sorgfalt erarbeitet. Dennoch können

Fehler nicht vollständig ausgeschlossen werden, und die Diplomarbeiten Agentur, die

Autoren oder Übersetzer übernehmen keine juristische Verantwortung oder irgendeine

Haftung für evtl. verbliebene fehlerhafte Angaben und deren Folgen.

http://www.diplom.de, Hamburg 2008

Printed in Germany

Contents

Introduction

Stochastic Processes

1.1

Construction of a Stochastic Process . . . . . . . . . . . . . . . . . .

1.1.1

Construction of a Canonical Process . . . . . . . . . . . . . .

1.1.2

Markovian Semigroups and Projective Families . . . . . . . .

1.1.3

Continuous Modification of a Stochastic Process . . . . . . .

1.2

The Multi-Variate Normal Distribution

. . . . . . . . . . . . . . . .

1.3

The Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.1

Conditional Expectation . . . . . . . . . . . . . . . . . . . . .

1.4.2

Martingales and Filtrations . . . . . . . . . . . . . . . . . . .

1.4.3

Stopping Times and Local Martingales . . . . . . . . . . . . .

Stochastic Calculus

2.1

Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1

The Quadratic Variation Process . . . . . . . . . . . . . . . .

2.1.2

The Stochastic Integral w.r.t. Martingales and the It^

o Integral

2.2

Change of Variable Formula . . . . . . . . . . . . . . . . . . . . . . .

2.2.1

Semimartingales and It^

o's Formula . . . . . . . . . . . . . . .

2.2.2

It^

o Processes and It^

o's Formula . . . . . . . . . . . . . . . . .

2.3

Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . .

2.3.1

Solutions of Stochastic Differential Equations . . . . . . . . .

2.3.2

Solving Stochastic Differential Equations

. . . . . . . . . . .

2.3.3

Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Stochastic Control Theory . . . . . . . . . . . . . . . . . . . . . . . .

Stochastic Portfolio Theory

3.1

The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1

A Stock Market Model . . . . . . . . . . . . . . . . . . . . . .

3.1.2

Portfolios and Value Processes . . . . . . . . . . . . . . . . .

3.1.3

The Riskless Asset and the State Price Density . . . . . . . .

3.1.4

Consumption and Income Processes . . . . . . . . . . . . . .

3.2

Relative Return Processes and Portfolio Generating Functions . . . .

3.2.1

Relative Return Processes . . . . . . . . . . . . . . . . . . . .

3.2.2

The Market Portfolio and Portfolio Generating Functions . .

CONTENTS

3.3

The Long-Term Behaviour of Portfolios and Markets . . . . . . . . .

3.3.1

Long-Term Behaviour of Portfolios . . . . . . . . . . . . . . .

3.3.2

Market Long-Term Behaviour and Diversity . . . . . . . . . .

Stochastic Portfolio Optimisation

4.1

Decisions under Risk . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Some Optimisation Problems . . . . . . . . . . . . . . . . . . . . . .

4.2.1

Markowitz Portfolio Optimisation Problem . . . . . . . . . .

4.2.2

Mean-Variance Problems corresponding to Relative Returns .

4.2.3

Maximise the Probability of Reaching a Goal . . . . . . . . .

4.3

Consumption and Investment with Labour Earnings . . . . . . . . .

4.3.1

Maximising Expected Utility from Consumption . . . . . . .

4.3.2

Maximising Expected Utility from Terminal Wealth . . . . .

4.3.3

Utility from Consumption and Terminal Wealth

. . . . . . .

4.4

Solving Investment-Consumption Problems via Stochastic Control

4.5

Maximising Utility from Relative Return Processes . . . . . . . . . .

Bibliography

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-

Introduction

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.

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

)

) where (,

A, P ) is a probability space and

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

)

for a stochastic process (,

A, P, (X

)

), and sometimes just X.

For a fixed

we call the mapping t X

() a path of the process (X

)

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

1.1 Construction of a Stochastic Process

space. Given a family of probability spaces (

, P

)

there exists the product

probability space

, P

(1.1)

where

is the Cartesian product constructed by

:= : I

and the product -algebra

is constructed by the projections

from

onto

, namely

:= (

; j

I).

