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Credit Risk: Modelling, Valuation and Hedging

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Risk-Neutral Valuation: Pricing and Hedging of Finance Derivatives

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Credit Risk Valuation

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Visual Explorations in Finance with Self-Organizing Maps

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Efficient Methods for Valuing Interest Rate Derivatives

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Exponential Functionals of Brownian Motion and Related Processes

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Incomplete Information and Heterogeneous Beliefs

in Continuous-time Finance

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Jean-Lue Prigent

Weak

Convergence

of Financial

Markets

With 8 Figures

and 1 Table

,

Springer

Professor Jean-Luc Prigent

THEMA

University of Cergy

Boulevard du Port 33

95011 Cergy

France

Mathematics Subject Classification (2003): 91-02, 91B28, 93A3Q, 60-xx, 60G35,

62P05, 60BIO, 65CxX

ISBN 978-3-642-07611-4

ISBN 978-3-540-24831-6 (eBook)

DOI 10.1007/978-3-540-24831-6

Bibliographic information published by Die Deutsche Bibliothek

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© Springer- Verlag Berlin Heidelberg 2003

Originally published by Springer-Verlag Berlin Heidelberg New York in 2003.

Softcover reprint of the hardcover 1st edition 2003

The use of general descriptive names, registered names, trademarks, etc. in this publication

does not imply, even in the absence of a specific statement, that such names are exempt

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Cover design: design & production, Heidelberg

To My Family and S.

Preface

Motivation

• One of the main problem treated in this book is the following:

Continuous and discrete time financial models are at best approximations

of the reality. So, it seems important to connect them and to compare their

predictions when, in the discrete time setting, periods between trades shrink

to zero. But does the convergence of stocks prices imply the convergence of

optimal portfolio strategies, derivatives prices and hedging strategies ?

We can alternatively ask the following question :

Consider two investors who estimate stock prices from statistical data, one

in a discrete time setting, the other one in continuous time. Suppose they

agree on the stock price distribution, for example a GARCH model in discrete time for the first one and a Hull and White type model for the second

one (it means that the discrete time periods are sufficiently small to accept

that the distributions of the stock log returns are equal). When for instance

they determine the no-arbitrage prices, each one for his own model, do they

necessarily agree also for example on options prices or spreads ?

As it will be seen in this book, the answer is not straightforward. For example, convergence of options prices is typically proved for binomial tree

(for example, the well known Cox-Ross-Rubinstein derivation of the BlackScholes formula) or suitable multinomial trees and is established in the

complete case. However, real markets are usually incomplete. This may induce "instability" of financial variables or instruments due to convergence

problems within various financial models. This point is illustrated in chapter 2 for optimal portfolio policies, option pricing and hedging strategies.

This lack of robustness for some basic approximations shows that we must

be particularly cautious when dealing with convergence problems.

VIII

• It is often easier to derive analytic or numerical results in discrete time

than in continuous time (or vice versa). Hence a second purpose is to recall

some basic approximations which are of particular interest to build numerical algorithms. They can be applied for the pricing of American, Asian

and barrier options on stocks or indexes and to approximate bonds and

interest rates derivatives .

• Finally, we have to make a choice: what type of convergence should we use?

As it is well-known, convergence in distribution, also called "weak convergence" is a convenient tool in many statistical studies. It further allows to

analyze stochastic phenomena without specifying a particular probability

space: often in practice, only the set of values of the observed stochastic

processes is involved. "Weak convergence" refers here to the convergence in

distribution for stochastic processes treated as random elements of function

spaces. Despite its greater complexity (due to "tightness condition") when

compared to the weak convergence for finite-dimensional distributions, the

"functionar' weak convergence is useful: contrary to the former mode, it

can guarantee convergence of exotic option prices, such as Asian options

which involve the whole path of the stock process.

To summarize, the purpose of this book is to apply the theory of weak

convergence of stochastic processes to the study of financial markets.

Readership

This book assumes the reader has a good knowledge of probability theory

in continuous time. It is aimed at an audience with a sound mathematical

background. It supposes also that basic financial theory, such as valuation

and hedging of derivatives, is already known. However:

- In the first chapter, basic notions and definitions of stochastic processes

are first recalled. Second, an overview of the theory of weak convergence of

semimartingales is provided. In particular, a guideline is given for the weak

convergence of stochastic integrals and contiguity properties.

- Along the second chapter, the standard notions and properties of the financial markets theory are recalled (but not detailed).

- Finally, the emphasis throughout the third chapter is on presenting the basic

discrete models and their continuous time limits. The focus remains on a survey about multinomial approximations and more generally about computing

problems with lattices for different types of options. Other approximations

such as ARCH models ... are also introduced and detailed. Nevertheless, a perfect knowledge of the first two chapters is not fully required.

IX

Book Structure

• The first chapter tries to answer the question :

How to prove that a sequence (Xn)n of stochastic processes

weakly converges to a given stochastic process X ?

This mathematical chapter is only a guide to the reader. While main results are included, the proofs are not provided, as they are already excellent

treatment of this theory readily available :

- The books of Dellacherie and Meyer [108] present the general properties of stochastic processes (volume I) and martingales (volume II), Ethier

and Kurtz [148] deals with Markov processes, Elliott [144], Kopp [250)

and Rogers and Williams [365] introduce the stochastic integration. The

books of McKean [286], Chung and Williams [75] and Karatzas and Shreve

[236] dal with Brownian motion and continuous martingales. The book of

Protter [351] gives a very clear presentation of semimartingales, stochastic

integration and stochastic differential equations.

- Concerning main results of weak convergence of semi martingales, it is

referred to Jacod and Shiryaev's book [214].

- Nevertheless, with respect to this latter book, two parts are added:

1) A special emphasis on the weak convergence of sequences of triangular arrays, which are of particular interest, when dealing with convergence

problems from discrete time to continuous time models.

2) A survey of main results concerning weak convergence of sequences of

stochastic integrals and solutions of stochastic differential equations (see

also the new version of Jacod and Shiryaev's book to appear in 2003) .

• The second chapter deals with the following question:

What are the main problems that we encounter

when examining weak convergence of financial markets ?

Thus this chapter introduced the results about weak convergence of :

- Optimal portfolio policies for utility maximizing investors.

- Option prices, in particular convergence problems of bid-ask spreads.

- Hedging strategies which duplicate options in complete financial markets

or hedging strategies which minimize the locally quadratic risk when facing

incomplete markets.

x

A survey of basic results of financial theory is included but not detailed

since many books are also available:

- For the main notions, among others, Duffie [124][125][126], Bingham and

Kiesel [37], Kwok [261]' Elliott and Kopp [145], Lamberton and Lapeyre

[264], Nielsen [321], Bjork [39J.

- More particularly, Pliska [339J introduces all of the main financial concepts for the discrete time case. Musiela and Rutkowski [312J deal in particular with the theory of bond markets and term structure models. In

Jeanblanc-Picque and Dana [220], the equilibrium approach is detailed.

Shiryaev [381J introduces a large variety of stochastic models.

Hence, in this chapter, we focus on convergence results. Some particular

proofs are fully detailed to show how weak convergence results of the first

chapter can be applied .

• The third chapter reviews a list of results to solve the following problem:

How to construct in practice a sequence (Xn)n of stochastic processes

which weakly converges to a given stochastic process X ?

Although it is not a purely "numerical" chapter, many of standard approximations of basic continuous time processes are recalled :

- Its first part contains some general results about approximations of solutions of stochastic differential equations (standard and backward).

- Its second part is devoted to standard lattice models, when approximating

diffusions. Binomial and trinomial schemes are especially examined when

the continuous time limit process is driven by a Brownian motion. For

most of these models, the discrete time subdivision of the time interval is

deterministic.

- A third part proposes other models for example diffusions with jumps.

This class of processes contains Levy processes and in particular subordinators which are of particular interest when examining dynamics of high

frequency data. Both deterministic and random discretizations are studied.

By considering sequences of random times, the latter ones allow in particular to examine problems of portfolio rebalancing.

- Finally, a list of some standard approximations of interest rate models is

provided: factor models as well as Heath-Jarrow-Morton type models and

Market models are briefly reviewed.

XI

Final Word and Acknowledgments

This book is an attempt to summarize the main convergence results about

financial markets which are known at present. It is focussed on robustness

of financial instruments under convergence of discrete time to continuous

time financial markets. In particular, it indicates option pricing rules that

are stable under convergence of the underlying assets. This feature reduces

the model risk when we must choose between discrete time or continuous

time to describe asset prices dynamics.

While some of the quoted results concern more academics than practitioners, it seems important to underline main features of convergence, first

of all that approximating models do converge. Both "pure mathematical"

speed (in the spirit of the famous Central Limit Theorem) and computational speed must be analyzed. Obviously, all convergence problems are not

yet solved. Further extensions are still in progress both in the mathematical

field (to take more dependency properties into account, to obtain functional

speed of convergence ... ) and also in the financial theory (search of other algorithms to simulate financial variables, study of more general option pricing

or portfolio problems taking account of market imperfections such as trading

strategies which are unavailable in continuous time ... ).

I hope that this book will contribute to stimulate new research on the

sometimes awkward (nevertheless fascinating ?) weak convergence world and

financial theory.

While I cannot thank all the people for supports and useful discussions

since I began to study the financial theory, I want to mention in particular

my colleagues of the research department THEMA, the members of HSBCCCF and in particular Eric Baesen and Jean-Fran<;ois Boulier, also Nicole El

Karoui who encouraged me some years ago to work in financial mathematics,

especially to examine weak convergence problems, Patrick Navatte, Patrice

Poncet and Yves Simon who inclined me to discover and study more applied

features of the financial market theory and Monique Jeanblanc whose legendary kindness is such that I have never hesitated before telling her with

my problems.

