enlargement of filtration and insider trading · 2020-06-02 · enlargement of filtration &...

Enlargement of Filtration and Insider Trading

A. Es-SaghouaniUnder supervision of

Dr. P.J.C. SpreijFaculteit der Natuurwetenschappen, Wiskunde en Informatica ,

Korteweg-de Vries Institute for Mathematics,Plantage Muidergrracht 24, 1018 TV Amsterdam, The Netherlands

A thesis submitted for candidacy for the degreeof Master of Mathematics

Amsterdam, 6 January 2006

Nederlandse Samenvatting

Het onderwerp van deze scriptie is Vergroting van Filtraties en Handelen met Voorkennis. Demeeste financiele verschijnselen worden bestudeerd met behulp van de theorie van martingalen.Een belangrijk onderdeel van deze theorie is de filtratie, dat is een stijgende rij van σ-algebra’s.In een financiele markt correspondeert een σ-algebra met alle publieke informatie die beschikbaaris voor alle handelaren tot en met een tijdstip t. De bedoeling van deze scriptie is een korteinleiding te geven in het vergroten van een filtratie en hoe dit gerelateerd is aan het handelenmet voorkennis. Wij doen dit onder de aanname dat de handelaar met voorkennis (iets) meerinformatie tot zijn beschikking heeft dan de gewone handelaar, dat kan bijvoorbeeld zijn deprijs van een aandeel in de toekomst. In wiskundige termen is die extra informatie gegevenals een stochastische variabele. Wij kijken naar de filtratie gegenereerd door de oorspronkelijkefiltratie en de σ-algebra gegenereerd door de stochast. We bestuderen hoe de objecten vande oorspronkelijke filtratie eruit zien in de grote filtratie. Onder andere bestuderen we hoe hetprobleem van het maximaliseren van de nutsfunctie van de twee handelaren kan worden opgelost,en we kijken ook naar het verschil tussen de twee.

i

Abstract

We consider a probability space (Ω,F ,P) equipped with two filtrations F = (Ft)t≥0 and G =(Gt = Ft ∨ σ(G))t≥0, where G is a random variable taking values in a Polish space. We givea condition on G such that every F-semimartingale remains a G-semimartingale. Then wetransfer martingale representation theorems from F to G. We then use these theorems to solvethe problem of maximizing the expected logarithmic utility for an investor having the filtrationG at his disposal, and rewrite his additional expected logarithmic utility, with respect to aninvestor hoe has only the filtration F at his disposal, in terms of relative entropy. At last wegive another approach for the problem of enlargement of filtrations.

ii

Contents

Nederlandse Samenvatting i

Abstract ii

Introduction iv

1 Enlargement of filtrations 11.1 Initial enlargement of filtrations (I.E.F) . . . . . . . . . . . . . . . . . . . . . . . 11.2 Stochastic exponential representation of the process 1/pG . . . . . . . . . . . . . 61.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Martingale representation theorems for I.E.F 102.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 The martingale preserving probability measure . . . . . . . . . . . . . . . . . . . 112.3 Martingale representation theorems for I.E.F . . . . . . . . . . . . . . . . . . . . 14

3 Insider trading and utility maximization 213.1 The ordinary investor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Solution of the logarithmic utility maximization problem . . . . . . . . . 263.2.3 Insider’s additional expected logarithmic utility . . . . . . . . . . . . . . . 293.2.4 Explicit calculations of the insider’s expected logarithmic utility . . . . . 30

4 Enlargement of filtrations and Girsanov’s theorem 334.1 Preliminaries and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Girsanov-type for change of filtrations . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A Some Important Theorems and Lemma’s 38A.1 Appendix Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.2 Appendix Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3 Appendix Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iii

Introduction

In the past decades an extensive mathematical theory using martingale techniques has beendeveloped for the problems of characterization of no arbitrage, hedging and pricing of financialderivatives and utility maximization of investors in financial markets. One of the importantfeatures of this theory is the assumption of one common information flow (filtration) on whichthe portfolio decisions of all economic agents are based. In this thesis we give a short introductionto enlargement of filtration and insider trading. We do this by considering a financial market thatis a probability space (Ω,F ,P) equipped with a filtration F = (Ft)t∈[0,T ] (public information).While the ordinary trader makes its decisions based on the information flow (Ft)t∈[0,T ], theinsider possesses from the beginning extra information about the outcome of some randomvariable G taking values in a Polish space (U,U), foe example, the future price of a stock. Theinsider’s information flow is therefore described by the enlarged filtration G = (Gt)t∈[0,T ] =(Ft∨σ(G))t∈[0,T ]. This thesis is based on articles of Amendinger [1, 2, 3], Jacod [7, 9], Pikovskyand Karatzas [13] and Ankirchner [4].

The problem of the enlargement of filtrations consists of the following three important issues:

1. Give conditions on the random variable G such that every F-semimartingale become aG-semimartingale.

2. If part 1 is satisfied, give the decomposition of G-semimartingales.

3. If a version of the martingale representation holds under the filtration F, find a version ofthe martingale representation theorem with respect to the enlarged filtration G.

For the insider trading, we will be interested in its expected logarithmic utility maximizationand give some example where we obtain explicit formulae for the utility gain.

The outline of this thesis is as follows. In Chapter 1 we give most of the results of Jacod [9]and Amendinger [3]. For this we fix a time horizon T . If the regular conditional distributionsof the random variable G given Ft, t ∈ [0, T ] are absolutely continuous with respect to the lawof G, Jacod [9] proved that every (P,F)-semimartingale remains a (P,G)-semimartingale onthe interval [0, T ], and gave the canonical decomposition of (P,F)-semimartingales in G whichinvolves the conditional density process ql, l ∈ U . For most of other results we will assumethe equivalence between the regular conditional probabilities of G given Ft, t ∈ [0, T ) and theunconditional law of G. Based on this assumption we give also an exponential representation oftheir conditional density process pG.

In Chapter 2, we give sufficient conditions such that the existence of martingale measures,under which the stock price process S is a martingale, in the filtration F implies their existence

iv

Enlargement of Filtration & Insider Trading v

in the enlarged filtration G. Moreover, we show that the density process of an equivalent G-martingale measure that decouples the σ-algebras F and σ(G) is the product of the densityprocess of an equivalent G-martingale measure and the process 1/pG. And we use these tworesults to show the inheritance of the martingale representation theorem in the enlarged filtrationG.

In Chapter 3, we treat the insider’s problem of maximizing his expected logarithmic utility. Inthis case we give the optimal strategy and also the expression of the utility gain. We establishalso a relationship between the additional expected logarithmic utility of the insider and therelative entropy of the probability measure P with respect to the probability measure P definedon (Ω,GT ) by the process 1/pG.

In Chapter 4 we treat another aspect of the problem of enlargement of filtrations, studiedby Ankirchner [4]. He considered enlargement of the filtration F by an other filtration K, this isdone by studying the filtration

G = Gt := ∩s>t(Fs ∨ Ks), t ≥ 0,

and he replaces Jacod’s condition by the a condition inspired from the notion of the decouplingmeasure. The idea is that the enlargement of the filtration can be interpreted as a change fromthe decoupling measure to the original measure. Then Girsanov’s theorem is used to obtain thesemimartingale decomposition relative to the enlarged filtration G.The notations in this thesis will be the same for all the chapters.

Chapter 1

Enlargement of filtrations

1.1 Initial enlargement of filtrations (I.E.F)

Let (Ω,F ,P) be a probability space with a filtration F = (Ft)t∈[0,T ] satisfying the usual con-ditions, i.e., the filtration F is right-continuous (Ft =

⋂ε>0Ft+ε) and each Ft contains all

(F ,P)-null sets). T ∈ (0,∞] is a fixed time horizon, and we assume that the σ-algebra F0 istrivial. Let G be an F-measurable random variable with values in a Polish space (U,U).

Definition 1.1.1. A continuous adapted stochastic process X is called a semimartingale if ithas a representation of the form X = X0 +M + A. Where M is a continuous local martingaleand A is a continuous adapted process of locally bounded variation and M0 = A0 = 0.

Definition 1.1.2. Given the notations above, we call the filtration G defined by :

Gt := Ft ∨ σ(G), t ∈ [0, T ]

the initially enlarged filtration of F.

In this chapter we will assume that the enlarged filtration G satisfies the usual conditions.Therefore we redefine G as follows : for every t,

Gt = ∩ε>0(Ft+ε ∨ σ(G)).

The following theorem is due to J. Jacod. Before we give the content of the theorem we in-troduce the following hypothesis called ”l’hypothese (H’)”: every F-semimartingale is a G-semimartingale.

Theorem 1.1.3. L’hypothese (H’) is satisfied under the following condition:(A) For every t there exists a positive measure σ-finite ηt on (U,U) such that P[G ∈ ·|Ft](ω) ηt(·) almost surely in ω, where P[G ∈ ·|Ft](ω) stands for a regular version of the conditional lawof G with respect to Ft.

Proof. For a detailed proof of this result we refer the reader to the proof of Theorem 1.1,Jacod [9].

Remark 1.1.4. For the existence of the regular conditional probabilities of G with respect toeach Ft see Theorem A.1.1 and Corollary A.1.2 in the Appendix, or Shiryaev [15].

1

Enlargement of Filtration & Insider Trading 2

Proposition 1.1.5. The condition (A) is equivalent to the condition (A’):There exists a positive measure σ-finite η on (U,U) such that P[G ∈ ·|Ft](ω) η(·) for allt > 0, ω ∈ Ω.In this case we can take for η the law of the variable G.

Proof. It is clear that we only need to prove that (A) implies (A’), with η the law of G. Fixt > 0 and suppose that (A) is satisfied. Then by Doob’s Theorem, see Theorem A.1.3 inthe Appendix, there exists a Ft ⊗ U-measurable positive function: (ω, x) 7→ qxt (ω) such thatP[G ∈ dx|Ft](ω) = ηt(dx)qxt (ω). Let bt(x) = E

[qxt

]and

qxt (ω) =

qxt (ω)bt(x)

if bt(x) > 0,0 otherwise

Because qxt = 0 a.s. if bt(x) = 0, we have qxt = qxt bt(x) a.s., and the measure ηt(dx)bt(x)qxt (ω) isstill a version of P[G ∈ dx|Ft](ω). Hence for every Ft-measurable positive function g we have∫

g(x)η(dx) = E[g(G)

]= E

[ ∫g(x)P[G ∈ dx|Ft](dω)

]=

∫g(x)E

[qxt

]ηt(dx) =

∫g(x)bt(x)ηt(dx),

whence ηt(dx)bt(x) = η(dx), thus P[G ∈ dx|Ft](ω) = η(dx)qxt (ω).

N.B. Doob’s derivation theorem gives joint measurability of (ω, x) 7→ q(ω, x), whereas theRadon-Nikodym Theorem only gives x 7→ q(ω, x) is measurable for all ω.Now for the rest we will assume the following:

Assumption 1.1.6. The condition (A’) is satisfied: there exists a σ-finite positive measure ηon (U,U) such that for all t ∈ [0, T ), the regular condition distribution of G given Ft is absolutelycontinuous with respect to η for P-almost all ω ∈ Ω, i.e,

P[G ∈ ·|Ft](ω) η(·) for P-a.a. ω ∈ Ω. (1.1)

Remark 1.1.7. The measure η is not necessary the law of G, because sometimes another simplemeasure could be taken, like the Lebesgue measure on U = Rd.