The probability measure P =

is a probability measure on the measure space

(

) such that for any projection

from

onto a finite subspace

I we have that

(P ) = P

holds. Thereby,

(P ) denotes the image measure of the mapping

. The right-

hand side is the finite probability measure on the finite measure space

(

Such a probability measure

does indeed exist and is unique.

Theorem 1.1.2. Let (

, P

)

be a family of probability spaces with I =

Then there exists exactly one probability measure P on

such that for all

finite subsets J

(P ) = P

The construction of the measurable space (

) 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

with

:= E

and

with

(1.2)

Let us take the stochastic process as defined above (,

A, P, (X

)

) with state space

(E,

B). We define

and

1.1 Construction of a Stochastic Process

:= X

(P ).

(1.3)

Clearly, P

is the joint distribution of the random variables (X

)

. This means

taking a finite subset J =

, ..., t

} I and m events B

B, j J, the finite

dimensional distribution of the process (,

A, P, (X

)

) is given by

× ... × B

) = P (X

-1

× ... × B

))

= P (X

, ..., X

(1.4)

The above equation (1.4) gives us the probability that the process hits the set B

for each j

J at time t

Now, let

denote for J

K I the projection from

onto

, which is

simply the restriction of

to the space

, i.e.

(

) =

. We denote with

H = H(I) the set of all finite subsets of I.

Definition 1.1.3. Let (P

)

be the family of the finite dimensional distributions

of a stochastic process (,

A, P, (X

)

) with state space (E,

B). We call (P

)

a projective family of probability measures over (E,

B) if for all J, K H with

= J K we have

) = P

It is possible to extend such a projective family of probability measures to the whole

measure space (E

). 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

)

of probability measures over

(E,

B) there exists a unique probability measure P

on (E

) such that

) = P

holds.

The unique probability measure P

determined as above is called the projective limit

of the projective family (P

)

. It is often also denoted by

lim

:= P

Let us now construct the canonical process. Let (P

)

be a projective family

of probability measures and let P

be its projective limit. The canonical process

corresponding to the projective family (P

)

is defined as the process

A, P, (X

)

) := (E

, P

, (

)

Thus, the canonical process is simply a projection. We have

() := (t) :=

().

(1.5)

1.1 Construction of a Stochastic Process

The definition of the canonical process does make sense since (E

, P

) is, as seen

earlier, a probability space and the projections are measurable.

Taking J

H and B B

we obtain

P (X

B) = P (

-1

(B)) =

(B)) = P

(B).

Thus, the probability that X

hits B is given by P

(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

)

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

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

u)() :=

u( )K(, d ).

(1.6)

It can be shown that K

is positivity preserving and K

is actually an operator from

A) into itself. Moreover, K

is linear and for every sequence of functions

)

A) that converges to u M

A) we have K

u, see

1.1 Construction of a Stochastic Process

[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=1,2,3

and two kernels K

from (

) to (

i+1

), for i = 1, 2, and let K

be the

corresponding kernel operators for i = 1, 2. Then we obtain the composition K

and K

from

(

) to

(

) by

u)(

)

(

, d

)

(

, d

)

(

, d

(

, d

)

(

, d

)

(1.7)

for a function u

(

). Thus, for the composition K

we get a new kernel

(

, A

) =

(

, A

(

, d

which is a kernel from (

) to (

). In particular, K

is Markovian (or

sub-Markovian) if both kernels, K

and K

, are Markovian (or sub-Markovian).

A family of kernels (P

)

on the measurable space (E,

B) forms a semigroup of

kernels if for all t, s

s+t

= P

Such a semigroup is called (sub-)Markovian if all kernels P

are (sub-)Markovian.

Important to mention are the Chapman-Kolmogorov-equations which hold for all

semigroups of kernels, that is

s+t

(x, B) =

(x, dy)P

(y, B)

for any (x, B)

E × B and s, t 0.