I am grateful also for fruitful and enjoyable collaborations to Mohamed

Ahnani, Mondher Bellalah, Philippe Bertrand, Jean-Philippe Lesne, Olivier

Renault, Raphael Sobotka, Christophe Villa and in particular Olivier Scaillet

with whom I have written some of the papers about convergence quoted in

this book.

XII

I express a thought of gratitude for those who were my professors when I

was a young student learning the Probability theory and especially for Jean

Memin who has been my mathematical PhD thesis advisor, some years ago,

and to whom we are all indebted for some of the important results about

weak convergence of stochastic processes quoted in this book.

I would like also to thank the staff of Spinger-Verlag, in particular Martina

Bihn for her wonderful patience since she accepted to be in charge of this

book.

Paris, January 2003

Jean-Luc Prigent

Contents

1.

Weak Convergence of Stochastic Processes ...............

1

1.1 Basic Properties of Stochastic Processes . . . . . . . . . . . . . . . . . . . 2

1.1.1 Stochastic Basis, Filtration, Stopping Times. . . . . . . . . 2

1.1.2 Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Martingales......................................

7

1.1.4 Semimartingales and Stochastic Integrals. . . . . . . . . . .. 14

1.1.5 Markov Processes and Stochastic Differential Equations 40

1.1.6 The Discrete Time Case. . . . . . . . . . . . . . . . . . . . . . . . . .. 58

1.2 Weak Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64

1.2.1 The Skorokhod Topology. . . . . . . . . . . . . . . . . . . . . . . . .. 64

1.2.2 Continuity for the Skorokhod Topology ............. 67

1.2.3 Definition of Weak Convergence. . . . . . . . . . . . . . . . . . .. 69

1.2.4 Criteria for Tightness in ][])k. . . . . . . . . . . . . . . . . . . . . . .. 72

1.2.5 The Meyer-Zheng Topology. . . . . . . . . . . . . . . . . . . . . . .. 74

1.3 Weak Convergence to a Semimartingale ..... . . . . . . . . . . . . .. 75

1.3.1 Functional Convergence and Characteristics ......... 75

1.3.2 Limits of Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87

1.3.3 Limit Theorems for Markov Processes. . . . . . . . . . . . . .. 88

1.3.4 Convergence of Triangular Arrays .................. 92

1.4 Weak Convergence of Stochastic Integrals ................. 100

1.4.1 Introduction ..................................... 100

1.4.2 The Uniform Tightness Condition U.T .............. 101

1.4.3 Functional Limit Theorems for Sequences of Stochastic Integrals and Stochastic Differential Equations. . . . . 104

1.5 Limit Theorems, Density Processes and Contiguity ......... 108

1.5.1 Hellinger Integral and Hellinger Process ............. 108

1.5.2 Contiguity and Entire Separation ................... 115

1.5.3 Convergence of the Density Processes ............... 121

1.5.4 The Statistical Invariance Principle ................. 125

XIV

Contents

2.

Weak Convergence of Financial Markets ..................

2.1 Convergence of Optimal Consumption-Portfolio Strategies ...

2.1.1 Weak Convergence of Controlled Processes ..........

2.1.2 The Martingale Approach .........................

2.2 Convergence of Options Prices ...........................

2.2.1 Problems and Examples ...........................

2.2.2 Contiguity Properties .............................

2.2.3 The Case of Incomplete Markets ...................

2.2.4 Transaction Costs ................................

2.2.5 American Options ................................

2.3 Convergence of Hedging Strategies ........................

2.3.1 Binomial Case and Clark-Hauss man Formula ........

2.3.2 Weak Convergence of Integrands ...................

2.3.3 The Local Risk-Minimizing Strategy ................

129

130

131

161

185

185

194

198

218

230

240

243

251

256

3.

The Basic Models of Approximations ..................... 267

3.1 General Remarks ....................................... 267

3.1.1 Some numerical methods for forward and backward

stochastic differential equations .................... 268

3.1.2 Some numerical methods for computations of Greeks .. 272

3.2 Lattice ................................................ 274

3.2.1 Simple Binomial Processes as Diffusion Approximations274

3.2.2 Correction Terms for Path-Dependent Options ....... 287

3.2.3 Adjustment Prior to Maturity and Smoothing of the

Payoff Functions ........... . . . . . . . . . . . . . . . . . . . . .. 295

3.2.4 Fast Accurate Binomial Pricing .................... 301

3.2.5 Approximating a Diffusion by a Trinomial Tree ...... 305

3.3 Alternative Approximations .............................. 309

3.3.1 ARCH Approximations ........................... 309

3.3.2 Levy Processes ................................... 321

3.3.3 Convergence for Random Time Intervals ............ 345

3.3.4 Deterministic or Random Discretizations of ContinuousTime Processes .................................. 359

3.4 Approximations of Term Structure Models ................. 371

3.4.1 Bonds and Interest Rate Derivatives ................ 371

3.4.2 Basic Interest Models and their Approximations ...... 377

3.4.3 Two-factors Model ............................... 387

3.4.4 Market Models: Discretization of Lognormal Forward

Libor and Swap Rate Models ...................... 388

3.4.5 Discretization of Deflated Bond Prices .............. 388

3.4.6 Pricing Interest Rate or Equity Derivatives and Discretization ....................................... 395

Index ......................................................... 419

1. Weak Convergence of Stochastic Processes

The processes of interest are assumed to take values in the Euclidian space

JRd. More general complete and separable metric spaces can in fact be considered but the applications developed here concern dynamics of processes

taking values in JRd like for example d-dimensional stock prices. Moreover,

paths of these processes are sufficiently regular: the space IID(JRd) of the rightcontinuous functions having left limits (rcll) is in particular important.

"Weak convergence" means convergence in distribution. Results of weak

convergence for sequences of JRd-valued random variables are well-known:

central limit theorem, laws of large numbers (see for example [379]). However,

the study which is developed here is more complicated: it deals with weak

convergence of sequences of entire processes. Fortunately, these ones can be

considered as "simple" random variables no longer with values in JRd but

(very often) in IID(JRd). Thus, similar concepts can be introduced and results

be proved (with more effort ... ). For example, the definition is of the same

kind: a sequence of processes (Xn)n is said to weakly converge to the limit

process X if and only if

lim E[J(Xn)] = E[J(X)] ,

n--+CX)

for each f in the space C(IID(JR d)) which is the space of bounded and continuous mappings from IID(JRd) to JR, where IID(JRd) is equipped with a "good"

topology. Conditions are given under which the above property is satisfied,

especially when both Xn and X are semimartingales.

Since the purpose is more ambitious, technical tools must be used. First,

the general theory of stochastic processes must be introduced. It is briefly

reviewed in this chapter (see for example the books of DeUacherie and Meyer

[108] for a complete exposition of this theory). Secondly, the theory of convergence in distribution of semi martingales must be developed. A survey of

the main results of this theory is presented. A more general and systematic

exposition can be found in particular in Jacod and Shiryaev [214] to which

this chapter refers. Concerning the special case of the weak convergence of

stochastic integrals, the main results are those of Jakubowski, Memin and

Pages [215], Memin and Slominsky [298] and also Kurtz and Protter [256].

J.-L. Prigent, Weak Convergence of Financial Markets

© Springer-Verlag Berlin Heidelberg 2003

2

1. Weak Convergence of Stochastic Processes

1.1 Basic Properties of Stochastic Processes

In this and the succeeding chapters, the time parameter takes values in either

a finite interval [0, T] or [0,(0). Nevertheless, an autonomous treatment for

the discrete time is provided at the end of the first subsection of this chapter.

The set T denotes the time parameter set in each case.

1.1.1 Stochastic Basis, Filtration, Stopping Times

The concept of filtration is used to model the acquisition of information as

time evolves. This a key concept in financial theory: it represents for example the flow of possible observations of prices on a financial market which is

available for traders and portfolio managers.

Here are some standard notations to be used in the whole book.

Suppose (st, F, IP') is a probability space.

Definition 1.1.1. A filtration IF = (Ft, t E T) is an increasing family of

sub-sigma-fields F t C :F. A stochastic basis is a probability space equipped

with a filtration.

Increasing means that if s ::; t, then Fs C Ft. F t is usually interpreted as

the set of events that occur before or at time t. Generally, F t represents the

history of some process observed up to time t but other possible histories are

allowed. It is assumed that IF satisfies the "usual conditions" (see Dellacherie

and Meyer [108]). This means:

i) Complete: every IP'-null set in F belongs to Fo and so to all Ft.

ii) Right continuous: F t

= ns>tFs.

Definition 1.1.2. A stochastic basis B = (st, F, IF, IP') is a probability space

equipped with a filtration and is also called a filtered probability space.

To deal with problems like for example "what is the optimal time to invest

or to exercise an American option?" when taking available information into

account, it is natural to introduce the notion of stopping time.

Definition 1.1.3. (Stopping time).

a) A stopping time is a mapping T : st --+ T such that {T ::; t} E F t for all t.

b) If T is a stopping time, FT denotes the collection of all sets A E F such

that An {T ::; t} E F t for all t E T.

c) 1fT is a stopping time, F T - denotes the (J-field generated by Fo and all

the sets of the form An {t < T} EFt where t E T and A EFt.

1.1 Basic Properties of Stochastic Processes

3

Remark 1.1.1. The event {T ::; t} depends only on the history up to time

t. A stopping time is a random time T such that at each time t, one may

decide whether T ::; t or T > t from what one knows up to time t. FT is

interpreted as the set of events that occur before or at time T. For example,

the first time an asset price reaches a given level is a stopping time with

respect to a filtration which takes account of the information delivered by

the asset prices. But the last time it reaches a given level is not a stopping

time since information about the future is necessary to define this last time.

Let us now examine some basic and very useful features of stopping times.

Proposition 1.1.1. (Stopping times properties).

i) If T is a stopping time and t E T then T + t is a stopping time.

ii) 1fT is a stopping time and A E FT then TA defined by

TA(W)

= {T(W)

ifw E A,

+00 zfw tJ. A,

is also a stopping time.

iii) Assume Sand T are stopping times.