We introduce the following notations : let H := (Ht)t∈[0,T ], where H ∈ F,G, be a genericfiltration, H0 := (Ht)t∈[0,T ), Ω := Ω × U , Ht :=

⋂ε>0(Ht+ε ⊗ U), H := (Ht)t∈[0,T ] and H0 :=

(Ht)t∈[0,T ). The fact that the time horizon T is included or excluded is of importance, as weshall see in the section of examples that are given later on.Let K = (Kt)t∈[0,T ] = (K1

t , . . . ,Kdt )>t∈[0,T ]

1 be a d-dimensional continuous local F-martingalewith quadratic variation 〈K〉 =

(〈Ki,Kj〉

)i,j=1,...,d

taken with respect to F.

Definition 1.1.8. We call the optional σ-algebra on Ω× [0, T ), the σ-algebra generated by thecadlag H0-adapted stochastic processes, and we denote it by O(H0).

1where by > we mean the transpose.


Definition 1.1.9. We call the predictable σ-algebra on Ω× [0, T ), the σ-algebra generated bythe left-continuous H0-adapted stochastic processes, and we denote it by P(H0).

The following lemma provides a ‘nice’ version of the conditional density process ql resultingfrom the absolute continuity in Equation (1.1).

Lemma 1.1.10 (Lemme 1.8 and corollaire 1.11 of Jacod [9]). Suppose Assumption 1.1.6is satisfied. Then:

1. There exists a non-negative O(F0)-measurable function (ω, l, t) 7→ qlt(ω) which is right-continuous with left limits in t and such that:

a. for all l ∈ U , ql is an F0-martingale, the processes ql, ql− are strictly positive on[0, T l), and ql = 0 on [T l, T ), where

T l := inft ≥ 0|qlt− = 0 ∧ T ; (1.2)

b. for all t ∈ [0, T ), the measure qlt(.)η(dl) on (U,U) is a version of the conditionaldistribution P[G ∈ dl|Ft].

2. TG = T P-a.s., where TG(ω) = TG(ω)(ω) = T l(ω) on G = l.

Remark 1.1.11. The conditional density process ql is the key to the study of continuouslocal F-martingales in the enlarged filtration G0. The following theorem shows that underAssumption 1.1.6, every continuous local F-martingale is a G0-semimartingale, and explicitlygives its canonical decomposition.

Theorem 1.1.12 (Theoreme 2.1 of Jacod [9]). Suppose Assumption 1.1.6 is satisfied. Fori = 1, . . . , d, there exists a P(F0)-measurable function (ω, l, t) 7→ (klt(ω))i such that

〈ql,Ki〉 =∫

(kl)iql−d〈Ki〉. (1.3)

For every such a function ki, we have

1.∫ t0 |(k

Gs )i|d〈Ki〉s <∞ P-a.s. for all t ∈ [0, T ), where kG = kl on G = l, and

2. Ki is a G0-semimartingale, and the continuous local G0-martingale in its canonical de-composition is given by:

Kit := Ki

t −∫ t

0(kGs )id〈Ki〉s, t ∈ [0, T ). (1.4)

Remark 1.1.13. If the absolute continuity in Assumption 1.1.6 holds for all t ∈ [0, T ], then Kis even a local G-martingale.

Let now take a look at the conditional density process ql. Since F0 is trivial, we have∫A

P[G ∈ dl] = P[G ∈ A] = P[G ∈ A|F0] =∫Aql0η(dl),


for all A ∈ U . By choosing U smaller if necessary, we can therefore assume that ql0 > 0 for alll ∈ U , so we obtain for P-a.a. ω and all t ∈ [0, T )

P[G ∈ A|Ft](ω) =∫Aqlt(ω)η(dl) =

∫ApltP[G ∈ dl],

where

plt(ω) :=qlt(ω)ql0

. (1.5)

From this, we observe that we can take p as the process q appearing in Lemma 1.1.10 and inTheorem 1.1.12 by choosing for η the law of G. In the following we will write just pG, but wemean by this that ”pG = pl on G = l”. By part 2 of Lemma 1.1.10, the first time pG hits 0is P-a.s. equal to T so that we can consider the process 1

pG on [0, T ). If the regular conditionaldistributions of G given Ft are equivalent to the law of G, then the process 1

pG turns out to be a

positive G0-martingale starting from 1 and thus defines a probability measure Pt on (Ω,Gt) forall t ∈ [0, T ). Pt coincides with P on Ft, and the σ-algebras Ft and σ(G) become independentunder Pt. These properties are shown in the following proposition due to Amendinger [3], p.267.

Proposition 1.1.14. Suppose that the regular conditional distributions of G given Ft are equiv-alent to the law of G for all t ∈ [0, T ), i.e., for all l ∈ U , the process (plt)t∈[0,T ) is strictly positiveP-a.s. then:

1. For t ∈ [0, T ), the σ-algebras Ft and σ(G) are independent under the probability measure

Pt(A) :=∫A

1pGt

dP for A ∈ Gt, (1.6)

i.e., for At ∈ Ft and B ∈ U ,

Pt[At ∩ G ∈ B] = P[At]P[G ∈ B] = Pt[At]Pt[G ∈ B]. (1.7)

2. 1pG is a G0-martingale.

Proof. To prove Equation (1.7), fix At ∈ Ft and B ∈ U . By conditioning on Ft, we obtain

E[1At∩G∈B

1pGt

]= E

[1AtE

[1G∈B

1pGt|Ft

]]=

∫At

E[1G∈B

1pGt|Ft

](ω)P(dω).

The definition of plt(ω) yields

E[1G∈B

1pGt|Ft

](ω) =

∫B

1plt(ω)

plt(ω)P[G ∈ dl] = P[G ∈ B],

and so we get the first equality in Equation (1.7). The second follows by choosing At = Ω orB = U .


Now fix 0 ≤ s ≤ t < T and choose A ∈ Gs of the form A = As ∩ G ∈ B with As ∈ Fs andB ∈ U . Then we obtain by Equation (1.7) and by reversing the above argument that

E[1A

1pGt

]= P[As]P[G ∈ B] = E

[1AtP[G ∈ B]

]=

∫As

∫B

1pls(ω)

pls(ω)P[G ∈ dl]P(dω)

= E[1AsE

[1G∈B

1pGs|Fs

]]= E

[1A

1pGs

].

Now let D =A ∈ Gs : E

[1A 1

pGt

]= E

[1A 1

pGs

], we show now that D is a d-system.

1. Ω ∈ D, indeed, from the first part of the proof we have that E[1Ω

1pG

t

]= E

[1Ω

1pG

s

].

2. Let A,B ∈ D such that A ⊂ B. Then we have that

E[1B\A

1pGt

]= E

[(1B − 1A)

1pGt

]= E

[1B

1pGt

]−E

[1B

1pGt

]= E

[1B

1pGs

]−E

[1B

1pGs

]= E

[(1B − 1A)

1pGt

]= E

[1B\A

1pGs

],

hence B\A ∈ D.

3. Let An ↑ A∞ with An ∈ D for all n ≥ 0. Since the process 1/pG is strictly positive, thenby the monotone convergence theorem we have that

E[1A∞

1pGt

]= lim

n→∞E

[1An

1pGt

]= lim

n→∞E

[1An

1pGs

]= E

[1A∞

1pGs

],

thus A∞ ∈ D.

Hence D is a d-system that contains the π-system

C =As ∩ G ∈ B; with As ∈ Fs and B ∈ U : E

[1As∩G∈B

1pGt

]= E

[1As∩G∈B

1pGs

]generating the σ-algebra Fs ∨ σ(G), and by Proposition 2.2.6 below, we have Gs = Fs ∨ σ(G).Therefore, Gs ⊂ D ⊂ Gs. Whence this extends to arbitrary sets A ∈ Gs. Hence the process 1

pG

is a G0-martingale with 1pG0

= 1, hence Equation (1.6) defines indeed a probability measure on(Ω,Gt).


1.2 Stochastic exponential representation of the process 1/pG

This section shows that under the assumption of Proposition 1.1.14, the processes pl and 1/pG

can be represented as stochastic exponentials of a particular form. More precisely, the F0-martingale pl is the stochastic exponential of the sum of a stochastic integral with respect to Kwith integrand κl and an orthogonal local F0-martingale, whereas the G0-martingale 1

pG can bewritten as a stochastic exponential of the sum of a stochastic integral with respect to κG withrespect to K and an orthogonal local F0-martingale. To do this, we give the following lemmawithout proof.

Lemma 1.2.1. Under Assumption 1.1.6, there exists an Rd-valued, P(F0)⊗U-measurable pro-cess (κlt)t∈[0,T ) such that for all l ∈ U ,

∫ t

0d〈K〉sκls =

∫ t0

(kls

)(1)d〈K(1)〉s...∫ t

0

(kls

)(d)d〈K(d)〉s

, t ∈ [0, T ). (1.8)

Proof. For the proof of this result we refer to the reader to Amendinger [3].

For further developments, we need a week integrability condition on κ.

Assumption 1.2.2. : The process κ from Lemma 1.2.1 satisfies∫ T

0

(κls

)>d〈K〉s(κls

)<∞ P-a.s. for all l ∈ U. (1.9)

Remark 1.2.3. The process κG is P(G0)-measurable, indeed we need only to show the mea-surability of the mapping

((ω, t);P(G0)

)→

((ω, t,G(ω));P(F0)⊗ U

).

For any A ∈ P(F0) and B ∈ U , we have

(ω, t) : (ω, t) ∈ A and G(ω) ∈ B = A ∩(ω : G(ω) ∈ B × [0, T )

)= A ∩

(ω : G(ω) ∈ B × 0

)∩

(ω : G(ω) ∈ B × (0, T )

),

and therefore we have the measurability of the mapping above. By the measurability of κlt weget the measurability of κG. And so the stochastic integral

∫ (κG

)>dK is well defined underAssumption 1.2.2. For each l ∈ U , the process κl is unique up to null sets with respect toP × 〈K〉, and so the stochastic integrals

∫ (κl

)>dK and∫ (κG

)>dK do not depend on thechoice of κ. Finally, we can now write K :=

(K1, . . . , Kd

)> more compactly as

K = K −∫

d〈K〉κG.


Proposition 1.2.4.

1. Suppose that the regular conditional distributions of G given Ft are equivalent to the lawof G for all t ∈ [0, T ). Then there exists a local G0-martingale N null at zero which isorthogonal to K from Equation (1.4) (i.e., 〈K(i), N〉 for i = 1, . . . , d) and such that

1pGt

= E(−

∫(κGs )>dKs + N

)t, t ∈ [0, T ). (1.10)

2. Fix l ∈ U . If pT l− > 0 P-a.s., then there exists a local F0-martingale N l null at zero whichis orthogonal to K and such that

plt = E( ∫

(κls)>dKs +N l

)t, t ∈ [0, T ). (1.11)

Proof. See Proposition 2.9, p. 270, of Amendinger [3].