Having a semigroup of Markovian kernels (P

)

then for (x, B)

E × B the

probability P

(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

)

be a Markovian semi-

group and let be a probability measure. Set for every set J =

, ..., t

} H with

< ... < t

and every B

(B) :=

...

, ..., x

-t

n-1

, dx

)...P

, dx

)(dx

(1.8)

Then (P

)

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 (

)

1.1 Construction of a Stochastic Process

(

) we can construct a translation invariant Markovian semigroup (P

)

(

) (and so a projective family) by

(x, B) :=

- x)

(1.9)

for any (x, B)

). On the other hand, it is possible to obtain a convo-

lution semigroup of probability measures given a translation invariant Markovian

semigroup by setting

(B) := P

(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

0,t

...

0,t

(d factors)

(1.10)

and

(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

k=0

-t

(t)

where ,

0 and

Example 1.1.9 (Cauchy semigroup). The Cauchy convolution semigroup on (

R, B)

defined by

:= c

where c

is the density given by c

(x) :=

+ x

)

-1

, and

Example 1.1.10 (Ornstein-Uhlenbeck process). The Ornstein-Uhlenbeck Marko-

vian semigroup,

(x, B) :=

(x, y)dy

(x,

·) being the density of the one-dimensional normal distribution N(

)

with

:= e

-t

and

:= c

- e

-2t

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

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

)

) and

( ,

A , P , (X

)

) with the same parameter set I and the same state space (E,

equivalent if they lead to the same projective family (P

)

Similarly, two processes are equivalent if they have the same finite distributions P

Definition 1.1.12. Let E be a Polish space and (P

)

a projective family of prob-

ability measures over E. A set

is called essential with respect to (P

)

whenever a stochastic process exists with state space E, time-parameter set I, finite-

dimensional distributions P

, and path-set .

Besides the notion of equivalence for stochastic processes we have also some stronger

notations.

Definition 1.1.13. Let (X

)

and (Y

)

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

= Y

a.s.,

and we call these two processes indistinguishable if almost surely we have that

= Y

for all t

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

)

is called a continuous modification of (X

)

if its paths are con-

tinuous and (X

)

is a modification of (X

)

. If such a continuous modification

exists for the process (X

)

then the set of all continuous paths C := C([0,

), E)

is essential, and the process (X

)

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

)

, because then we

know that we can find a continuous canonical process that is equivalent to (X

)

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

)

with state space

: For some real numbers > 0, > 0, and > 0 the inequality

- X

)

|t - s|

(1.11)

holds for all s, t

[0, +).

1.2 The Multi-Variate Normal Distribution

The next corollary gives us even more information.

Corollary 1.1.15 (Theorem of Kolmogorov and Chentov). Suppose the stochastic

process (X

)

with state space

satisfies condition (1.11) with positive real con-

stants , , . Then the process has a modification (X

)

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

(x) := (2

)

-1/2

(x-)2

(1.12)

The normal distribution is the probability measure on (

R, B

) that has density g

with respect to the Lebesgue measure

, i.e.

N (,

) :=

:= g

The expectation of a normally distributed random variable is and its variance

is given by

. If = 0 and = 1 then N (0, 1) is called the standard normal

distribution. For the case = 0 we have

and call it the degenerated

normal distribution. Instead of normal distribution we often also refer to

the Gauß-measure. For a random variable X that is normally distributed with mean

and variance

, we write

N(,

From now on let

denote the set of all one-dimensional normal distributions

together with the degenerated normal distributions, i.e.

{

R, 0}

Using the linear transformation

T (x) := x +

(1.13)

it is possible to transform every normal distribution to the standardised normal

distribution by

T (

0,1

) =

1.2 The Multi-Variate Normal Distribution

The n-th moment of a normally distributed random variable X is defined as

:= E(X

) =

0,1

(x)dx

for n

Using the transformation (1.13) we can also rewrite the moments as

E(X

) =

k=0

n-k

since

E(X

) =

(dx) =

T (

0,1

)(dx) =

(x + )

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

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 (

Definition 1.2.2. A probability measure on (

) is a d-dimensional normal

distribution if for every linear mapping h :

R the image measure of h is in

h()

From now on we denote the set of d-dimensional normally distributions with

A random vector X = (X

, ..., X

) :

is called normal distributed if its

distribution is in

. Clearly, this is the case if and only if for any choice of real

numbers

, ...,

we have that the distribution of

Y :=

+ ... +

is in

. To see this we only have to take Y = h(X) with the linear mapping

h(x) = h(x

, ..., x

) :=

+ ... +

. Thus, all components of the vector X are

one-dimensional normally distributed.