1) Their minimum S 1\ T and their maximum S V T are also stopping times.

2) If S::; T then Fs eFT'

3) If A E F s! then :

An {S ::; T} EFT, An {S

= T} EFT and A n {S < T}

E

iv) If (Tn)nEN is a sequence of stopping times then I\nENTn and

two stopping times.

FT-.

VnENTn

are

1.1.2 Stochastic Processes

Basic notions. Let us fix some terminology :

Definition 1.1.4. A continuous time stochastic process X taking values in

a measurable space (E, £) is a family of random variables (Xt)t defined on

(Sl,IF,JP'), indexed by t, which take values in (E,£).

Hence, for all t, X t is a random variable with values in E. A process can

also be considered as a mapping from Sl x T into E. Moreover, for each fixed

w (which represents a "state of the world"), t ---. Xt(w) is a function defined

on T, called a path or a trajectory of the process X.

Definition 1.1.5. A process X is called rell ("cadlag" ln french) if all its

paths are right-continuous and admit left limits. When X is rell, two other

processes can be defined:

X_ = (Xt-)tET with X t- = limsL1X = (L1Xt )tET with L1Xt = X t - X t -.

4

1. Weak Convergence of Stochastic Processes

Definition 1.1.6. When X is a process and T is a mapping

process stopped at time T, denoted by X T , is defined by

n

X[ = XTI\t.

-+

T, the

(1.1 )

A standard question is : when is it possible to say that two stochastic

processes model the same random phenomenon ? For this, three notions can

be introduced:

- The weakest is based on the fact that, in practice, a stochastic process is only

observed at a finite number of instants (discrete time observations). Therefore

it leads to the definition of the finite-dimensional distributions of a process X.

Consider for example the Euclidian case: E

H.

= JRd

with its Borelian O'-field

Definition 1.1.7. Consider for n E N, t1, ... , in E T and A E H, the measure

defined on ®nJR d by

JlDXt" ... ,Xt n

(A) = JID[{w En: (Xtl (w), ... , Xtn (W)) E A}].

Two processes X and Yare said equivalent (or have the same law), denoted

by X '" Y, if their families of finite-dimensional distributions coincide.

As it can be noted, this does not require that the two processes are defined on the same probability space (n, F, JID) and avoids the choice of a proper

probability space to describe a random phenomenon. Thus, if X is considered as a map X : n -+ (JRd) T with only Borelian subsets to be considered,

then a canonical version of X can be identified up to equivalence (this is the

well-known Kolmogorov Extension Theorem).

- However, if T is uncountable, stronger notions of "equivalence" must be

defined since trajectories have often values in none Borelian set of (JRd) T.

Two concepts can be introduced to make precise paths of a process. Note

that, in what follows, the two processes X and Y must be defined on the

same probability space (n, F, JID).

Definition 1.1.8. The process Y is said to be a modification of X if for

every t,

X t = yt a.s ..

Definition 1.1.9. A random set A is called evanescent if the set

{w : :3 t E Twith (w, t) E A} is JID-null; two processes X and Yare said

indistinguishable ~f the random set {X =1= Y} = {(w, t) : Xt(w) =1= yt(w)} is

evanescent, i. e. if almost all paths of X and Yare the same.

1.1 Basic Properties of Stochastic Processes

5

As it can be seen, the difference between the two definitions is that, if

Y is a modification of X, the set of zero measure on which X t and yt may

differ, may depend on the time t, which is not the case when they are indistinguishable since one has X t = yt a.s. for all t E T. Thus, if X and Y

are indistinguishable, they are modifications of each other but the converse

is not true, unless, for example, both X and Yare rc (right-continuous) or

lc (left-continuous). Note also that when T is countable, these two notions

coincide. As for random variables, in most cases, the notations X = Y (or

X :

The Optional u-Field. Consider the stochastic basis B

= (n, F, IF, JP').

Definition 1.1.10. a) A process X is adapted to the .filtration F if X t is

Ft-measurable for every t E T.

b) The optional IT-field is the IT-.field 0 on n x T generated by all rcll

adapted processes.

c) A process that is O-measurable is called optional.

d) A process X is F-progressive if for each t, the restriction of X to [0, tj x n

is B([O, t]) x Ft-measurable.

Note that, if X is F-progressive then X is F-adapted. The converse is

false but every right continuous process X which is F-adapted is also Fprogressive.

Notation: F X is the filtration generated by the process X itself

F{ = IT(X.,

S :

t).

Proposition 1.1.2. Let X an optional process.

1) Considered as a mapping on n x T, it is F ® T -measurable.

2) If T is a stopping time, the stopped process XT is also optional.

Another characterization of the optional IT-field is based on stochastic

intervals : if 8 and T are two stopping times, one may define four kinds of

stochastic intervals which are the following random sets:

[[8, T]]

{ [[8, T[[

]]8, T]]

]]8, T[[

=

=

=

=

{(w, t)

{(w, t)

{(w, t)

{(w, t)

:t

:t

:t

:t

E

E

E

E

T,8(w) :~~T,8(w) :~~~~T,8(w) < t :~~~~T,8(w) < t < T(u!)}.~~

Denote by 11A the indicator of the set A such that

11 ( )

A W

= { 1 if w

°ifw ~ A,A.

E

6

1. Weak Convergence of Stochastic Processes

Proposition 1.1.3. (Optional properties).

i) If Sand T are two stopping times and if Y is an F s-measurable random

variable, the four following processes

Yll[[S,Tll, Yll[[s,T[[, Ylllls,Tll andYlllls,T[[

are optional.

ii) If X is rell and adapted, the two processes X_ and L1X are optional.

iii) The cy-field 0 is generated by the stochastic intervals [[0, TJl, where T is

any stopping time.

Next, hitting times can be examined and in particular the following result

of Hunt (see for example Dellacherie and Meyer [108]).

Theorem 1.1.1. ff A is an optional random set, its first entry time TA

defined by

TA(W) = inf{t: (w, t) E A}

is a stopping time.

This result can be applied to the useful particular result :

Proposition 1.1.4. 1) If X is an JRd-valued process and if B is a Borelian

subset of IR d , then T defined by T(w) = inf{t : X(w, t) E B} is a stopping

time.

2) If X is an IR-valued adapted rc process with nondecreasing paths, then,

for every a E JR, T defined by T(w) = inf{t : X(w, t) ~ a} is a stopping time.

The Predictable u-Field. The notion of "predictable (J-field" is linked

to the idea that the value of a process at time t can be "obtained" from

observations just before time t (see the discrete case for a good intuition).

More precisely :

Definition 1.1.11. The predictable cy-field is the cy-field P on [2 x T that

is generated by all lc adapted processes. A process or a random set that is

P-measurable is called predictable.

It is obvious that the predictable cy-field P is included in the optional

cy-field O. It is also possible to get another characterization of P.

Theorem 1.1.2. 1) The predictable (J- field is also generated by anyone of

the following collections of random sets :

i) A x {O} where A E Fo and [[0, T]] where T is any stopping time.

ii) A x {O} where A E Fo and A x (s, t] where s < t, A E Fs.

2) The predictable (J-field is also generated by all adapted processes that have

continuous paths.

1.1 Basic Properties of Stochastic Processes

7

Proposition 1.1.5. (Predictable properties).

1) If X is a predictable process and if T is a stopping time then the stopped

process XT is also predictable.

2) If Sand T are two stopping times and if Y is an F s-measurable random

variable, the process Ylllls,Tll is predictable.

3) If X is a rcll adapted process then X_ is a predictable process. Consequently, if X is predictable itse~f then LlX is also predictable.

Definition 1.1.12. A predictable time is a mapping T :

the stochastic interval [[0, T[[ is predictable.

st

--+

T such that

Recall some standard properties of predictable times (see Dellacherie and

Meyer [108] for more details).

Proposition 1.1.6. (Predictable times).

1) Every predictable time is also a stopping time.

2) If (Tn)n is a sequence of predictable times then:

i) VnENTn is a predictable time.

ii) If S = AnENTn and UnEN{S = Tn} = st then S is a predictable time.

3) If T is a predictable time and A EFT~, the time TA (see Theorem 1.1.1)

is predictable.

Definition 1.1.13. A rcll process X is called quasi-left continuous if

LlXT = 0 a.s. on the set {T < oo} for every predictable t1me T.

Proposition 1.1. 7. Let X be a rcll adapted process. Then X is quasi-left

continuous if and only if for any increasing sequence (Tn)" of stopping times

with limit T, lim XTn = X T a.s. on the set {T < oo}.

Definition 1.1.14. A generalized stochastic process A is a process such

that:

(i) A is lR+ -valued, Ao = 0 and its paths are non-decreasing.

(ii) 1fT = inf{t: AT = oo} then A is right-continuous on [[O,T[[.

A generalized stochastic process A does not jump to infinity if AT- = +00

on T < 00, in which case its paths are everywhere right-continuous.

1.1.3 Martingales

As it is well-known, the concept of Martingale is crucial in the modern theory

of finance. If the price process M of a stock is a martingale, the conditional

expectation at time s of the future value M t of the stock at time t is given

by its current value Ms.

This subsection is a review of properties of martingales, submartingales

and supermartingales that are essentially due to Doob. The proofs may be

found in most standard books (see for example Doob [115], Dellacherie and

Meyer [108], Elliott [144], Williams [404] ... ).

8

1. Weak Convergence of Stochastic Processes

Notation: consider a random variable X, integrable under the probability

lP'. Denote lEIP[X] the expectation of X and if Q is a sub-field of F denote

lEIP[ X IQ] the conditional expectation of X under Q. For p > 1, let JLP be the

space of random variables such that lEll'[IXIP] is finite.