Remark 1.2.5. If the regular conditional distributions of G given Ft are equivalent to the lawof G for all t ∈ [0, T ), then the condition in the second part of Proposition 1.2.4 is automaticallysatisfied for all l ∈ U .

The next corollary gives an explicit expression for N in Equation (1.10) in terms of NG, ifpl is continuous for all l ∈ U . As a consequence, we obtain then in particular that 1/pG canbe written as a stochastic exponential of a stochastic integral with respect to K, if we have inaddition a martingale representation theorem for the filtration F.

Corollary 1.2.6.

1. If pl is continuous and strictly positive for all l ∈ U , then

1pGt

= E(−

∫(κGs )>dKs −NG + 〈NG〉

)t, t ∈ [0, T ). (1.12)

In particular, N from Equation (1.10) is given by

Nt = −NGt + 〈NG〉t, t ∈ [0, T ). (1.13)

2. In particular, if pl = E( ∫

(κl)>dK)

for all l ∈ U , then

1pGt

= E(−

∫(κG)>dK

)t, t ∈ [0, T ). (1.14)

Proof. See Corollary 2.10, p. 272, of Amendinger [3]


1.3 Examples

This section illustrates the preceding results by several examples for G. We will give an examplewhere we have equivalence between the regular conditional probability of G given Ft and the lawof G for t ∈ [0, T ), an example where we have absolute continuity for t ∈ [0, T ] and equivalenceonly for t ∈ [0, T ), and an example where we have equivalence for all t ∈ [0, T ].

Example 1.3.1. Let G be the end point WT of a one-dimensional F-Brownian motion W on[0, T ]. Then we have G = ∩ε>0(Ft+ε ∨ σ(WT )) and we have for all t < T

P[WT ∈ dl|Ft

]= P

[(WT −Wt +Wt) ∈ dl|Ft

]= P

[(WT −Wt) ∈ (dl − y)

]∣∣∣Wt=y

=1√

2π(T − t)exp

(− (l −Wt)2

2(T − t)

)dl

= pltP[WT ∈ dl

],

where

plt =

√T

T − texp

(− (l −Wt)2

2(T − t)+

l2

2T

), l ∈ R,

is strictly positive for all t < T . Furthermore, applying Ito’s formula to (l−Wt)2

(T−t) we get

plt = E( ∫

l −Ws

T − sdWs

)t.

and hence it is an F0-martingale by Novikov’s condition. In this example, the conditional law ofG given Ft is even equivalent to the law of G for all t ∈ [0, T ). On the other hand, the conditionallaw of WT given FT is the point mass in WT (ω) and therefore not absolutely continuous withrespect to the law of WT .

Example 1.3.2. Let G be a random variable with values in a countable set U such thatP[G = l] > 0 for all l ∈ U . Then every A ∈ σ(G) is of the form A = ∪l∈J

G = l

for some

J ⊆ U . therefore we have

P[G ∈ A|Ft

]=

∑l∈J

P[G = l

∣∣Ft] =∑l∈J

pltP[G = l

]=

∫ApltP

[G ∈ dl

]for all t ∈ [0, T ], where plt = P[G = l|Ft]/P[G = l], and so the conditional law of G givenFt is absolutely continuous with respect to the law of G for all t ∈ [0, T ]. Thus we obtain byTheorem 1.1.12 and the Remark 1.1.13 that every local F-martingale is a G-semimartingale.However, the conditional laws of G given Ft are equivalent to the law of G on Ft for t < T

only if P[G = l|Ft] > 0 P-a.s. for all l ∈ U . Moreover, there is no equivalence on FT if Gis FT -measurable, because in this case P[G = l|FT ] = 1G=l is zero with positive probability(unless G is a constant and equal to l).

As a special case, consider the situation in which G describes whether the endpoint of aone-dimensional F-Brownian motion lies in some given interval, i.e., G := 1WT∈[a,b] for some


a < b. Then we have

p1t =

P[G = 1

∣∣Ft]P[G = 1]

and p0t =

1−P[G = 1

∣∣Ft]1−P[G = 1]

,

and for t ∈ [0, T ) we have

P[G = 1

∣∣Ft] = P[1

WT∈[a,b] = 1|Ft

]= P

[WT ∈ [a, b]|Ft

]= P

[WT −Wt +Wt ∈ [a, b]|Ft

]= P

[WT −Wt ∈ [a− y, b− y]

]∣∣∣Wt=y

=1√

2π(T − t)

∫ b

aexp

(− (l −Wt)2

2(T − t)

)dl,

andP[G = 1] = P

[G = 1

∣∣F0

]= Φ(b/

√T )− Φ(a/

√T ),

where Φ is the standard normal distribution function. Hence, P[G ∈ .

∣∣Ft] is absolutely con-tinuous with respect to the law of G for t ∈ [0, T ] and equivalent to the law of G only for allt ∈ [0, T ).

Example 1.3.3. Let G = WT + ε, where WT is the endpoint of a one dimensional (P,F)-Brownian motion W and ε is a random variable independent of FT such that ε ∼ N (0, 1).

Then we have for all t ∈ [0, T ]

P[G ∈ dl

∣∣Ft] = P[(WT −Wt +Wt + ε) ∈ dl

∣∣Ft]= P

[(WT −Wt + ε) ∈ (dl − y)

]∣∣∣y=Wt

=1√

2π(T − t+ 1)exp

(− (l −Wt)2

2(T − t+ 1)

)dl

= pltP[(WT + ε) ∈ dl

],

where

plt =

√T + 1

T − t+ 1exp

(− (l −Wt)2

2(T − t+ 1)+

l2

2(T + 1)

), l ∈ R,

is strictly positive for all t ∈ [0, T ]. Furthermore, applying Ito’s formula to (l−Wt)2

(T−t+1) we get

plt = E( ∫

l −Ws

T − s+ 1dWs

)t,

and hence it is a F-martingale by Novikov’s condition.In this example, the conditional law of Ggiven Ft is even equivalent to the law of G for all t ∈ [0, T ], the end point included.

Chapter 2

Martingale representation theoremsfor I.E.F

2.1 Notations

Recall that we are working in a probability space (Ω,F ,P) equipped with a filtration F =(Ft)t∈[0,T ] satisfying the usual conditions. T > 0 is a fixed finite time horizon. We assume thatF0 is trivial and Fs = FT = F for all s ≥ T . For an F-measurable random variable G withvalues in a Polish space (U,U), we define the initially enlarged filtration G = (Gt)t∈[0,T ] by

Gt = Ft ∨ σ(G), t ∈ [0, T ].

Let H = (Ht)t∈[0,T ] ∈ F,G be a generic filtration and R be a generic probability measureon (Ω,HT ). The collection of uniformly integrable continuous (R,H)-martingales is denoted byM(R,H). For p ∈ [1,∞), Hp(R,H) denotes the set of the continuous (R,H)-martingales Msuch that :

‖M‖Hp(R,H) :=(E

[sups∈[0,T ]

|Ms|p]) 1

p <∞.

The set of bounded continuous (R,H)-martingales is denoted by H∞(R,H). For p ∈ [1,∞),Lp(M,R,H) denotes the space of d-dimensional H-predictable processes φ = (φ(1), . . . , φ(d)) suchthat:

‖φ‖Lp(M,R,H) := ER[( ∫ T

0φ>u d[M,M ]uφu

) p2]<∞.

For a d-dimensional continuous (R,H)-semimartingale S, Lsm(S,R,H) denotes the set of d-dimensional H-predictable processes φ that are integrable with respect to S, in the sense that∫φ>dS =

∑di=1

∫φidSi, where for every i = 1, . . . , d the stochastic integral

∫φidSi is well

defined. To emphasize the dependence of the stochastic integral on H, we shall then writeH-

∫φ>dS, where φ ∈ Lsm(S,R,H).

Throughout this chapter we fix a d-dimensional continuous process S = (S(1), . . . , S(d))>,and we assume that there exists a probability measure QF ∼ P on (Ω,FT ) such that eachcomponent of S is in H2

loc(QF,F). This assumption is motivated because as we shall see in

Section 3.1, the process S is not always a martingale in the real world, i.e. a (P,F)-martingale.Let ZF be the density process of QF with respect to P.

10


2.2 The martingale preserving probability measure

In this section we define the martingale preserving probability measure and also shows how itcan be used to transfer properties of stochastic processes and structures from F to G. Recallthat in Chapter 1, we have used The following assumption in some results and will be imposedin the remainder of this thesis.

Assumption 2.2.1. The regular conditional distributions of G given Ft are equivalent to thelaw of G, i.e.:

P[G ∈ .|Ft](ω) ∼ P[G ∈ .] for all t ∈ [0, T ] and P-a.a. ω ∈ Ω (2.1)

Theorem 2.2.2. If the Assumption (2.2.1) is satisfied, then

1. Let QG be the measure defined by

QG(A) :=∫A

ZFT

pGTdP for A ∈ GT (2.2)

has the following properties:

i. QG = QF on (Ω,FT ), and QG = P on (Ω, σ(G)), i.e. for AT ∈ FT and B ∈ U ,

QG[AT ∩ G ∈ B] = QF[AT ]P[G ∈ B] = QG[AT ]QG[G ∈ B]. (2.3)

ii. The σ-algebras FT and σ(G) are independent under QG.

2. ZG := ZF

pG is a (P,G)-martingale.

Proof. 1. To prove Equation (2.3), let AT ∈ FT and B ∈ U . By conditioning on FT , we get

E[1AT∩G∈B

ZFT

pGT

]= E

[1AT

E[1G∈B

ZFT

pGT|FT

]]=

∫AT

E[1G∈B

ZFT

pGT|FT

](ω)P(dω).

The definition of pGT yields

E[1G∈B

1pGT|FT

](ω) =

∫B

1plT (ω)

plT (ω)P[G ∈ dl]

Therefore

E[1G∈B

ZFT

pGT|FT

](ω) =

∫B

ZFT (ω)plT (ω)

plT (ω)P[G ∈ dl] =∫BZFT (ω)P[G ∈ dl]


and hence

QG[AT ∩ G ∈ B

]= E

[1AT∩G∈B

ZFT

pGT

]= E

[1AT

E[1G∈B

ZFT

pGT|FT

]]=

∫AT

E[IG∈B

ZFT

pGT|FT

](ω)P(dω)

=∫AT

∫BZFT (ω)P[G ∈ dl]P(dω)

=∫AT

ZFT (ω)

∫B

P[G ∈ dl]P(dω)

= QF[AT ]P[G ∈ B]

where in the last equality we used the fact that ZF is the density process of Q with respectto P. Thus we get the first equality in Equation (2.3). The second follows by choosingAT = Ω or B = U .

2. Now fix 0 ≤ s ≤ t ≤ T and choose A ∈ Gs of the form A = As∩G ∈ B with As ∈ Fs andB ∈ U . Then we obtain by Equation (2.3), using the fact that ZF is a (P,F)-martingaleand by reversing the above argument that

E[1A

ZFt

pGt

]= QF[As]P[G ∈ B] = E

[1AsZ

Fs (ω)P[G ∈ B]

]=

∫As

∫B

ZFs (ω)pls(ω)

pls(ω)P[G ∈ dl]P(dω)

= E[1AsE

[1G∈B

ZFs

pGs|Fs

]]= E

[1A

ZFs

pGs

].