We define the expectation of a random vector X = (X

, ..., X

) as

m := E(X) := (E(X

), ..., E(X

)).

The covariance of X

and X

Cov(X

, X

) = E((X

- E(X

))(X

- E(X

)))

and the covariance matrix C of X is defined as

C := Cov(X) := (Cov(X

, X

))

j,k=1,...,d

1.3 The Brownian Motion

It can be shown that the covariance matrix C is positive definite. We write analo-

gously to the 1-dimensional case N

(m, C) for a d-dimensional normal distribution

with expectation m and covariance matrix C.

Now consider the probability distribution given by

k=1

...

where m

R and

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

)-random variables X

, ..., X

with X

N(m

). Then

the random vector X := (X

, ..., X

) follows the distribution . Define

Y :=

+ ... +

then the summands are normally distributed with

, (

) and

since the random variables are independent we have

k=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

(m, C), with m

and C

being a symmetric positive definite matrix, has a density with respect to the Lebesgue

measure

. We have

m,C

= g

m,C

with

m,C

(x) := (2)

-d/2

|C|

-1/2

(x-m)

-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

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

)

) with state space (

)

is a d-dimensional Brownian motion if the following two properties are fulfilled:

(i) The process (B

)

has stationary and independent increments and the incre-

ments are normally distributed such that for 0

s < t

- B

0,t-s

...

0,t-s

(d factors).

(ii) The paths t

() are a.s. continuous.

If moreover, B

= 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

< ... < t

, the random variable

... B

has Gaussian distribution (i.e. normal distribution).

Taking the Brownian convolution semigroup in (1.10)

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

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

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

)

is a Brownian motion in (

)

and s

0, then (B

s+t

- B

)

is a normalised Brownian motion.

2. (Temporal translation) Along with (B

)

, the process (B

s+t

)

is also a Brow-

nian motion in (

) for every s

3. (Symmetry) Along with (B

)

, the process (

-B

)

is also a normalised Brow-

nian motion in (

4. (Scale changing) Along with (X

)

the process ( X

)

is also a Brownian

motion in (

) for every = 0.

5. (Coordinate process) If (B

)

is a Brownian motion in (

) and B

, ..., B

denote the coordinates of B

, then each of the coordinate processes (B

)

, j =

1, ..., d is a Brownian motion in (

). In the normalised case the random vari-

ables B

, ..., B

are independent for each t

6. (Covariance matrix) If (B

)

is a normalised real Brownian motion then for ev-

ery choice of finitely many instants 0

< ... < t

, the random vector B

...B

is an n-dimensional Gaussian random vector having the zero vector as expected value

and the covariance matrix

Cov(B

... B

) = (t

)

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

Theorem 1.3.3 (Law of the Iterated Logarithm). For every standard Brownian

motion (B

)

we have for almost every

lim sup

()

2t log log(1/t)

= 1,

lim inf

()

2t log log(1/t)

= 1,

lim sup

()

2t log log(t)

= 1,

lim inf

()

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

|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

)

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

)

is a

collection of sub--algebras of

A such that for every t > s 0 we have

(1.15)

A process (X

)

is adapted to a filtration (

)

if X

-measurable for every

0. The canonical filtration of the random variable (X

)

is defined by

; s

t). Trivially, every process (X

)

is adapted to its canonical filtration.

To emphasise that a process (X

)

is adapted to a filtration (

)

we often write

)

Definition 1.3.4. The stochastic process (B

)

:= (,

A, P, B

)

, adapted

to the filtration (

)

, and with state space (

), is called a d

-dimensional

Brownian motion with filtration if the following conditions are fulfilled:

(i) For each 0

s < t, the process B

- B

is independent of

and is normally

distributed, i.e.