Definition 1.1.15. A martingale (resp. sub martingale, resp. super martingale) is an adapted process X on the basis (ft, F, IF, lP') whose lP'- almost all

paths are rcll, such that every X t is integrable and that for s ::::; t :

Note that, if in the above definition, the paths of a martingale X are

not assumed to be rcll, it can be proved that there exists a unique (up to

indistinguishability) martingale Y which is rcll and indistinguishable from

X.

Proposition 1.1.8. (Jensen's Inequality).

Let ¢ : IR -+ IR be convex.

1) Let Y and ¢(Y) be integrable random variables. Then, for any a-field Q,

¢ (lE[YIQ]) ::::; lE[¢(Y)IQ]·

(1.2)

2) Consequently, if M is a martingale such that ¢(Md is integrable for any

t, then ¢(M) is a sub martingale (in particular IMI and M2).

3) Assume that X is a submartingale such that ¢(Xt ) is integrable for any t.

If moreover ¢ is nondecreasing then ¢(X) is also a submartingale.

A process is said to admit a terminal variable Xoo if X t converges a.s. to

a limit Xoo as t T00; in such a case, the variable XT is (a.s.) well defined for

any stopping time with X T = Xoo on {T = oo}.

Theorem 1.1.3. Let X be a supermartingale such that there exists an integrable random variable Y with X t ::::: lE[YIFt ] for all t E T. Then:

i) (Doob's limit Theorem) X t converges a.s. to a finite limit Xoo.

ii) (Doob's Stopping Theorem):

a) If Sand T are two stopping times, the random variables Xs and X T are

integrable, and Xs ::::: lE[XTIFs] on the set {S::::; T}. In particular, X T is a

supermaTtingale.

b) If X is a martingale such that there exists an integrable random variable

= lE[YIFt ] for all t E T, then Xs ::::: lE[XTIFs] a.s. (X is said also

"closed" by Y).

Y with X t

1.1 Basic Properties of Stochastic Processes

9

Locally Square-Integrable Martingales. Two classes of martingales can

be examined. The first concerns the property of uniform integrability, the

second is about the notion of local martingale.

Definition 1.1.16. An uniformly integrable martingale M is such that the

family of random variables (MdtET is uniformly integrable. It means:

SUPtET

1

jMt!>c

IMtldlP'

-+

0 whenc

-+ 00.

M denotes the class of all uniformly integrable martingales.

From Theorem 1.1.3, it can be deduced:

Theorem 1.1.4. Let M be an uniformly integrable martingale and let T a

stopping time. Then the stopped process MT is also an uniformly integrable

martingale.

A particular case of such martingale is the following :

Definition 1.1.17. A square-integrable martingale M is such that

1e denotes the class of all square-integrable martingales.

Obviously, 1{2 C M. Moreover, each of these two sets are stable under stopping: that is for any stopping time T and for any process X in one of these

sets, the stopped process XT is also in this set.

Theorem 1.1.5. 1) If M is an uniformly integrable martingale, then M t

converges a.s. and in ll) to a terminal variable Moo and fo'r all stopping times

T, MT = lE[MooIFTl.

2) Moreover, M is a square-integrable martingale if and only if Moo is squareintegrable, in which case MT -+ Moo takes place in ll}.

3) If Y is an integrable random variable, there exists an uniformly integrable

martingale M, and only one up to an evanescent set, such that M t = lE[YIFtl

for all t E T and Moo = lE[YIFoo-l.

Theorem 1.1.6. (Doob's Inequality)

If M is a square-integrable martingale,

Another characterization of the elements of M is of particular interest :

Proposition 1.1.9. If M is an adapted rcll process with a terminal variable

Moo, then M is an uniformly integrable martingale if and only if for each

stopping time T, the variable MT is integrable and satisfies lE[MTl = lE[Mol,

10

1. Weak Convergence of Stochastic Processes

Now, examine the notion of localization of martingales.

Definition 1.1.18. A local martingale (resp. a locally square-integrable

martingale) is a process such that there exists an increasing sequence (Tn)n

of stopping times satisfying: limn Tn = 00 a.s. and every stopped process

XTn is an uniformly integrable martingale (resp. a locally square-integrable

martingale) .

Denote 1{~oc the space of all locally square-integrable martingales.

Definition 1.1.19. A process X is of class (D) if the set of random variables

{XT : T any finite-value stopping time} is uniformly integrable.

Proposition 1.1.10. (Class (D }).

1} Each martingale is a local martingale .

2} Each uniformly integrable martingale is a process of class (D).

3} A local martingale is an uniformly integrable martingale if and only if it

is a process of class (D).

To see the differences between these notions of martingales, examine the

next examples (see Jacod and Shiryaev [214]) :

Example 1.1.1. Let (Zn)n be a sequence of i.i.d. (independent and identically

distributed) random variables such that IP'[Zn = 1] = IP'[Zn = -1] = 1/2.

Consider the filtration F t = a(Zp : p E N*, p:::; t) and let M t = LI:S;P:S;[tJ Zp

where ttl denotes the integer part of t. Then, M is obviously a martingale

but, by the central limit theorem, M t does not converge a.s. as t T 00 and

thus M is not uniformly integrable.

Example 1.1.2. Let (An)nEN* be a measurable partition with IP'[An] = 2- n .

Let (Zn)n be a sequence of random variables that are independent of the An

with IP'[Zn = 2n] = IP'[Zn = _2n] = 1/2. Consider the filtration

Ft

= a(An : n E N*) ift E [0,1) and F t = a(An, Zn : pEW) ift E [1, (0).

Consider the following random variables :

y,

n

=

"Z

~

l:S;p:S;n

pllAp' X t =

°

{oYo",

ift E [0,1)

ift E [1, (0)

T

,n

=

°

{+oo on the set UI:S;p:S;n Ap

elsewhere

(Tn)n is a sequence of stopping times that increases to 00. The process XTn

is equal to

(resp. Y n ) on [[0,1[[ (resp. [[1,00[[) and Y n is bounded and

independent from F I - which implies that X T " is an uniformly integrable

martingale and X is a local martingale. But X is not a martingale since

Xl = Y00 is not integrable.

1.1 Basic Properties of Stochastic Processes

11

Doob-Meyer decomposition. Compensators. The stochastic basis

B = (st, F, IF, JP') is fixed throughout. Recall some basic properties about

processes with finite variation.

Definition 1.1.20. (Finite variation).

1) Denote V+ the set of all real-valued processes A that are rcll, adapted with

Ao = 0 and whose each path is non-decreasing.

A real-valued process A is said having a finite variation if for each .finite interval [0, t], there exists a constant C such that for all .finite partitions (ti)i

of [0, t], L:i IA ti + 1 - Ati I is dominated by C.

2) Denote V the set of all real-valued processes A that are rcll, adapted with

Ao = 0 and whose each path has .finite variation. Denote Var( A) the variation

process of A that is the process such that Var(AMw) is the total variation of

the function s ---> As (w) on the interval (0, tj de.fined by:

Var(A)t(w)

=

li~

L

IAtk/n(W) - At(k-l)/n(w)l·

l~k~n

Let A E V. For each wEst, the path t ---> At(w) is the distribution

function of a signed measure (positive if A is increasing) that is finite on each

interval [0, t] and that is finite on lR+ if and only if Var(A)oo(w) < 00. This

measure is denoted by dA t (w). If A E V and B E V and if the measure dA t (w)

is absolutely continuous to the measure dBt(w), this relation is denoted by

dA~dB.

It is immediately possible to define a Riemann-Stieljes integral with respect to such measure dA t (w): for each optional process H, define the integral

process denoted by H.A or by J~ HsdAs as follows on T:

H.At(w)

= {J~ Hs(w)dAs(w)

+00

if J~ l~s(w)ld[Var(A)]s(w) <

otherwise.

00,

Proposition 1.1.11. 1) Let B E V and let H be an optional process such

that B.A is finite-valued. Then B.A is also in V.

moreover, Band Hare

predictable then B.A is also predictable.

2) Let A E V and BE V be such that dB ~ dA, then there exists an optional

process H such that A = H.B up to an evanescent set. Aforeover, if A and

B are predictable then H may be chosen predictable.

rr

Recall now a very powerful result due to Meyer (see Dellacherie and Meyer

[108]), known as "the Doob-Meyer decomposition of submartingales". This

property allows to introduce martingales in many problems and underlies the

stochastic calculus of semimartingales.

12

1. Weak Convergence of Stochastic Processes

Theorem 1.1.7. If X is a sub martingale of class (D) (see Definition 1.1.19),

there exists a unique (up to indistinguishability) increasing integrable predictable process A with Ao = 0 such that X - A is an uniformly integrable

martingale.

Corollary 1.1.1. If X is a predictable local martingale whose paths have

finite variations then X = 0 (up to an evanescent set).

Another important result can be deduced:

Denote Aloe (resp. AtJ the space of all adapted processes A such that there

exists an increasing sequence (Tn)n of stopping times satisfying: limn Tn = 00

a.s. and every stopped process ATn has integrable variation: IEWar(A)ool <

00 (resp. is a locally integrable adapted increasing process).

Corollary 1.1.2. Let A be in Aloe (resp. At). Then, there exists a unique

(up to an evanescent set) predictable process in Aloe (resp.At), called the

compensator of A and denoted by AP, such that A - AP is a local martingale.

Example 1.1.3. Consider the Extended Poisson Process N on (Q,F,JP'). It is

an adapted simple point process (N is increasing and its jumps are equal to

1) such that :

i) IE[Ntl < 00 for each t E IR+.

ii) Nt - Ns is independent of the a-field Fs for all 0 ::; s ::; t.

iii) The function at = IE[Ntl is continuous (called the intensity of N).

In fact, the distribution of the variable Nt - Ns is a Poisson distribution with

mean at - as. Moreover, the compensator of N is its intensity: (Nt - at)t is

a martingale.

Let us study more explicitly the structure of 1-{2 and 1-{~oe'

Theorem 1.1.8. (Predictable Quadratic Covariation).