Then arguing as in Proposition 1.1.14 , this extends to arbitrary sets A ∈ Gs. Hence theprocess ZF

pG is a (P,G)-martingale with ZF0

pG0

= 1 because ZF0 = 1 = pG0 and so Equation (2.2)

defines indeed a probability measure on (Ω,Gt).

The following theorem shows that the martingale property is preserved under an initialenlargement of filtration and a simultaneous change to the measure QG.

Theorem 2.2.3. If the Assumption 2.2.1 is satisfied, then for all p ∈ [1,∞]

Hp(loc)

(QF,F

)= Hp

(loc)

(QG,F

)⊆ Hp

(loc)

(QG,G

)(2.4)

and in particularM(loc)

(QF,F

)= M(loc)

(QG,F

)⊆M(loc)

(QG,G

). (2.5)

Proof. Let M be a (QF,F)-martingale we have then

EQG[Mt|Gs] = EQG

[Mt|Fs ∨ σ(G)] = EQG[Mt|Fs] = EQF

[Mt|Fs] = Ms,

where in the second equality we used independence of σ(G) and FT under QG and in the thirdequality we used the equality of QG and QF on (Ω,FT ). Therefore M is a (QG,G)-martingale.


Since F-stopping times are also G-stopping times, any localizing sequence (τn)n for a process Mwith respect to (QF,F) will then also be a localizing sequence for the process M with respect to(QG,F) and (QG,G). The integrability properties in Equations (2.4) and (2.5) follow from theequality of QG and QF on (Ω,FT ).

Remark 2.2.4. Any (QF,F)-Brownian motion W is a (QG,G)-Brownian motion. Indeed, Sincethe quadratic variation of continuous martingales can be computed pathwise without involvingthe filtration and since QG = QF on (Ω,FT ), we obtain for all t ∈ [0, T ] that

〈W 〉(QG,G)

t = 〈W 〉(QG,F)

t = 〈W 〉(QF,F)

t = t

and therefore W is also a (QG,G)-Brownian motion, by Levy’s characterization theorem.

Definition 2.2.5. The probability measure QG defined by Equation (2.2) on (Ω,GT ), is calledthe martingale preserving probability measure under initial enlargement of filtration. This ter-minology is justified by Theorem 2.2.3.

Using the decoupling property of QG, the following proposition shows that G inherits theright-continuity from F.

Proposition 2.2.6. If the Assumption 2.2.1 is satisfied, then G is right-continuous.

Proof. Define Gs+ :=⋂ε>0 Gs+ε for s ∈ [0, T ]. Fix t ∈ [0, T ) and δ ∈ (0, T − t). Because of the

independence of Ft+δ and σ(G) under QG, and using Theorem A.2.3, it is enough to show thatfor Gt+δ-measurable random variables Yt+δ of the form Yt+δ = h(G)Ht+δ, where h is a boundedU-measurable function and Ht+δ is a bounded Ft+δ-measurable random variable, we have

EQG[Yt+δ|Gt+] = EQG

[Yt+δ|Gt].

For all ε ∈ (0, δ), we get

EQG[Yt+δ|Gt+] = h(G)EQG

[Ht+δ|Gt+]

= h(G)EQG[EQG

[Ht+δ|Ft+ε ∨ σ(G)]|Gt+]

= h(G)EQG[EQG

[Ht+δ|Ft+ε]|Gt+], (2.6)

since Ht+δ and Ft+ε are independent of G under QG. And because of the right-continuity of F,we can always choose right-continuous versions of F-martingales. This implies

limε0

EQG[Ht+δ|Ft+ε] = EQG

[Ht+δ|Ft],

and since Ht+δ is bounded, passing in Equation (2.6) to the limit as ε decreases to 0 and applyingthe dominated converging theorem, we obtain

EQG[Yt+δ|Gt+] = h(G)EQG[

EQG[Ht+δ|Ft|Gt+

]= h(G)EQG

[Ht+δ|Ft]

= h(G)EQG[Ht+δ|Gt] = EQG

[Yt+δ|Gt],

because Ht+δ and Ft are independent of G under QG.


In particular, we have for all Gt+-measurable random variables X that

X = EQG[X|Gt+ ] = EQG

[X|Gt] QG − a.s.

Since QG ∼ P and G0 contains all (P,F)-null sets, X is therefore Gt-measurable. This completesthe proof.

The following proposition shows that the stochastic integrals defined under F remain un-changed under an initial enlargement that satisfies Assumption 2.2.1.

Proposition 2.2.7. If the Assumption 2.2.1 is satisfied. For a d-dimensional (QF,F)- semi-martingale Y , the following equalities then hold:

Lsm(Y,QF,F) = Lsm(Y,QG,F) (2.7)

= ϑ : ϑ is F-predictable and ϑ ∈ Lsm(Y,QG,G) (2.8)

and for ϑ ∈ Lsm(Y,QF,F) the stochastic integrals F-∫ϑdY and G-

∫ϑdY have a common version.

Proof. Because QF = QG on (Ω,FT ), we get then the first equality.For the second equality we have by Theorem 2.2.3, the (QG,F)-semimartingale Y is also a

(QG,G)-semimartingale and thus

Lsm(Y,QG,F) ⊇ ϑ : ϑ is F-predictable and ϑ ∈ Lsm(Y,QG,G)

by Theoreme 7 of Jacod [8], see Theorem A.2.1 in the Appendix. For the other inclusion, letϑ ∈ Lsm(Y,QG,F), i.e. there exists a local (QG,F)-martingale M and an F-adapted processA of finite variation such that Y = M + A, and such that ϑ ∈ Lloc(M,QG,F) and

∫ϑ>dA

exists. By Theorem 2.2.3, M is a (QG,F)-martingale. Since F ⊆ G, the process A is G-adapted. Therefore Y = M+A is also a (QG,G)-semimartingale decomposition. And since M ∈Mloc(QG,F) ∩Mloc(QG,G), Corollaire 9.21 of Jacod [7], see corollary A.2.2 in the Appendix,implies that ϑ ∈ Lloc(M,QG,G), and since the

∫ϑ>dA can be computed pathwise without

involving the filtrations, we get that ϑ ∈ Lsm(Y,QG,G) thus the proof is complete.

2.3 Martingale representation theorems for I.E.F

In this section we transfer martingale representation theorems from F to the initially enlargedfiltration G. For these purpose we suppose throughout this section that the following represen-tation property holds with respect to S ∈ H2

loc(QF,F):

Assumption 2.3.1. For any ψ ∈ L∞(FT ), there exists φ ∈ L2(S,QF,F) such that

ψ = EQF[ψ] +

∫ T

0φ>s dSs.

Remark 2.3.2. By Theorem 13.4 of He, Wang and Yan [6], see Theorem A.2.5 in the Appendix,the assumption above is equivalent to the representation property of a local (QF,F)-martingale.That is for every local (QF,F)-martingale M there exists a φ ∈ L1

loc(S,QF,F) such that K =

K0 +∫φ>dS


Theorem 2.3.3. Suppose Assumptions 2.2.1 and 2.3.1 are satisfied.

1. For any M ∈ H2(QG,G), there exists a process ψ ∈ L2(S,QG,G) such that

Mt = M0 +∫ t

0ψ>s dSs, t ∈ [0, T ].

2. For any M ∈Mloc(QG,G), there exists a process ψ ∈ L1(S,QG,G) such that

Mt = M0 +∫ t

0ψ>s dSs, t ∈ [0, T ].

Proof. To prove the first claim it is sufficient to show that any random variable X ∈ L2(QG,GT )can be written in the form

X = EQG[X|G0] +

∫ T

0ψ>s dSs,

for some ψ ∈ L2(S,QG,G). Since GT = FT ∨ σ(G), Theorem IV. 3.5.1 of Malliavin [11], seeTheorem A.2.3 in the Appendix, implies that the vector subspace of L∞(GT ) defined by

V =X ∈ L∞(GT ) : X =

m∑i=1

IiJi, with Ii ∈ L∞(FT ), Ji ∈ L∞(σ(G)), m ∈ N

is dense in L2(QG,GT ). Thus there exists a sequence (Xn)n∈N = (∑mn

i=1 Ii,nJi,n)n∈N in V , withIi,n ∈ L∞(FT ) and Ji,n ∈ L∞(σ(G)), such that Xn converges to X in L2(QG). By Assumption2.3.1, and the fact that QG = QF on (Ω,FT ) (Theorem 2.2.2), then there exists a sequence(φi,n)n∈N ∈ L2(S,QG,F) such that

Ii,n = EQG[Ii,n] +

∫ T

0(φi,ns )>dSs. (2.9)

Since S is a local (QG,F)-martingale and thus a local (QG,G)-martingale by Theorem 2.2.3,Proposition 2.2.7 implies that the value of the stochastic integral

∫ T0 (φi,ns )>dSs is not changed

when it is considered under G. Because Ji,n is bounded and G0-measurable since

G0 = F0 ∨ σ(G) = σ(G)

because F0 is trivial, we have

Ji,n

∫ T

0(φi,ns )>dSs =

∫ T

0Ji,n(φi,ns )>dSs. (2.10)

The independence of FT and G0 under QG yields

EQG[Ii,nJi,n|G0] = Ji,nEQG

[Ii,n|G0] = Ji,nEQG[Ii,n] (2.11)

By Equations (2.9), (2.10) and (2.11) we then obtain

Xn =mn∑i=1

Ji,n(EQG

[Ii,n] +∫ T

0(φi,ns )>dSs

)= EQG

[Xn|G0] +∫ T

0(ψns )>dSs, (2.12)


where ψn :=∑mn

i=1 gi,nφi,n is in L2(S,QG,G) due to the boundedness of Ji,n and since φi,n is

in L2(S,QG,G). Since Xn converges to X in L2(QG), EQG[Xn|G0] converges to EQG

[X|G0] inL2(S,QG,G) and thus Equation (2.12) yields that

∫ T0 (ψns )>dSs converges in L2(QG) as well.

Since each component of S is in H2loc(Q

G,G), and since the mapping ϑ 7→∫ϑ>dS is an isom-

etry from (L2(S,QG,G), ‖.‖L2(S,QG,G)) to (H2loc(Q

G,G), ‖.‖H2loc(Q

G,G)), the space of stochasticintegrals ∫ T

0ϑ>s dSs : ϑ ∈ L2(S,QG,G)

,

is closed in L2(QG). This implies the existence of a process ψ ∈ L2(S,QG,G) such that∫ T0 (ψns )>dSs converges to

∫ T0 ψ>s dSs in L2(QG). Hence Equation (2.12) yields

X = L2 − limn→∞

(EQG

[Xn|G0] +∫ T

0(ψns )>dSs

)= EQG

[X|G0] +∫ T

0ψ>s dSs,

and thus the first claim. For the proof of the second part of the theorem, we will proceed insteps. For convenience we will denote by Gt the filtration (Gs)s∈[0,t] for all t ∈ [0, T ] and ofcourse we have GT = G.