- B

0,t-s

...

0,t-s

(d factors).

(ii) Almost every path t

() is continuous.

Again, we say (B

)

is normalised if it starts at 0 almost surely, i.e. B

= 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

1.4.1

Conditional Expectation

Let X be a integrable random variable and let us interpret every sub--algebra of

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

I) be a partition of such that

P (B

)

0, for every j I. Taking one of the B

, recall the definition of the

conditional probability measure P

under the hypothesis B

(A) := P (A

) :=

P (A

)

P (B

)

P (B

)

P )(A)

for all A

P (B

)

P ).

The conditional expectation of X under the hypothesis B

is the expectation of X

w.r.t. the probability measure P

, i.e.

(X) :=

XdP

P (B

)

XdP.

(1.16)

We use the conditional expectation under B

(1.16) to introduce a new random

variable defined by

(X)1

assigns to each

the conditional expectation under B

, and is measurable

with respect to the -algebra

C := (B

; j

I). Moreover, we have the characteris-

ing property that for every C

C, we have that

dP =

XdP.

(1.17)

The random variable X

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

on which is

C-measurable and fulfils

dP =

XdP

for all C

(1.18)

Now it makes sense to give the following definition of the conditional expectation.

1.4 Martingales

Definition 1.4.2. Assume that the circumstances of the last theorem are given and

let X

be the P -a.s. determined random variable such that (1.18) is fulfilled. Then

we call X

the conditional expectation under the hypothesis

C and define

E(X

|C) := X

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 (depending

on whether X and Y are non-negative or integrable). We have

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.

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

is a sub--algebra of

, then

E(E(X|C

)

) =

E(E(X|C

)

) =

E(X|C

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,

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

(P ). Then it can be shown

that the conditional expectation X

is the best approximation in the Gauß mean-

square sense. The conditional expectation X

of X is the random variable that

minimises

E((X - X

)

) =

E(X

)

- E(X

1.4 Martingales

A rather geometric idea is to see the conditional expectation as a projection. Let

(P ) denote the space of all equivalence classes of

(P ), and let two random

variables be equivalent if they are almost surely equal. Then we can obtain L

(P )

as a direct sum of orthogonal spaces, i.e.

(P ) = L

(P )

Here, L

(P ) denotes the subspace of L

(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

(P ) onto

(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 (

)

the -algebra

:= (

We define the -algebra of events strictly prior to t as

:= (

s<t

where we set

:= 0, and the -algebra of events immediately after t as

:= (

s>t

Furthermore, we call the filtration (

)

right-continuous if (

)

= (

)

and

left-continuous if (

)

= (

)

. If (X

)

is adapted to the left-continuous

filtration (

)

then we can obtain X

by knowing all the prior variables X

[0, t). Similar, right-continuity means that if X

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

is right-

continuous and

contains all null sets.

Definition 1.4.4. A stochastic process (,

A, P, (X

)

) that is adapted to the fil-

tration (

)

and for which X

(P ), t

0, is called a martingale if

E(X

) = X

for t

s 0, a.s.,

(1.20)

a supermartingale if

E(X

)

for t

s 0, a.s.,

(1.21)

Details

Seiten
Erscheinungsform: Originalausgabe
Jahr: 2007
ISBN (eBook): 9783836612876
DOI: 10.3239/9783836612876
Dateigröße: 784 KB
Sprache: Englisch
Institution / Hochschule: Swansea University – Mathematics, Studiengang Finanzmathematik
Erscheinungsdatum: 2008 (Mai)
Schlagworte: multi-asset portfolio relative return stochastic control theory utility functions mathematics

Autor

Daniel Michelbrink (Autor:in)

Stochastic Calculus with Applications to Stochastic Portfolio Optimisation

Zusammenfassung

Leseprobe

Inhaltsverzeichnis

Details

Autor

Daniel Michelbrink (Autor:in)