1) Let (M, N) any pair of locally square-integrable martingales. Then, there

exists a predictable process (M, N), unique up to an evanescent set, such that

M N - (M, N) is a local martingale. Moreover:

(M,N) = 1/4«(M +N,M

+ N)

- (M - N,M - N)).

2) If (M, N) is a pair of square-integrable martingales then (M, N) has integrable variation and M N - (M, N) is an uniformly integrable martingale.

3) (M, M) is non-decreasing and admits a continuous version if and only if

M is quasi-left continuous (see Definition 1.1.13).

Definition 1.1.21. The process (M, N) is called the predictable quadratic

covariation or also the angle bracket of the pair (M, N).

Springer-Verlag Berlin Heidelberg GmbH

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Jean-Lue Prigent

Weak

Convergence

of Financial

Markets

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To My Family and S.

Preface

Motivation

• One of the main problem treated in this book is the following:

Continuous and discrete time financial models are at best approximations

of the reality. So, it seems important to connect them and to compare their

predictions when, in the discrete time setting, periods between trades shrink

to zero. But does the convergence of stocks prices imply the convergence of

optimal portfolio strategies, derivatives prices and hedging strategies ?

We can alternatively ask the following question :

Consider two investors who estimate stock prices from statistical data, one

in a discrete time setting, the other one in continuous time. Suppose they

agree on the stock price distribution, for example a GARCH model in discrete time for the first one and a Hull and White type model for the second

one (it means that the discrete time periods are sufficiently small to accept

that the distributions of the stock log returns are equal). When for instance

they determine the no-arbitrage prices, each one for his own model, do they

necessarily agree also for example on options prices or spreads ?

As it will be seen in this book, the answer is not straightforward. For example, convergence of options prices is typically proved for binomial tree

(for example, the well known Cox-Ross-Rubinstein derivation of the BlackScholes formula) or suitable multinomial trees and is established in the

complete case. However, real markets are usually incomplete. This may induce "instability" of financial variables or instruments due to convergence

problems within various financial models. This point is illustrated in chapter 2 for optimal portfolio policies, option pricing and hedging strategies.

This lack of robustness for some basic approximations shows that we must

be particularly cautious when dealing with convergence problems.

VIII

• It is often easier to derive analytic or numerical results in discrete time

than in continuous time (or vice versa). Hence a second purpose is to recall

some basic approximations which are of particular interest to build numerical algorithms. They can be applied for the pricing of American, Asian

and barrier options on stocks or indexes and to approximate bonds and

interest rates derivatives .

• Finally, we have to make a choice: what type of convergence should we use?

As it is well-known, convergence in distribution, also called "weak convergence" is a convenient tool in many statistical studies. It further allows to

analyze stochastic phenomena without specifying a particular probability

space: often in practice, only the set of values of the observed stochastic

processes is involved. "Weak convergence" refers here to the convergence in

distribution for stochastic processes treated as random elements of function

spaces. Despite its greater complexity (due to "tightness condition") when

compared to the weak convergence for finite-dimensional distributions, the

"functionar' weak convergence is useful: contrary to the former mode, it

can guarantee convergence of exotic option prices, such as Asian options

which involve the whole path of the stock process.

To summarize, the purpose of this book is to apply the theory of weak

convergence of stochastic processes to the study of financial markets.

Readership

This book assumes the reader has a good knowledge of probability theory

in continuous time. It is aimed at an audience with a sound mathematical

background. It supposes also that basic financial theory, such as valuation

and hedging of derivatives, is already known. However:

- In the first chapter, basic notions and definitions of stochastic processes

are first recalled. Second, an overview of the theory of weak convergence of

semimartingales is provided. In particular, a guideline is given for the weak

convergence of stochastic integrals and contiguity properties.

- Along the second chapter, the standard notions and properties of the financial markets theory are recalled (but not detailed).

- Finally, the emphasis throughout the third chapter is on presenting the basic

discrete models and their continuous time limits. The focus remains on a survey about multinomial approximations and more generally about computing

problems with lattices for different types of options. Other approximations

such as ARCH models ... are also introduced and detailed. Nevertheless, a perfect knowledge of the first two chapters is not fully required.

IX

Book Structure

• The first chapter tries to answer the question :

How to prove that a sequence (Xn)n of stochastic processes

weakly converges to a given stochastic process X ?

This mathematical chapter is only a guide to the reader. While main results are included, the proofs are not provided, as they are already excellent

treatment of this theory readily available :

- The books of Dellacherie and Meyer [108] present the general properties of stochastic processes (volume I) and martingales (volume II), Ethier

and Kurtz [148] deals with Markov processes, Elliott [144], Kopp [250)

and Rogers and Williams [365] introduce the stochastic integration. The

books of McKean [286], Chung and Williams [75] and Karatzas and Shreve

[236] dal with Brownian motion and continuous martingales. The book of

Protter [351] gives a very clear presentation of semimartingales, stochastic

integration and stochastic differential equations.

- Concerning main results of weak convergence of semi martingales, it is

referred to Jacod and Shiryaev's book [214].

- Nevertheless, with respect to this latter book, two parts are added:

1) A special emphasis on the weak convergence of sequences of triangular arrays, which are of particular interest, when dealing with convergence

problems from discrete time to continuous time models.

2) A survey of main results concerning weak convergence of sequences of

stochastic integrals and solutions of stochastic differential equations (see

also the new version of Jacod and Shiryaev's book to appear in 2003) .

• The second chapter deals with the following question:

What are the main problems that we encounter

when examining weak convergence of financial markets ?

Thus this chapter introduced the results about weak convergence of :

- Optimal portfolio policies for utility maximizing investors.

- Option prices, in particular convergence problems of bid-ask spreads.

- Hedging strategies which duplicate options in complete financial markets

or hedging strategies which minimize the locally quadratic risk when facing

incomplete markets.

x

A survey of basic results of financial theory is included but not detailed

since many books are also available:

- For the main notions, among others, Duffie [124][125][126], Bingham and

Kiesel [37], Kwok [261]' Elliott and Kopp [145], Lamberton and Lapeyre

[264], Nielsen [321], Bjork [39J.

- More particularly, Pliska [339J introduces all of the main financial concepts for the discrete time case. Musiela and Rutkowski [312J deal in particular with the theory of bond markets and term structure models. In

Jeanblanc-Picque and Dana [220], the equilibrium approach is detailed.

Shiryaev [381J introduces a large variety of stochastic models.

Hence, in this chapter, we focus on convergence results. Some particular

proofs are fully detailed to show how weak convergence results of the first

chapter can be applied .

• The third chapter reviews a list of results to solve the following problem:

How to construct in practice a sequence (Xn)n of stochastic processes

which weakly converges to a given stochastic process X ?

Although it is not a purely "numerical" chapter, many of standard approximations of basic continuous time processes are recalled :

- Its first part contains some general results about approximations of solutions of stochastic differential equations (standard and backward).

- Its second part is devoted to standard lattice models, when approximating

diffusions. Binomial and trinomial schemes are especially examined when

the continuous time limit process is driven by a Brownian motion. For

most of these models, the discrete time subdivision of the time interval is

deterministic.

- A third part proposes other models for example diffusions with jumps.

This class of processes contains Levy processes and in particular subordinators which are of particular interest when examining dynamics of high

frequency data. Both deterministic and random discretizations are studied.

By considering sequences of random times, the latter ones allow in particular to examine problems of portfolio rebalancing.

- Finally, a list of some standard approximations of interest rate models is

provided: factor models as well as Heath-Jarrow-Morton type models and

Market models are briefly reviewed.

XI

Final Word and Acknowledgments

This book is an attempt to summarize the main convergence results about

financial markets which are known at present. It is focussed on robustness

of financial instruments under convergence of discrete time to continuous

time financial markets. In particular, it indicates option pricing rules that

are stable under convergence of the underlying assets. This feature reduces

the model risk when we must choose between discrete time or continuous

time to describe asset prices dynamics.

While some of the quoted results concern more academics than practitioners, it seems important to underline main features of convergence, first

of all that approximating models do converge. Both "pure mathematical"

speed (in the spirit of the famous Central Limit Theorem) and computational speed must be analyzed. Obviously, all convergence problems are not

yet solved. Further extensions are still in progress both in the mathematical

field (to take more dependency properties into account, to obtain functional

speed of convergence ... ) and also in the financial theory (search of other algorithms to simulate financial variables, study of more general option pricing

or portfolio problems taking account of market imperfections such as trading

strategies which are unavailable in continuous time ... ).

I hope that this book will contribute to stimulate new research on the

sometimes awkward (nevertheless fascinating ?) weak convergence world and

financial theory.

While I cannot thank all the people for supports and useful discussions

since I began to study the financial theory, I want to mention in particular

my colleagues of the research department THEMA, the members of HSBCCCF and in particular Eric Baesen and Jean-Fran<;ois Boulier, also Nicole El

Karoui who encouraged me some years ago to work in financial mathematics,

especially to examine weak convergence problems, Patrick Navatte, Patrice

Poncet and Yves Simon who inclined me to discover and study more applied

features of the financial market theory and Monique Jeanblanc whose legendary kindness is such that I have never hesitated before telling her with

my problems.

I am grateful also for fruitful and enjoyable collaborations to Mohamed

Ahnani, Mondher Bellalah, Philippe Bertrand, Jean-Philippe Lesne, Olivier

Renault, Raphael Sobotka, Christophe Villa and in particular Olivier Scaillet

with whom I have written some of the papers about convergence quoted in

this book.

XII

I express a thought of gratitude for those who were my professors when I

was a young student learning the Probability theory and especially for Jean

Memin who has been my mathematical PhD thesis advisor, some years ago,

and to whom we are all indebted for some of the important results about

weak convergence of stochastic processes quoted in this book.