1. Let t ∈ [0, T ) and K ∈ H10(Q

G,Gt), i.e. K ∈ H1(QG,Gt) and K0 = 0. The filtration Gt

is right-continuous by Proposition 2.2.6. Then Theorem 10.5 of He, Wang and Yan [6]implies that H2

0(QG,Gt) is dense in H1

0(QG,Gt), thus there exists a sequence (Kn)n≥0 in

H20(Q

G,Gt) such that limn→∞ ||Kn−K||H10(QG,Gt) = 0. For all n ≥ 0, the first part of the

theorem yields the existence of ψn ∈ L2(S,QG,Gt) such that

Knt =

∫ t

0(ψns )>dSs, t ∈ [0, T ). (2.13)

Since Kn is in L1(S) := ∫

ϑ>dS : ϑ ∈ L1(S,QG,Gt), and since L1(S) is closed in

H10(Q

G,Gt) by Theorem 4.60 of Jacod [7], we conclude that K ∈ L1(S).

2. Let K ∈ H10(Q

G,G). By part 1, on the interval [0, T ) K is of the form

Kt =∫ t

0ψ>s dSs, t ∈ [0, T ), (2.14)

where ψ is Gt-predictable, t ∈ [0, T ), and for all t ∈ [0, T ),

EQG[( ∫ t

0ψ>s d〈S〉sψs

)1/2]<∞.

We now extend Equation (2.14) to the interval [0, T ]. Since for all t ∈ [0, T ), the filtrationGt is right-continuous, the Burkholder-Davis-Gundy inequalities imply for t ∈ [0, T )

EQG[( ∫ t

0ψ>s d〈S〉sψs

)1/2]= EQG

[〈K〉1/2t

]≤ C EQG

[sup

0≤s≤t|Ks|

]≤ C EQG

[sup

0≤s≤T|Ks|

],


where C is a positive constant. Hence by monotone convergence we obtain

EQG[( ∫ T

0ψ>s d〈S〉sψs

)1/2]≤ C EQG

[sup

0≤s≤T|Ks|

]≤ ∞,

because K ∈ H10(Q

G,G). Hence∫ψ>dS is a (QG,G)-martingale, and Equation (2.14)

implies that for all t ∈ [0, T )

Kt = EQG[ ∫ T

0ψ>s dSs

∣∣∣Gt].Since K ∈ H1

0(QG,G) and limt→T ||Kt−KT ||L1 = 0, by martingale convergence we obtain

KT =∫ T0 ψ>s dSs.

3. Now let K ∈ M0,loc(QG,G), i.e. there is a sequence of G-stopping times (σn)n∈N suchthat for each n ∈ N, Kσn ∈ M0(QG,G). Since for all t ∈ [0, T ), Gt is right-continuous,we have then that for all n ∈ N, τn := σn ∧ inft : |Kt| ≥ n ∧ T is a G-stopping time.Hence supt∈[0,T ] |Kτn

t | ≤ n + |Kτn |, hence Kτn ∈ H10(Q

G,G). Therefore part 2 yields theexistence of ψn ∈ L1(S,QG,G) such that Kτn =

∫(ψn)>dS. With τ0 = 0, we get that

ψ :=∑∞

n=1 ψn1]τn−1,τn] is in L1

loc(S,QG,G) and that K =

∫ψ>dS.

We will make the following assumption on S which we will need for the prove of a martingalerepresentation theorem with respect to G and the original probability measure P.

Assumption 2.3.4. The (P,F)-semimartingale S is continuous and can be written as

S = M +∫

d〈M〉α, (2.15)

where M is a d-dimensional continuous local (P,F)-martingale and α is a d-dimensional processin L1(M,P,F).

Application of Theorem 1.1.12 and Lemma 1.2.1 to the d-dimensional continuous local (P,F)-martingale M from Assumption 2.3.4 yields the existence of a P(F) ⊗ U-measurable function(ω, t, l) 7→ κlt(ω) such that

M := M −∫

d〈M〉κG

is a d-dimensional continuous local (P,G)-martingale. And we need the following assumptionon the integrability of κ.

Assumption 2.3.5. For all l ∈ U the process κl ∈ L1loc(M,P,F).


Lemma 2.3.6.

1. If Assumptions 2.3.1 and 2.3.4 are satisfied, then for t ∈ [0, T ]

ZFt = E

(−

∫α>s dMs

)t. (2.16)

2. If Assumptions 2.2.1, 2.3.1 and 2.3.4 are satisfied then for t ∈ [0, T ]

1pGt

= E(−

∫(κGs )>dMs

)t, (2.17)

ZGt =

ZFt

pGt= E

(−

∫(αs + κGs )>dMs

)t. (2.18)

Proof. Since 1ZF is a strictly positive (QF,F)-martingale, there exists a local (QF,F)-martingale

L with L0 = 0 such that 1ZF = E(L). By Assumption 2.3.1 and Remark 2.3.2, there exists a

process φ ∈ L1loc(S,Q

F,F) such that

1ZF = E(

∫φ>dS).

Then by Assumption 2.3.4, the density process ZF can be written as

ZF = exp−

∫φ>dM −

∫φ>d〈M〉α+

12

∫φ>d〈M〉α

, (2.19)

hence Ito’s formula implies that for i = 1, . . . , d

d(ZFSi) = ZFdSi + SidZF + d〈ZF, Si〉

= ZFdM i + ZF(d〈M〉α)i + SidZF + ZFd〈M i,−∫φ>dM〉

= ZFdM i + SidZF + ZF(d〈M〉(α− φ)

)i.

Since ZFSi, ZFdM i and SidZF are continuous local (P,G)-martingales,∫ZFd〈M〉(α− φ) is a

continuous local (P,G)-martingale of finite variation and thus vanishes. And since ZF > 0, weget that

∫d〈M〉α =

∫d〈M〉φ and so

∫α>dM =

∫φ>dM . By Equation (2.19), we get then

Equation (2.16).To prove Equation 2.17, let l ∈ U . Since pl is a strictly positive (P,F)-martingale by

Lemma 1.1.10 and Remark 1.1.11 (we have F instead of F0 because we have equivalence in-stead of absolute continuity), whence pl/ZF is a strictly positive (QF,F)-martingale. Because ofAssumption 2.3.1 there exists a process φl ∈ L1

loc(S,QF,F) such that

pl

ZF = E(∫

(φl)>dS).


By Assumption 2.3.4 and the first claim we have

pl =pl

ZFZF = E

( ∫(φl)>dS

)E(−

∫α>dM

)= E

( ∫(φl)>dS −

∫α>dM −

∫(φl)>d〈M〉α

)= E

( ∫(φl)>dM +

∫(φl)>d〈M〉α−

∫α>dM −

∫(φl)>d〈M〉α

)= E

( ∫(φl − α)>dM

).

On the other hand, because of Assumptions 2.2.1, 2.3.5 and part 2 of Proposition 1.2.4 can beapplied to get

pl = E( ∫

(κl)>dM +N l),

where N l is a local (P,F)-martingale with N l0 = 0 and orthogonal to M . the uniqueness of the

stochastic exponential thus implies that∫(φl − α)>dM =

∫(κl)>dM +N l. (2.20)

Now taking the covariation of both sides of Equation (2.20) with respect to N l, we get by theorthogonality of M and N l that 〈N l〉 = 0. By Assumption 2.3.4 and Equation (2.20) the processN l is continuous. Hence, N l is a continuous local (P,F)-martingale of finite variation and thusvanishes. Therefore we obtain that

pl = E( ∫

(κl)>dM).

Now part 2 of corollary 1.2.6 yields Equation (2.17). By Theorem 2.2.2 we have that ZG = ZF/pG

is a (P,F)-martingale, hence

ZG =ZF

pG= E

(−

∫α>s dMs

)tE(−

∫(κGs )>dMs

)t

= E(−

∫α>s dMs

)t−

∫(κGs )>dMs + 〈−

∫α>s dMs,−

∫(κGs )>dMs〉

)t

= E(−

∫α>s dMs −

∫(κGs )>dMs +

∫α>s d〈M〉sκGs

)t

= E(−

∫α>s (dMs + d〈M〉sκGs )−

∫(κGs )>dMs

)t

= E(−

∫α>s dMs −

∫(κGs )>dMs

)t

= E(−


)t.

Then the proof is complete.

We now show a martingale representation theorem for local (P,G)-martingales with respectto the continuous (P,G)-martingale M .


Theorem 2.3.7. Suppose Assumptions 2.2.1, 2.3.1, 2.3.4 and 2.3.5 are satisfied. For anyK ∈Mloc(P,G), there exists then φ ∈ L1

loc(M,P,G) such that

Kt = K0 +∫ t

0φ>s dMs, t ∈ [0, T ]. (2.21)

Proof. Since K ∈ Mloc(P,G), we have that KZG ∈ Mloc(QG,G), hence Theorem 2.3.3 implies

that for all t ∈ [0, T ],K

ZG = K0 +∫ t

0φ>s dSs

for some φ ∈ L1loc(S,Q

G,G). Now applying Ito’s formula and using the fact that

S = M +∫

d〈M〉(α+ κG) and ZG = E(−


)t,

we get for t ∈ [0, T ]

dKt = d(ZG K

ZG

)t=

( K

ZG

)tdZG

t + ZGt d

( K

ZG

)t+ d〈ZG,

K

ZG 〉t

= ZGt φ

>t dSt +

(K0 +

∫ t

0φ>s dSs

)dZG

t + d〈ZG,

∫φ>dS〉t

= ZGt φ

>t dMt + ZG

t φ>t d〈M〉t(α+ κG)t−

− ZGt

(K0 +

∫ t

0φ>s dSs

)(α+ κG)>t dMt−

− ZGt φ

>t d〈M〉t(α+ κG)t

= ZGt

(φt −

(K0 +

∫ t

0φ>s dSs

)(α+ κG)t

)>dMt.

By setting φ := ZGt

(φ−

(K0 +

∫ t0 φ

>s dSs

)(α+ κG)t

), we therefore obtain Equation (2.21). The

integrability property of φ is a consequence of the integrability of φ, the continuity of ZG and Sand the integrability assumptions on α and κG.

Chapter 3

Insider trading and utilitymaximization

In this chapter we consider two types of investors on different information levels trading ina general continuous-time security market. The market is described by the given probabilityspace (Ω,F ,P) with the filtration F = (Ft)t∈[0,R] satisfying the usual conditions and we assumealso that F0 is trivial. The price process S = St, t ∈ [0, R] that models the stock price isassumed to be a positive continuous (P,F)-semimartingale. While the ordinary investor, whohas as information at time t all the Ft-measurable random variables, makes his decisions basedon these random variables, the insider investor observes the same process S but he/she holds asinformation a bigger filtration G then the ordinary investor (that is, Ft ⊂ Gt, t ∈ [0, R]). Theadditional information of the insider could be for example the knowledge at time t = 0 of theoutcome of some F-measurable random variable G. For instance, G might be the price of Sat time t = R, or the value of some external source of uncertainty, etc. As in the precedingchapter G is an F-measurable random variable with values in some Polish space (U,U), andthen the insider information is modelled by the initially enlarged filtration G = (Gt)t∈[0,R] withGt = Ft ∨ σ(G), t ∈ [0, R].