I would like also to thank the staff of Spinger-Verlag, in particular Martina

Bihn for her wonderful patience since she accepted to be in charge of this

book.

Paris, January 2003

Jean-Luc Prigent

Contents

1.

Weak Convergence of Stochastic Processes ...............

1

1.1 Basic Properties of Stochastic Processes . . . . . . . . . . . . . . . . . . . 2

1.1.1 Stochastic Basis, Filtration, Stopping Times. . . . . . . . . 2

1.1.2 Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Martingales......................................

7

1.1.4 Semimartingales and Stochastic Integrals. . . . . . . . . . .. 14

1.1.5 Markov Processes and Stochastic Differential Equations 40

1.1.6 The Discrete Time Case. . . . . . . . . . . . . . . . . . . . . . . . . .. 58

1.2 Weak Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64

1.2.1 The Skorokhod Topology. . . . . . . . . . . . . . . . . . . . . . . . .. 64

1.2.2 Continuity for the Skorokhod Topology ............. 67

1.2.3 Definition of Weak Convergence. . . . . . . . . . . . . . . . . . .. 69

1.2.4 Criteria for Tightness in ][])k. . . . . . . . . . . . . . . . . . . . . . .. 72

1.2.5 The Meyer-Zheng Topology. . . . . . . . . . . . . . . . . . . . . . .. 74

1.3 Weak Convergence to a Semimartingale ..... . . . . . . . . . . . . .. 75

1.3.1 Functional Convergence and Characteristics ......... 75

1.3.2 Limits of Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87

1.3.3 Limit Theorems for Markov Processes. . . . . . . . . . . . . .. 88

1.3.4 Convergence of Triangular Arrays .................. 92

1.4 Weak Convergence of Stochastic Integrals ................. 100

1.4.1 Introduction ..................................... 100

1.4.2 The Uniform Tightness Condition U.T .............. 101

1.4.3 Functional Limit Theorems for Sequences of Stochastic Integrals and Stochastic Differential Equations. . . . . 104

1.5 Limit Theorems, Density Processes and Contiguity ......... 108

1.5.1 Hellinger Integral and Hellinger Process ............. 108

1.5.2 Contiguity and Entire Separation ................... 115

1.5.3 Convergence of the Density Processes ............... 121

1.5.4 The Statistical Invariance Principle ................. 125

XIV

Contents

2.

Weak Convergence of Financial Markets ..................

2.1 Convergence of Optimal Consumption-Portfolio Strategies ...

2.1.1 Weak Convergence of Controlled Processes ..........

2.1.2 The Martingale Approach .........................

2.2 Convergence of Options Prices ...........................

2.2.1 Problems and Examples ...........................

2.2.2 Contiguity Properties .............................

2.2.3 The Case of Incomplete Markets ...................

2.2.4 Transaction Costs ................................

2.2.5 American Options ................................

2.3 Convergence of Hedging Strategies ........................

2.3.1 Binomial Case and Clark-Hauss man Formula ........

2.3.2 Weak Convergence of Integrands ...................

2.3.3 The Local Risk-Minimizing Strategy ................

129

130

131

161

185

185

194

198

218

230

240

243

251

256

3.

The Basic Models of Approximations ..................... 267

3.1 General Remarks ....................................... 267

3.1.1 Some numerical methods for forward and backward

stochastic differential equations .................... 268

3.1.2 Some numerical methods for computations of Greeks .. 272

3.2 Lattice ................................................ 274

3.2.1 Simple Binomial Processes as Diffusion Approximations274

3.2.2 Correction Terms for Path-Dependent Options ....... 287

3.2.3 Adjustment Prior to Maturity and Smoothing of the

Payoff Functions ........... . . . . . . . . . . . . . . . . . . . . .. 295

3.2.4 Fast Accurate Binomial Pricing .................... 301

3.2.5 Approximating a Diffusion by a Trinomial Tree ...... 305

3.3 Alternative Approximations .............................. 309

3.3.1 ARCH Approximations ........................... 309

3.3.2 Levy Processes ................................... 321

3.3.3 Convergence for Random Time Intervals ............ 345

3.3.4 Deterministic or Random Discretizations of ContinuousTime Processes .................................. 359

3.4 Approximations of Term Structure Models ................. 371

3.4.1 Bonds and Interest Rate Derivatives ................ 371

3.4.2 Basic Interest Models and their Approximations ...... 377

3.4.3 Two-factors Model ............................... 387

3.4.4 Market Models: Discretization of Lognormal Forward

Libor and Swap Rate Models ...................... 388

3.4.5 Discretization of Deflated Bond Prices .............. 388

3.4.6 Pricing Interest Rate or Equity Derivatives and Discretization ....................................... 395

Index ......................................................... 419

1. Weak Convergence of Stochastic Processes

The processes of interest are assumed to take values in the Euclidian space

JRd. More general complete and separable metric spaces can in fact be considered but the applications developed here concern dynamics of processes

taking values in JRd like for example d-dimensional stock prices. Moreover,

paths of these processes are sufficiently regular: the space IID(JRd) of the rightcontinuous functions having left limits (rcll) is in particular important.

"Weak convergence" means convergence in distribution. Results of weak

convergence for sequences of JRd-valued random variables are well-known:

central limit theorem, laws of large numbers (see for example [379]). However,

the study which is developed here is more complicated: it deals with weak

convergence of sequences of entire processes. Fortunately, these ones can be

considered as "simple" random variables no longer with values in JRd but

(very often) in IID(JRd). Thus, similar concepts can be introduced and results

be proved (with more effort ... ). For example, the definition is of the same

kind: a sequence of processes (Xn)n is said to weakly converge to the limit

process X if and only if

lim E[J(Xn)] = E[J(X)] ,

n--+CX)

for each f in the space C(IID(JR d)) which is the space of bounded and continuous mappings from IID(JRd) to JR, where IID(JRd) is equipped with a "good"

topology. Conditions are given under which the above property is satisfied,

especially when both Xn and X are semimartingales.

Since the purpose is more ambitious, technical tools must be used. First,

the general theory of stochastic processes must be introduced. It is briefly

reviewed in this chapter (see for example the books of DeUacherie and Meyer

[108] for a complete exposition of this theory). Secondly, the theory of convergence in distribution of semi martingales must be developed. A survey of

the main results of this theory is presented. A more general and systematic

exposition can be found in particular in Jacod and Shiryaev [214] to which

this chapter refers. Concerning the special case of the weak convergence of

stochastic integrals, the main results are those of Jakubowski, Memin and

Pages [215], Memin and Slominsky [298] and also Kurtz and Protter [256].

J.-L. Prigent, Weak Convergence of Financial Markets

© Springer-Verlag Berlin Heidelberg 2003

2

1. Weak Convergence of Stochastic Processes

1.1 Basic Properties of Stochastic Processes

In this and the succeeding chapters, the time parameter takes values in either

a finite interval [0, T] or [0,(0). Nevertheless, an autonomous treatment for

the discrete time is provided at the end of the first subsection of this chapter.

The set T denotes the time parameter set in each case.

1.1.1 Stochastic Basis, Filtration, Stopping Times

The concept of filtration is used to model the acquisition of information as

time evolves. This a key concept in financial theory: it represents for example the flow of possible observations of prices on a financial market which is

available for traders and portfolio managers.

Here are some standard notations to be used in the whole book.

Suppose (st, F, IP') is a probability space.

Definition 1.1.1. A filtration IF = (Ft, t E T) is an increasing family of

sub-sigma-fields F t C :F. A stochastic basis is a probability space equipped

with a filtration.

Increasing means that if s ::; t, then Fs C Ft. F t is usually interpreted as

the set of events that occur before or at time t. Generally, F t represents the

history of some process observed up to time t but other possible histories are

allowed. It is assumed that IF satisfies the "usual conditions" (see Dellacherie

and Meyer [108]). This means:

i) Complete: every IP'-null set in F belongs to Fo and so to all Ft.

ii) Right continuous: F t

= ns>tFs.

Definition 1.1.2. A stochastic basis B = (st, F, IF, IP') is a probability space

equipped with a filtration and is also called a filtered probability space.

To deal with problems like for example "what is the optimal time to invest

or to exercise an American option?" when taking available information into

account, it is natural to introduce the notion of stopping time.

Definition 1.1.3. (Stopping time).

a) A stopping time is a mapping T : st --+ T such that {T ::; t} E F t for all t.

b) If T is a stopping time, FT denotes the collection of all sets A E F such

that An {T ::; t} E F t for all t E T.

c) 1fT is a stopping time, F T - denotes the (J-field generated by Fo and all

the sets of the form An {t < T} EFt where t E T and A EFt.

1.1 Basic Properties of Stochastic Processes

3

Remark 1.1.1. The event {T ::; t} depends only on the history up to time

t. A stopping time is a random time T such that at each time t, one may

decide whether T ::; t or T > t from what one knows up to time t. FT is

interpreted as the set of events that occur before or at time T. For example,

the first time an asset price reaches a given level is a stopping time with

respect to a filtration which takes account of the information delivered by

the asset prices. But the last time it reaches a given level is not a stopping

time since information about the future is necessary to define this last time.

Let us now examine some basic and very useful features of stopping times.

Proposition 1.1.1. (Stopping times properties).

i) If T is a stopping time and t E T then T + t is a stopping time.

ii) 1fT is a stopping time and A E FT then TA defined by

TA(W)

= {T(W)

ifw E A,

+00 zfw tJ. A,

is also a stopping time.

iii) Assume Sand T are stopping times.

1) Their minimum S 1\ T and their maximum S V T are also stopping times.

2) If S::; T then Fs eFT'

3) If A E F s! then :

An {S ::; T} EFT, An {S

= T} EFT and A n {S < T}

E

iv) If (Tn)nEN is a sequence of stopping times then I\nENTn and

two stopping times.

FT-.