We fix a time T ∈ (0, R], and we assume that the financial market S on the time interval[0, T ] is arbitrage free and complete for the ordinary investor in the following sense: there existsa unique probability measure QF equivalent to P on ((Ω,FT ) such that S is a local (QF,F)-martingale (arbitrage free), and any bounded FT -measurable random variable can be written asa some of a constant and a stochastic integral with respect to S (completeness), i.e. Assump-tion 2.3.1. Furthermore we suppose that the random variable G satisfies Assumption 2.2.1.

Remark 3.0.8.

1. The intuitive meaning of Assumption 2.2.1 is that at all times up to time T the insiderhas an informational advantage over the ordinary investor consisting in the knowledge ofall the outcomes of G as possible before and at time T . For the public the outcome of G isrevealed only after time T . As we have seen in Example 1.3.3, if G contains a noise termthat is independent from FR, then we can choose T = R. For the other Examples 1.3.1and 1.3.2 we can choose any T < R.

2. Assumption 2.2.1 combined with the existence of an equivalent local F-martingale measure

21


for S ensures the existence of an equivalent local G-martingale measure for S, i.e. underthe probability measure QG given in Theorem 2.2.2 with density process ZG. The priceprocess S is then by Theorem 2.2.3 a local (QG,G)-martingale, moreover by Theorem 2.3.3any local (QG,G)-martingale can be written as a sum of a G0-measurable random variableand a stochastic integral with respect to S. Whence Assumption 2.2.1 is sufficient to placethe insider in a complete market free of arbitrage.

3.1 The ordinary investor problem

In this section we will give an exposition of the problem for the ordinary investor in the Black-Scholes framework. A general model will be treated in the next section.As in the introduction of this chapter we suppose that the market is described by the givenprobability space (Ω,F ,P) and we consider a Brownian motion W = Wt; 0 ≤ t ≤ T definedon (Ω,F ,P) and we denote by F = (Ft)t∈[0,T ] the natural filtration generated by W and the(P,F)-sets of measure zero. The price process S = St; t ∈ [0, T ] that models the stockprice is assumed to be a positive continuous (P,F)-semimartingale evolving according to thestochastic equation

dSt = St(µdt+ σdWt

), t ∈ [0, T ], (3.1)

with S0 > 0, and µ is a constant and σ a strictly positive constant. Besides we denote byB = Bt, t ∈ [0, T ] the risk free asset, and we suppose that it evolves according to the stochasticdifferential equation, for given positive constant r,

dBt = Btrdt, t ∈ [0, T ] and B0 = 1. (3.2)

Before we continue, we need some definitions concerning trading strategies, self-financing strate-gies.

Definition 3.1.1. A trading strategy (portfolio) is a two-dimensional predictable, locally boundedprocess π = πt = (φt, ψt), t ∈ [0, T ] with values in R2.

Remark 3.1.2. The conditions on π ensure that the stochastic integrals∫ T0 φtdBt and

∫ T0 ψtdSt

are well defined. Where φt denotes the money that the investor invests in the riskless asset, andψ denotes the number of stocks held in the portfolio at time t.

Definition 3.1.3. Let π be a trading strategy.

1. The value of the portfolio π at time t is given by

Vt = V πt = φtBt + ψtSt.

The process V πt is called the value process, or the wealth process, of the trading strategy

π.

2. The gains process denoted by Gπt is defined by

Gπt =∫ t

0φsdBs +

∫ t

0ψsdSs.


3. A trading strategy π is called self-financing if the wealth process V πt satisfies

V πt = V π

0 +Gπt for all t ∈ [0, T ].

Now we return to our model and we observe that the Equations (3.1) and (3.2) have a uniquesolutions and by Ito’s formula we have that the solutions are given by

Bt = exp(rt), (3.3)

St = S0 exp((µ− 1

2σ2)t+ σWt

). (3.4)

The process modelling the stock price is not a F-martingale under the measure P. Therefore weneed another measure equivalent to P under which the process S is an F-martingale. We definethe process S by St = St/Bt = e−rtSt, the process S is called the discounted stock process. ByIto’s formula and Equation (3.1) we have

dSt =1Bt

dSt + Std( 1B

)t

= e−rtSt

(µdt+ σdWt

)+ St(−re−rt)dt

= St

(µ− r

σσe−rtdt+ σe−rtdWt

)(3.5)

dSt = σSt

(µ− r

σdt+ dWt

)(3.6)

Under P the processW is a Brownian motion and hence the process S is a local (P,F)-martingaleif and only if µ ≡ r. But this will rarely be the case in the real world.On the other hand, under another equivalent measure P the discounted stock process S maywell be an F-martingale, if viewed as a process on the filtered space (Ω,F ,F, P). Because thedrift term in Equation (3.5) causes the problem, we could first rewrite the equation as

dSt = σe−rtStdWt, (3.7)

for the process W defined byWt = Wt − αt.

Then we need to find a measure under which the process W is a martingale. By Novikov’scondition the process E(

∫αdW ) for αt = µ−r

σ is a P-martingale, with mean 1. Therefore, wecan define a probability measure P by dP = E(

∫αdW )TdP. By Girsanov’s theorem the process

W is then a P-Brownian motion (for the time parameter restricted to the interval [0, T ], andhence a P-martingale. And we have also that the Brownian motion possesses the martingalerepresentation property. Therefore we are working in an arbitrage free complete market.

3.2 Utility Maximization

The underlying principle for modelling economic behavior of investors (or economic agents ingeneral) is the maximization of expected utility, that is one assumes that agents have a utilityfunction U(.) and base their economic decisions on expected utility considerations.


Definition 3.2.1. A function U : (0,∞) → R is called a utility function, if

1. it is strictly concave, strictly increasing and continuously differentiable, and

2. U ′(0+) = limt→0+ U ′(t) = ∞ and U ′(∞) = limt→∞ U ′(t) = 0 ( called Inada Conditions).

Example 3.2.2. As examples of a utility function we have

1. u(x) = log x,

2. u(x) = xα, 0 < α < 1.

Since we will work a lot with exponential processes, we will assume a logarithmic utilityfunction because it enables us to obtain explicit formulae.

3.2.1 The model

We return now to our setup as introduced in the beginning of his chapter. We will work with theone dimensional case. We fix a continuous local F-martingale M with M0 = 0 and a predictableprocess α with

E[ ∫ T

0α2td〈M〉t

]<∞. (3.8)

The discounted price process of the stock denoted again by S is assumed to evolve according tothe stochastic differential equation

dSt = St(dMt + αtd〈M〉t

), t ∈ [0, T ], (3.9)

with S0 > 0. Using the Ito’s formula we get

St = S0E(M +

∫αd〈M〉

)t= S0 exp

( ∫ t

0(αs −

12)d〈M〉s +Mt

)(3.10)

By Theorem 1.1.12, and Lemma 1.2.1 we have that M is a G-semimartingale, and the localG-martingale M in its canonical G-decomposition has the form

Mt = Mt −∫ t

0κGs d〈M〉s, t ∈ [0, T ], (3.11)

with κ = (κlt) is a P(F) ⊗ U-measurable process. And so the discounted stock price evolutionfrom the insider’s point of view is

dSt = St(dMt + (κG + αt)d〈M〉t

)Using the Ito’s formula we get

St = S0 E(M +

∫(α+ κG)d〈M〉

)t

(3.12)

= S0 exp( ∫ t

0(αs + κGs −

12)d〈M〉s + Mt

)


Definition 3.2.3. Let x > 0 and denote by H ∈ F,G a generic filtration.

1. An H-portfolio process is an R-valued and H-adapted predictable process π = (πt)t∈[0,T ]

such that∫ T0 π2

t d〈M〉t <∞ P-a.s.

2. For an H-portfolio process π, the discounted wealth (value) process denoted by V (x, π) isdefined by V0(x, π) = x and satisfies

dVt(x, π) := πtdSt := ψtVt(x, π)dStSt

for t ∈ [0, T ]. (3.13)

3. The class of admissible H-portfolio processes up is defined by

AH(x, T ) :=π : π is an H-portfolio process and E

[log− VT (x, π)

]<∞

, (3.14)

where log− x = max0,− log x.

Remark 3.2.4. The process ψt describes the proportion of total wealth at time t invested is therisky asset S, and Equation (3.13) is the well known self-financing condition. For convenience,we will consider the process ψ, but we will always keep in our mind that the portfolio is theprocess π, and we obtain ψ by introducing the change of variables ψt = πt

StVt

. And from now onwe will denote the discounted wealth process by Vt(x, ψ) instead of Vt(x, π).

By Ito’s formula, for a trading strategy ψ ∈ AH(x, T ) with x > 0, the wealth process isstrictly positive and explicitly given by

Vt(x, ψ) = x E( ∫

ψsdSsSs

)t= x E

( ∫ψsdMs +

∫ψsαsd〈M〉s

)t

(3.15)

for t ∈ [0, T ]. From the insider’s point of view this can also be written, similar to Equation (3.12),as

Vt(x, ψ) = x E( ∫

ψsdMs +∫ψs(κGs + αs)d〈M〉s

)t, t ∈ [0, T ]. (3.16)

Definition 3.2.5. (Optimization Problems). Let the initial wealth x > 0.

1. The ordinary investor’s optimization problem is to solve:

maxψ∈AF(x,T )

E[log VT (x, ψ)

].

2. The insider’s optimization problem is to solve:

maxψ∈AG(x,T )

E[log VT (x, ψ)

].


3.2.2 Solution of the logarithmic utility maximization problem

Let us first work out the expression log VT (x, ψ) for ψ ∈ AG(x, T ) and x > 0, Equation (3.16)then gives

log VT (x, ψ) = log x+∫ T

0ψtdMt +

∫ T

0ψt(κGt + αt)d〈M〉t −

12

∫ T

0ψ2t d〈M〉t

= log x+∫ T

0ψtdMt +

∫ T

0ψt(κGt + αt −

12ψt)d〈M〉t

= log x+∫ T

0ψtdMt +

12

∫ T

0(κGt + αt)2d〈M〉t−

− 12

∫ T

0(κGt + αt − ψt)2d〈M〉t. (3.17)

Now if we had E[ ∫ T

0 ψ2t d〈M〉t

]<∞, the local G-martingale

∫ψtdMt would be a true martingale

and hence would have expectation zero. Then ψt = κGt +αt, t ≤ T , would be an optimal strategyfor the insider up to time T , yielding a maximal expected logarithmic utility up to time T of

log x+12E

[ ∫ T

0(κGt + αt)2d〈M〉t

].

Setting κG ≡ 0, and of course M = M we get the optimal strategy and maximal expectedlogarithmic utility for the ordinary investor.

Using the connection between the martingale density processes ZF and ZG and the logarith-mic optimization problem, the solution of the optimization problems is of the above form. Butfirst we give the following proposition.

Proposition 3.2.6.

1. The processes ZFS and ZFV (x, φ) with φ ∈ AF(x, T ) and x > 0 are local (P,F)-martingaleson [0, T ].

2. The processes ZGS and ZGV (x, ψ) with ψ ∈ AG(x, T ) and x > 0 and x > 0 are local(P,F)-martingales on [0, T ].

Proof. We will give the prove for the second claim only as the proof of the first one is just thesame by taking F instead of G and setting κG ≡ 0. Then using Ito’s formula we get

d(ZGS

)t= StdZG

t + ZGt dSt + d〈ZG, S〉t

= −ZFt St

(αt + κG

)dMt + ZG

t St(dMt + (αt + κGt )d〈M〉t

)+

+ ZGt Std

⟨−

∫(α+ κG)dM, M +

∫(α+ κG)d〈M〉

⟩t

= −ZGt St(αt + κGt )dMt + ZG

t StdMt+

+ (αt + κGt )ZFt Std〈M〉t − (αt + κGt )ZF

t Std〈M〉t= ZF

t St(1− (αt + κGt ))dMt,


thus ZGS is a local G-martingale. Using Ito’s formula again we get

d(ZGV (x, ψ)

)t= Vt(x, ψ)dZG

t + ZGt dVt(x, ψ) + d〈ZG, V (x, ψ)〉t

= −ZGt Vt(x, ψ)(αt + κGt )dMt + ZG

t ψtVt(x, ψ)dStSt

+ d〈ZG, V (x, ψ)〉t

= −ZGt Vt(x, ψ)(αt + κGt )dMt + ZF

t Vt(x, ψ)ψt(dMt + (αt + κGt )d〈M〉t

)+

+ ZGt Vt(x, ψ)d

⟨−

∫(αt + κGt )dM,

∫ψdM +

∫(αt + κGt )ψd〈M〉

⟩t

= −ZGt Vt(x, ψ)(αt + κGt )dMt + ZG

t Vt(x, ψ)ψtdMt+

+ ZGt Vt(x, ψ)ψt(αt + κGt )d〈M〉t − ZG

t Vt(x, ψ)ψt(αt + κGt )d〈M〉t= ZG

t Vt(x, ψ)(ψt − (αt + κGt )

)dMt,

whence the process ZGt Vt(x, ψ) is a local G-martingale.

The next theorem gives explicit solutions for the two optimization problems.

Theorem 3.2.7.

1. An optimal strategy up to time T for the ordinary investor is given by

φordt := αt, t ∈ [0, T ], (3.18)

and the corresponding maximal expected logarithmic utility up to time T is

E[log VT (x, φord)

]= log x+

12E

[ ∫ T

0α2t d〈M〉t

]. (3.19)

2. An optimal strategy up to time T for the insider is given by

ψinst := αt + κGt , t ∈ [0, T ], (3.20)

and the corresponding maximal expected logarithmic utility up to time T is

E[log VT (x, ψins)

]= log x+

12E

[ ∫ T

0(α2

t + (κGt )2)d〈M〉t]. (3.21)

Proof. We prove only the second part of the theorem because the first claim is a copy of thesecond by taking ZF instead of ZG and setting κG ≡ 0.

Let ψ ∈ AG(x, T ) be fixed. We will use the following well-known inequality, for concaveC1-function f such that f

′has an inverse g, we have then for all a, b

f(a) ≤ f(g(b))− b(g(b)− a).

Whence setting f(a) = log a, g(b) = 1b , a = VT (x, ψ) and b = yZG

T for some constant y > 0 weobtain

log(VT (x, ψ)) ≤ log(1

yZGT

)− yZGT (

1yZG

T

− VT (x, ψ))

= − log y − logZGT − 1 + yZG

T VT (x, ψ).


By Proposition 3.2.6, ZGV (x, ψ) is a local G-martingale, and is non-negative because V (x, ψ)and ZG are nonnegative, hence ZGV (x, ψ) is a G-supermartingale starting in x because ZG

0 = 1and V0(x, ψ) = x, whence

E[log VT (x, ψins)

]≤ −1− log y −E

[logZG

T

]+ xy (3.22)

for all ψ ∈ AG(x, T ) and y > 0. To find an optimal portfolio, it is then enough to findψ ∈ AF(x, T ) and y > 0 such that we have equality in Equation (3.22). Now we claim that withψins defined by Equation (3.20) and taking y = 1/x > 0 we will have equality in Equation (3.22).Indeed, Equation (3.17) yields

log VT (x, ψins) = log x+∫ T

0ψinst dMt +

12

∫ T

0(κGt + αt)2d〈M〉t−

− 12

∫ T

0(κGt + αt − ψinst )2d〈M〉t

= log x+∫ T

0(κGt + αt)dMt +

12

∫ T

0(κGt + αt)2d〈M〉t (3.23)

Moreover, because of Equation (3.8), Assumption 2.3.5, and the fact that ZGT > 0 P-a.s. so that

log−(ZGT ) we have ψins is in AG(x, T ). Again by Equation (3.8), Doob’s inequality implies that

both sup0≤t≤T∣∣ ∫ t

0 αsdMs

∣∣ and sup0≤t≤T∣∣ ∫ t

0 (αs+κGs )dMs

∣∣ are integrable. Indeed: since∫αdM

is a continuous local (P,F)-martingale we have by Doob’s inequality that(E

[sup

0≤t≤T

∣∣∣ ∫ t

0αsdMs

∣∣∣])2≤ E

[sup

0≤t≤T

( ∫ t

0αsdMs

)2]≤ 4E

[ ∫ T

0α2td〈M〉t

]<∞

hence∫αdM and

∫αdM are (P,F) and (P,G)-martingales on [0, T ] respectively. Then by the

definition of M and κG, we therefore obtain

0 = E[ ∫ t

0αsdMs −

∫ t

0αsdMs

]= E

[ ∫ t

0αsκ

Gs d〈M〉s

],

and hence taking Expectations in both sides of Equation (3.23) we get Equation (3.21):

E[log VT (x, ψins)

]= log x+ E

[ ∫ T

0(κGt + αt)dMt +

12

∫ T


]= log x+

12E

[ ∫ T


]= log x+

12E

[ ∫ T

0α2td〈M〉t +

∫ T

0(κGt )2d〈M〉t+

+ 2∫ T

0(κGt αt)d〈M〉t

]= log x+

12E

[ ∫ T

0α2td〈M〉t

]+

12E

[ ∫ T

0(κGt )2d〈M〉t.

]


Remark 3.2.8. By the above theorem we observe that given the optimal portfolio the optimalterminal wealths of the ordinary investor and the insider are given by

VT (x, φord) =x

ZFTand VT (x, ψins) =

x

ZGT,

respectively.

3.2.3 Insider’s additional expected logarithmic utility

In this subsection we will establish a relationship between the additional expected logarithmicutility of an insider and the relative entropy of the probability measure P with respect to theprobability measure defined by the process 1/pG in Proposition 1.1.14. First we give the followingdefinitions.

Definition 3.2.9. Given a utility function u, the insider’s additional expected utility up to timeT is defined by

maxψ∈AG(x,T )

E[u(VT (x, ψ)

)]− maxφ∈AF(x,T )

E[u(VT (x, φ)

)].

Remark 3.2.10. As a particular case, for u = log we have the insider’s utility gain up to timeT is given by E[aT ] where

aT :=12

∫ T

0(κGt )2d〈M〉t (3.24)

Definition 3.2.11. Let P and Q two probabilities on (Ω,F), such that P Q. The relativeentropy of P with respect to Q on F is defined as

HF(P|Q

):= EP

[log

dPdQ

∣∣∣F

]Remark 3.2.12. It is well-known that HF

(P|Q

)is always non negative and, equal to zero if

and only if P = Q on F , and increasing in F .

Proposition 3.2.13. The insider’s utility gain up to time T is given by

EP[aT ] = HGT

(P|PT

), (3.25)

where the probability P is given by Equation (1.6).

Proof. By taking log of both sides of Equation (2.17) we get for t = T

log pGT =∫ T

0κGt dMt +

12

∫ T

0(κGt )2d〈M〉t (3.26)

Thanks to Equation (3.8) and Assumption 2.3.5, Doob’s inequality implies that

sup0≤t≤T

∣∣ ∫ t

0κGs dMs

∣∣is integrable, hence

∫κGdM is an (P,G)-martingale on [0, T ], therefore taking expectation of

both sides of Equation (3.26) we get

EP[log pGT

]=

12EP

[ ∫ T

0(κGt )2d〈M〉t

].

From the definition of PT we have that pGT = dP/dPT , whence we obtain our result.


3.2.4 Explicit calculations of the insider’s expected logarithmic utility

In this subsection we will calculate the insider’s expected terminal logarithmic utility in theexamples given Chapter 1.

Example 3.2.14. Example 1.3.2 revisited.Recall that G = 1WR∈[a,b], and as observed we have Assumption 2.2.1 is satisfied for everyT < R and we have for t ∈ [0, T ]

p1t =

P[G = 1|Ft]P[G = 1]

and p0t =

1−P[G = 1|Ft]1−P[G = 1]

,

and P[G = 1] = P[G = 1|F0] = Φ(b/√R)−Φ(a/

√R), where Φ is the standard normal distribu-

tion function. In this example the random variable G is discrete. The entropy of G is definedby

H(G) := −P[G = 0] log P[G = 0]−P[G = 1] log P[G = 1], (3.27)

and the conditional entropy of G given Ft, is defined by

H(G|Ft) := −EP[P[G = 0|Ft] log P[G = 0|Ft] + P[G = 1|Ft] log P[G = 1|Ft]

], t ∈ [0, R]

(3.28)Therefore conditioning on FT yields

EP[aT ] = HGT

(P|PT

)= EP

[log pGT

]= EP

[EP

[log pGT |FT

]]= EP

[log(p0

T )P[G = 0|FT ] + log(p1T )P[G = 1|FT ]

]= EP

[P[G = 0|FT ] log P[G = 0|FT ] + P[G = 1|FT ] log P[G = 1|FT ]

]−P[G = 0] log P[G = 0]−P[G = 1] log P[G = 1]

= H(G)−H(G|FT ). (3.29)

Example 3.2.15. Example 1.3.1 revisited in a general setup.Suppose that the insider’s information about the outcome of the Brownian motion W is possiblydistorted by some independent noise, that is he knows the value of

G := γWR + (1− γ)ε,

where the noise ε is a random variable independent of FR and standard normally distributed,and γ ∈ [0, 1]. For T < R, the conditional distribution of G given FT is then normal distributedwith mean mT = γWT and variance σT = γ2(R−T )+(1−γ)2. Indeed repeating the calculationsof Example 1.3.3 with γWR + (1− γ)ε instead of WR + ε we have that

P[G ∈ dl|FT ] = P[(γWR − γWT + γWT + (1− γ)ε) ∈ dl|FT ]

= P[(γ(WR −WT ) + (1− γ)ε) ∈ (dl − y)]|y=γWT

=1√

2π(γ2(R− T ) + (1− γ)2)exp

(− (l − γWT )2

2(γ2(R− T ) + (1− γ)2))dl

= plTP[(γWT + (1− γ)ε) ∈ dl]

= plTP[G ∈ dl],


where

plT =

√γ2R+ (1− γ)2

γ2(R− T ) + (1− γ)2exp

(− (l − γWT )2

2(γ2(R− T ) + (1− γ)2)+

+l2

2(γ2R+ (1− γ)2)

), l ∈ R.