VnENTn

are

1.1.2 Stochastic Processes

Basic notions. Let us fix some terminology :

Definition 1.1.4. A continuous time stochastic process X taking values in

a measurable space (E, £) is a family of random variables (Xt)t defined on

(Sl,IF,JP'), indexed by t, which take values in (E,£).

Hence, for all t, X t is a random variable with values in E. A process can

also be considered as a mapping from Sl x T into E. Moreover, for each fixed

w (which represents a "state of the world"), t ---. Xt(w) is a function defined

on T, called a path or a trajectory of the process X.

Definition 1.1.5. A process X is called rell ("cadlag" ln french) if all its

paths are right-continuous and admit left limits. When X is rell, two other

processes can be defined:

X_ = (Xt-)tET with X t- = lims

4

1. Weak Convergence of Stochastic Processes

Definition 1.1.6. When X is a process and T is a mapping

process stopped at time T, denoted by X T , is defined by

n

X[ = XTI\t.

-+

T, the

(1.1 )

A standard question is : when is it possible to say that two stochastic

processes model the same random phenomenon ? For this, three notions can

be introduced:

- The weakest is based on the fact that, in practice, a stochastic process is only

observed at a finite number of instants (discrete time observations). Therefore

it leads to the definition of the finite-dimensional distributions of a process X.

Consider for example the Euclidian case: E

H.

= JRd

with its Borelian O'-field

Definition 1.1.7. Consider for n E N, t1, ... , in E T and A E H, the measure

defined on ®nJR d by

JlDXt" ... ,Xt n

(A) = JID[{w En: (Xtl (w), ... , Xtn (W)) E A}].

Two processes X and Yare said equivalent (or have the same law), denoted

by X '" Y, if their families of finite-dimensional distributions coincide.

As it can be noted, this does not require that the two processes are defined on the same probability space (n, F, JID) and avoids the choice of a proper

probability space to describe a random phenomenon. Thus, if X is considered as a map X : n -+ (JRd) T with only Borelian subsets to be considered,

then a canonical version of X can be identified up to equivalence (this is the

well-known Kolmogorov Extension Theorem).

- However, if T is uncountable, stronger notions of "equivalence" must be

defined since trajectories have often values in none Borelian set of (JRd) T.

Two concepts can be introduced to make precise paths of a process. Note

that, in what follows, the two processes X and Y must be defined on the

same probability space (n, F, JID).

Definition 1.1.8. The process Y is said to be a modification of X if for

every t,

X t = yt a.s ..

Definition 1.1.9. A random set A is called evanescent if the set

{w : :3 t E Twith (w, t) E A} is JID-null; two processes X and Yare said

indistinguishable ~f the random set {X =1= Y} = {(w, t) : Xt(w) =1= yt(w)} is

evanescent, i. e. if almost all paths of X and Yare the same.

1.1 Basic Properties of Stochastic Processes

5

As it can be seen, the difference between the two definitions is that, if

Y is a modification of X, the set of zero measure on which X t and yt may

differ, may depend on the time t, which is not the case when they are indistinguishable since one has X t = yt a.s. for all t E T. Thus, if X and Y

are indistinguishable, they are modifications of each other but the converse

is not true, unless, for example, both X and Yare rc (right-continuous) or

lc (left-continuous). Note also that when T is countable, these two notions

coincide. As for random variables, in most cases, the notations X = Y (or

X :

The Optional u-Field. Consider the stochastic basis B

= (n, F, IF, JP').

Definition 1.1.10. a) A process X is adapted to the .filtration F if X t is

Ft-measurable for every t E T.

b) The optional IT-field is the IT-.field 0 on n x T generated by all rcll

adapted processes.

c) A process that is O-measurable is called optional.

d) A process X is F-progressive if for each t, the restriction of X to [0, tj x n

is B([O, t]) x Ft-measurable.

Note that, if X is F-progressive then X is F-adapted. The converse is

false but every right continuous process X which is F-adapted is also Fprogressive.

Notation: F X is the filtration generated by the process X itself

F{ = IT(X.,

S :

t).

Proposition 1.1.2. Let X an optional process.

1) Considered as a mapping on n x T, it is F ® T -measurable.

2) If T is a stopping time, the stopped process XT is also optional.

Another characterization of the optional IT-field is based on stochastic

intervals : if 8 and T are two stopping times, one may define four kinds of

stochastic intervals which are the following random sets:

[[8, T]]

{ [[8, T[[

]]8, T]]

]]8, T[[

=

=

=

=

{(w, t)

{(w, t)

{(w, t)

{(w, t)

:t

:t

:t

:t

E

E

E

E

T,8(w) :

Denote by 11A the indicator of the set A such that

11 ( )

A W

= { 1 if w

°ifw ~ A,A.

E

6

1. Weak Convergence of Stochastic Processes

Proposition 1.1.3. (Optional properties).

i) If Sand T are two stopping times and if Y is an F s-measurable random

variable, the four following processes

Yll[[S,Tll, Yll[[s,T[[, Ylllls,Tll andYlllls,T[[

are optional.

ii) If X is rell and adapted, the two processes X_ and L1X are optional.

iii) The cy-field 0 is generated by the stochastic intervals [[0, TJl, where T is

any stopping time.

Next, hitting times can be examined and in particular the following result

of Hunt (see for example Dellacherie and Meyer [108]).

Theorem 1.1.1. ff A is an optional random set, its first entry time TA

defined by

TA(W) = inf{t: (w, t) E A}

is a stopping time.

This result can be applied to the useful particular result :

Proposition 1.1.4. 1) If X is an JRd-valued process and if B is a Borelian

subset of IR d , then T defined by T(w) = inf{t : X(w, t) E B} is a stopping

time.

2) If X is an IR-valued adapted rc process with nondecreasing paths, then,

for every a E JR, T defined by T(w) = inf{t : X(w, t) ~ a} is a stopping time.

The Predictable u-Field. The notion of "predictable (J-field" is linked

to the idea that the value of a process at time t can be "obtained" from

observations just before time t (see the discrete case for a good intuition).

More precisely :

Definition 1.1.11. The predictable cy-field is the cy-field P on [2 x T that

is generated by all lc adapted processes. A process or a random set that is

P-measurable is called predictable.

It is obvious that the predictable cy-field P is included in the optional

cy-field O. It is also possible to get another characterization of P.

Theorem 1.1.2. 1) The predictable (J- field is also generated by anyone of

the following collections of random sets :

i) A x {O} where A E Fo and [[0, T]] where T is any stopping time.

ii) A x {O} where A E Fo and A x (s, t] where s < t, A E Fs.

2) The predictable (J-field is also generated by all adapted processes that have

continuous paths.

1.1 Basic Properties of Stochastic Processes

7

Proposition 1.1.5. (Predictable properties).

1) If X is a predictable process and if T is a stopping time then the stopped

process XT is also predictable.

2) If Sand T are two stopping times and if Y is an F s-measurable random

variable, the process Ylllls,Tll is predictable.

3) If X is a rcll adapted process then X_ is a predictable process. Consequently, if X is predictable itse~f then LlX is also predictable.

Definition 1.1.12. A predictable time is a mapping T :

the stochastic interval [[0, T[[ is predictable.

st

--+

T such that

Recall some standard properties of predictable times (see Dellacherie and

Meyer [108] for more details).

Proposition 1.1.6. (Predictable times).

1) Every predictable time is also a stopping time.

2) If (Tn)n is a sequence of predictable times then:

i) VnENTn is a predictable time.

ii) If S = AnENTn and UnEN{S = Tn} = st then S is a predictable time.

3) If T is a predictable time and A EFT~, the time TA (see Theorem 1.1.1)

is predictable.

Definition 1.1.13. A rcll process X is called quasi-left continuous if

LlXT = 0 a.s. on the set {T < oo} for every predictable t1me T.

Proposition 1.1. 7. Let X be a rcll adapted process. Then X is quasi-left

continuous if and only if for any increasing sequence (Tn)" of stopping times

with limit T, lim XTn = X T a.s. on the set {T < oo}.

Definition 1.1.14. A generalized stochastic process A is a process such

that:

(i) A is lR+ -valued, Ao = 0 and its paths are non-decreasing.

(ii) 1fT = inf{t: AT = oo} then A is right-continuous on [[O,T[[.

A generalized stochastic process A does not jump to infinity if AT- = +00

on T < 00, in which case its paths are everywhere right-continuous.

1.1.3 Martingales

As it is well-known, the concept of Martingale is crucial in the modern theory

of finance. If the price process M of a stock is a martingale, the conditional

expectation at time s of the future value M t of the stock at time t is given

by its current value Ms.

This subsection is a review of properties of martingales, submartingales

and supermartingales that are essentially due to Doob. The proofs may be

found in most standard books (see for example Doob [115], Dellacherie and

Meyer [108], Elliott [144], Williams [404] ... ).

8

1. Weak Convergence of Stochastic Processes

Notation: consider a random variable X, integrable under the probability

lP'. Denote lEIP[X] the expectation of X and if Q is a sub-field of F denote

lEIP[ X IQ] the conditional expectation of X under Q. For p > 1, let JLP be the

space of random variables such that lEll'[IXIP] is finite.

Definition 1.1.15. A martingale (resp. sub martingale, resp. super martingale) is an adapted process X on the basis (ft, F, IF, lP') whose lP'- almost all

paths are rcll, such that every X t is integrable and that for s ::::; t :

Note that, if in the above definition, the paths of a martingale X are

not assumed to be rcll, it can be proved that there exists a unique (up to

indistinguishability) martingale Y which is rcll and indistinguishable from

X.

Proposition 1.1.8. (Jensen's Inequality).

Let ¢ : IR -+ IR be convex.

1) Let Y and ¢(Y) be integrable random variables. Then, for any a-field Q,

¢ (lE[YIQ]) ::::; lE[¢(Y)IQ]·

(1.2)

2) Consequently, if M is a martingale such that ¢(Md is integrable for any

t, then ¢(M) is a sub martingale (in particular IMI and M2).

3) Assume that X is a submartingale such that ¢(Xt ) is integrable for any t.

If moreover ¢ is nondecreasing then ¢(X) is also a submartingale.

A process is said to admit a terminal variable Xoo if X t converges a.s. to

a limit Xoo as t T00; in such a case, the variable XT is (a.s.) well defined for

any stopping time with X T = Xoo on {T = oo}.

Theorem 1.1.3. Let X be a supermartingale such that there exists an integrable random variable Y with X t ::::: lE[YIFt ] for all t E T. Then:

i) (Doob's limit Theorem) X t converges a.s. to a finite limit Xoo.

ii) (Doob's Stopping Theorem):

a) If Sand T are two stopping times, the random variables Xs and X T are

integrable, and Xs ::::: lE[XTIFs] on the set {S::::; T}. In particular, X T is a

supermaTtingale.

b) If X is a martingale such that there exists an integrable random variable

= lE[YIFt ] for all t E T, then Xs ::::: lE[XTIFs] a.s. (X is said also

"closed" by Y).

Y with X t

1.1 Basic Properties of Stochastic Processes

9

Locally Square-Integrable Martingales. Two classes of martingales can

be examined. The first concerns the property of uniform integrability, the

second is about the notion of local martingale.

Definition 1.1.16. An uniformly integrable martingale M is such that the

family of random variables (MdtET is uniformly integrable. It means:

SUPtET

1

jMt!>c

IMtldlP'

-+

0 whenc

-+ 00.

M denotes the class of all uniformly integrable martingales.

From Theorem 1.1.3, it can be deduced:

Theorem 1.1.4. Let M be an uniformly integrable martingale and let T a

stopping time. Then the stopped process MT is also an uniformly integrable

martingale.

A particular case of such martingale is the following :

Definition 1.1.17. A square-integrable martingale M is such that

1e denotes the class of all square-integrable martingales.

Obviously, 1{2 C M. Moreover, each of these two sets are stable under stopping: that is for any stopping time T and for any process X in one of these

sets, the stopped process XT is also in this set.

Theorem 1.1.5. 1) If M is an uniformly integrable martingale, then M t

converges a.s. and in ll) to a terminal variable Moo and fo'r all stopping times

T, MT = lE[MooIFTl.

2) Moreover, M is a square-integrable martingale if and only if Moo is squareintegrable, in which case MT -+ Moo takes place in ll}.

3) If Y is an integrable random variable, there exists an uniformly integrable

martingale M, and only one up to an evanescent set, such that M t = lE[YIFtl

for all t E T and Moo = lE[YIFoo-l.

Theorem 1.1.6. (Doob's Inequality)

If M is a square-integrable martingale,

Another characterization of the elements of M is of particular interest :

Proposition 1.1.9. If M is an adapted rcll process with a terminal variable

Moo, then M is an uniformly integrable martingale if and only if for each

stopping time T, the variable MT is integrable and satisfies lE[MTl = lE[Mol,

10

1. Weak Convergence of Stochastic Processes

Now, examine the notion of localization of martingales.

Definition 1.1.18. A local martingale (resp. a locally square-integrable

martingale) is a process such that there exists an increasing sequence (Tn)n

of stopping times satisfying: limn Tn = 00 a.s. and every stopped process

XTn is an uniformly integrable martingale (resp. a locally square-integrable

martingale) .

Denote 1{~oc the space of all locally square-integrable martingales.

Definition 1.1.19. A process X is of class (D) if the set of random variables

{XT : T any finite-value stopping time} is uniformly integrable.

Proposition 1.1.10. (Class (D }).

1} Each martingale is a local martingale .

2} Each uniformly integrable martingale is a process of class (D).

3} A local martingale is an uniformly integrable martingale if and only if it

is a process of class (D).

To see the differences between these notions of martingales, examine the

next examples (see Jacod and Shiryaev [214]) :

Example 1.1.1. Let (Zn)n be a sequence of i.i.d. (independent and identically

distributed) random variables such that IP'[Zn = 1] = IP'[Zn = -1] = 1/2.

Consider the filtration F t = a(Zp : p E N*, p:::; t) and let M t = LI:S;P:S;[tJ Zp

where ttl denotes the integer part of t. Then, M is obviously a martingale

but, by the central limit theorem, M t does not converge a.s. as t T 00 and

thus M is not uniformly integrable.

Example 1.1.2. Let (An)nEN* be a measurable partition with IP'[An] = 2- n .

Let (Zn)n be a sequence of random variables that are independent of the An

with IP'[Zn = 2n] = IP'[Zn = _2n] = 1/2. Consider the filtration

Ft

= a(An : n E N*) ift E [0,1) and F t = a(An, Zn : pEW) ift E [1, (0).

Consider the following random variables :

y,

n

=

"Z

~

l:S;p:S;n

pllAp' X t =

°

{oYo",

ift E [0,1)

ift E [1, (0)

T

,n

=

°

{+oo on the set UI:S;p:S;n Ap

elsewhere

(Tn)n is a sequence of stopping times that increases to 00. The process XTn

is equal to

(resp. Y n ) on [[0,1[[ (resp. [[1,00[[) and Y n is bounded and

independent from F I - which implies that X T " is an uniformly integrable

martingale and X is a local martingale. But X is not a martingale since

Xl = Y00 is not integrable.

1.1 Basic Properties of Stochastic Processes

11

Doob-Meyer decomposition. Compensators. The stochastic basis

B = (st, F, IF, JP') is fixed throughout. Recall some basic properties about

processes with finite variation.

Definition 1.1.20. (Finite variation).

1) Denote V+ the set of all real-valued processes A that are rcll, adapted with

Ao = 0 and whose each path is non-decreasing.

A real-valued process A is said having a finite variation if for each .finite interval [0, t], there exists a constant C such that for all .finite partitions (ti)i

of [0, t], L:i IA ti + 1 - Ati I is dominated by C.

2) Denote V the set of all real-valued processes A that are rcll, adapted with

Ao = 0 and whose each path has .finite variation. Denote Var( A) the variation

process of A that is the process such that Var(AMw) is the total variation of

the function s ---> As (w) on the interval (0, tj de.fined by:

Var(A)t(w)

=

li~

L

IAtk/n(W) - At(k-l)/n(w)l·

l~k~n

Let A E V. For each wEst, the path t ---> At(w) is the distribution

function of a signed measure (positive if A is increasing) that is finite on each

interval [0, t] and that is finite on lR+ if and only if Var(A)oo(w) < 00. This

measure is denoted by dA t (w). If A E V and B E V and if the measure dA t (w)

is absolutely continuous to the measure dBt(w), this relation is denoted by

dA~dB.

It is immediately possible to define a Riemann-Stieljes integral with respect to such measure dA t (w): for each optional process H, define the integral

process denoted by H.A or by J~ HsdAs as follows on T:

H.At(w)

= {J~ Hs(w)dAs(w)

+00

if J~ l~s(w)ld[Var(A)]s(w) <

otherwise.

00,

Proposition 1.1.11. 1) Let B E V and let H be an optional process such

that B.A is finite-valued. Then B.A is also in V.

moreover, Band Hare

predictable then B.A is also predictable.

2) Let A E V and BE V be such that dB ~ dA, then there exists an optional

process H such that A = H.B up to an evanescent set. Aforeover, if A and

B are predictable then H may be chosen predictable.

rr

Recall now a very powerful result due to Meyer (see Dellacherie and Meyer

[108]), known as "the Doob-Meyer decomposition of submartingales". This

property allows to introduce martingales in many problems and underlies the

stochastic calculus of semimartingales.

12

1. Weak Convergence of Stochastic Processes

Theorem 1.1.7. If X is a sub martingale of class (D) (see Definition 1.1.19),

there exists a unique (up to indistinguishability) increasing integrable predictable process A with Ao = 0 such that X - A is an uniformly integrable

martingale.

Corollary 1.1.1. If X is a predictable local martingale whose paths have

finite variations then X = 0 (up to an evanescent set).

Another important result can be deduced:

Denote Aloe (resp. AtJ the space of all adapted processes A such that there

exists an increasing sequence (Tn)n of stopping times satisfying: limn Tn = 00

a.s. and every stopped process ATn has integrable variation: IEWar(A)ool <

00 (resp. is a locally integrable adapted increasing process).

Corollary 1.1.2. Let A be in Aloe (resp. At). Then, there exists a unique

(up to an evanescent set) predictable process in Aloe (resp.At), called the

compensator of A and denoted by AP, such that A - AP is a local martingale.

Example 1.1.3. Consider the Extended Poisson Process N on (Q,F,JP'). It is

an adapted simple point process (N is increasing and its jumps are equal to

1) such that :

i) IE[Ntl < 00 for each t E IR+.

ii) Nt - Ns is independent of the a-field Fs for all 0 ::; s ::; t.

iii) The function at = IE[Ntl is continuous (called the intensity of N).

In fact, the distribution of the variable Nt - Ns is a Poisson distribution with

mean at - as. Moreover, the compensator of N is its intensity: (Nt - at)t is

a martingale.

Let us study more explicitly the structure of 1-{2 and 1-{~oe'

Theorem 1.1.8. (Predictable Quadratic Covariation).

1) Let (M, N) any pair of locally square-integrable martingales. Then, there

exists a predictable process (M, N), unique up to an evanescent set, such that

M N - (M, N) is a local martingale. Moreover:

(M,N) = 1/4«(M +N,M

+ N)

- (M - N,M - N)).

2) If (M, N) is a pair of square-integrable martingales then (M, N) has integrable variation and M N - (M, N) is an uniformly integrable martingale.

3) (M, M) is non-decreasing and admits a continuous version if and only if

M is quasi-left continuous (see Definition 1.1.13).

Definition 1.1.21. The process (M, N) is called the predictable quadratic

covariation or also the angle bracket of the pair (M, N).

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