Now applying Proposition 3.25 we obtain

EP[aT ] = HGT

(P|PT

)= EP

[log pGT

]= EP

[log

(√γ2R+ (1− γ)2

γ2(R− T ) + (1− γ)2exp

(− (G− γWT )2

2(γ2(R− T ) + (1− γ)2)+

+G2

2(γ2R+ (1− γ)2)

))]= EP

[log

(√γ2R+ (1− γ)2

γ2(R− T ) + (1− γ)2)− (G− γWT )2

2(γ2(R− T ) + (1− γ)2)+

+G2

2(γ2R+ (1− γ)2)

]=

12

log(1 +

γ2T

γ2(R− T ) + (1− γ)2)−EP

[(γ(WR −WT ) + (1− γ)ε)2

2(γ2(R− T ) + (1− γ)2)

]+

+ EP[(γWR + (1− γ)ε)2

2(γ2R+ (1− γ)2)

]EP[aT ] =

12

log(1 +

γ2T

γ2(R− T ) + (1− γ)2). (3.30)

If γ = 1 then we have Example 1.3.1, and we have

EP[aT ] =12

log(1 +

T

(R− T )

),

and if we let T R we have that EP[aT ] = ∞. Intuitively, this is true. Because the more theinsider knows about the outcomes of the random variable G, the bigger his expected utility gainbecomes.

Example 3.2.16. Example 1.3.3 revisited.Recall that we suppose that the insider’s information about the outcome of the Brownian motionW is distorted by an independent noise, that is he knows the value of

G := WR + ε,

where the noise ε is a random variable independent of FR and standard normally distributed.For T ≤ R, the conditional distribution of G given FT is then normal distributed with meanmT = WT and variance σT = (R − T ) + 1 and is equivalent to the law of G and their densityprocess is given by the calculations of Example 1.3.3, that is

plT =

√R+ 1

R− T + 1exp

(− (l −WT )2

2(R− T + 1)+

l2

2(R+ 1)), l ∈ R.


Now applying Proposition 3.25 we obtain

EP[aT ] = HGT

(P|PT

)= EP

[log pGT

]= EP

[log

(√R+ 1

R− T + 1exp

(− (G−WT )2

2(R− T + 1)+

G2

2(R+ 1)

))]= log

(√R+ 1

R− T + 1

)+ EP

[− (G−WT )2

2(R− T + 1)+

G2

2(R+ 1)

]=

12

log(1 +

T

R− T + 1

)+ EP

[− (WR −WT + ε)2

2(R− T + 1)

]+ EP

[(WR + ε)2

2(R+ 1)

]EP[aT ] =

12

log(1 +

T

2(R− T + 1)

). (3.31)

If we let T R we have that the utility gain of the insider up to time R is

EP[aR] =12

log(1 +

R

2

).

Chapter 4

Enlargement of filtrations andGirsanov’s theorem

In the previous first two chapters we have treated the case of initial enlargement of filtrationsby the information represented by a random variable G. In this chapter we will reconsider thesame problem but in a more general perspective, from a different angle.

4.1 Preliminaries and notations

Let (Ω,F ,P) be a probability space with right continuous filtrations F = (Ft)t≥0 and K =(Kt)t≥0. Moreover, let F∞ =

∨t≥0Ft and K∞ =

∨t≥0Kt. The objective is to study the

enlarged filtration G defined by

Gt := ∩s>t(Fs ∨ Ks), t ≥ 0.

We relate this enlargement to a measure change on the product space Ω := Ω×Ω equipped withthe σ-algebra F := F∞ ⊗K∞. We endow Ω with the filtration F = (Ft)t≥0 where

Ft := ∩s≥t(Fs ⊗Ks), t ≥ 0.

We embed Ω into Ω by the following map

ϕ : (Ω,F) → (Ω, F), ω 7→ (ω, ω).

Moreover, we define a probability measure on the measurable space (Ω, F) denoted by P as theimage of the probability measure P under ϕ, i.e. P := Pϕ. Hence for all F-measurable functionsf : Ω → R we have ∫

f(ω, ω′)dP(ω, ω′) =∫f(ω, ω)dP(ω). (4.1)

At last we define the following probability measure Q on the measurable space (Ω, F) by

Q = P|F∞ ⊗P|K∞ ,

as the product measure of the restrictions of P to F∞ and K∞ respectively.Since we are dealing with different probability measures, and most of the results only hold for

33


completed filtrations, we use the following notation to specify to which probability measurecompletion is taken. Let (Ct)t≥0 be a filtration and R a probability measure. We denote by(CRt )t≥0 the filtration (Ct)t≥0 completed by the R-negligible sets.We first start with a simple observation.

Lemma 4.1.1. If f : Ω → R is F Pt -measurable, then the map f ϕ is GP

t -measurable.

Proof. First, observe that

Gt = ∩s>tσ(A ∩B : A ∈ Fs, B ∈ Ks)= ∩s>tσ(ϕ−1(A×B) : A ∈ Fs, B ∈ Ks)= ϕ−1

(∩s>t σ(A×B : A ∈ Fs, B ∈ Ks)

)= ϕ−1

(∩s≥t (Fs ⊗Ks)

)= ϕ−1(Ft).

Let f = 1A with A ∈ F Pt . Then there is a set B ∈ Ft such that P(A M B) = 0. From the

first part we deduce that the map 1B ϕ is Gt-measurable. And since we have P-almost surly1A ϕ = 1B ϕ, the map 1A ϕ is GP

t -measurable. Define S by

S := g : g is bounded F Pt -measurable and g ϕ is GP

t -measurable

S is a vector space and we have 1 ∈ S. Now let (hn)n be a sequence of function in S such that0 ≤ hn ↑ h and h is bounded. Therefore h is F P

t -measurable and we have that 0 ≤ hn ϕ ↑ hϕhence h ϕ is GP

t -measurable, whence h ∈ S. Since S contains the indicator functions, then thestatement is true for arbitrary F P

t -measurable functions.

Lemma 4.1.2. If X is FP-predictable, then X ϕ is GP-predictable.

Proof. Let 0 < s ≤ t, A ∈ F Ps and ϑ = 1A1]s,t], then by Lemma 4.1.1, ϑ ϕ = (1A ϕ)1]s,t] is

GP-predictable. Then the proof is completed by a monotone class argument.

Lemma 4.1.3. Let X be an FP-adapted process, then X = X ϕ is an GP-adapted process.Moreover, if X is a local FP-martingale, then X is a local GP-martingale.

Proof. The first part of the Lemma concerning adaptation is a consequence of Lemma 4.1.1. Forthe second part, let X be an FP-martingale, and let 0 ≤ s < t and A ∈ Gs. Then there exists aset B ∈ Fs such that A = ϕ−1(B) and hence

EP[1A(Xt −Xs)] = EP[1A(Xt − Xs)] = 0.

Whence X is a GP-martingale.Now for the case of a local martingales it is enough to show that if (Tn)n≥0 is a localizing sequenceof F P

t -stopping times, then Tn = Tn ϕ for all n ≥ 0 is a localizing sequence of GPt -stopping

times. Indeed, this is the case since

Tn ≤ t = ϕ−1(Tn ≤ t) ∈ ϕ−1(F Pt ) ⊂ GP

t .


Theorem 4.1.4. Let S be a cadlag (P, FP)-semimartingale, then S := Sϕ is a cadlag (P,GP)-semimartingale.

Proof. Let S be a FP-semimartingale and S = S ϕ. The process S has cadlag paths P-a.s.and by Lemma 4.1.3 we have that S is GP-adapted. By the Theorem of Bichteler-Dellacherie-Mokobodsky, see Theorem A.3.1 in the Appendix, it is sufficient to show that if (θn) is a sequenceof simple G-adapted integrands converging uniformly to 0, then the sequence of simple integrals(θn · S) converges to 0 in probability relative to P. We recall that any G-simple integrand is ofthe form

∑1≤i≤n 1]ti,ti+1]θi, with θi is Gti-measurable. Since Gt = ϕ−1(Ft) we can find simple

(F)-adapted processes (θn) converging uniformly to zero such that θn ϕ = θn. Since S is asemimartingale this implies that the sequence (θn · S) converges to zero in probability relativeto P, and hence (θn · S) converges to 0 in probability relative to P.

Until now we have seen how objects can be translated from Ω to Ω. Now we look at thereverse transfer.

Lemma 4.1.5. Let M be a right-continuous local (P,FPt )-martingale, then the process defined

by M(ω, ω′) := M(ω) is a right continuous local (Q, FQt )-martingale.

Proof. That M is FQ-adapted is obvious. For A ∈ Fs, B ∈ Ks we have

EQ[1A(ω)1B(ω′)(Mt − Ms)] = P(B)EP[1A(Mt −Ms)] = 0.

Then by the monotone class theorem, for all bounded Fs ⊗Ks-measurable functions θ we haveEQ[θ(Mt− Ms)] = 0. Since M is right-continuous, this remains true for all bounded ∩u>s(Fu⊗Ku)-measurable θ, whence M is a (Q, FQ

t )-martingale.The stopping times can be extended to the product space via the transformation T (ω, ω′) =T (ω). Therefore we get That

MT (ω,ω′)t (ω, ω′) = M

T (ω,ω′)t (ω) = M

T (ω)t (ω)

whence the local martingale property translates to the product space Ω with respect to Q.

4.2 Girsanov-type for change of filtrations

In the rest of this chapter we will assume the following

Assumption 4.2.1. The probability measure P is absolutely continuous with respect to themeasure Q on F , i.e. P Q.

Now let M be a local (P,FP)-martingale and M its extension to Ω as given in Lemma 4.1.5.By Assumption 4.2.1, M is a (P, FP)-semimartingale and hence, by theorem 4.1.4, M is a(P,GP)-semimartingale. Thus, l’hypothese (H’) is satisfied. But what does the local martingalepart of M relative to (P,GP) look a like?

The change of filtrations corresponds to changing the measure from Q to P on the productspace Ω. Then using Assumption 4.2.1 Girsanov’s theorem applies on Ω. As a consequence we


obtain a Girsanov-type result for the corresponding change of filtrations. To get this, we firstintroduce the density process. Let Z = (Zt)t≥0 denote a cadlag FQ-adapted process with

Zt :=dPdQ

∣∣∣FQ

t

, t ≥ 0.

Note that we consider the completed filtration in order to insure the existence of a cadlag densityprocess. Then Theorem 4.1.4 implies that the process Z defined by Z = Z ϕ is a (P,GP)-semimartingale. Before giving the Girsanov-type results, we study the behavior of quadraticvariation processes under the projection ϕ.

Lemma 4.2.2. Let X and Y be continuous (P, FP)-semimartingales. If X = X ϕ and Y =Y ϕ, then

〈X, Y 〉 ϕ = 〈X,Y 〉

up to indistinguishability relative to P.

Proof. Set X = X ϕ and Y = Y ϕ. Let t > 0 and tni = t i2n for i = 0, 1, . . . , 2n. We know thatthe sums

X0Y0 +2n∑i=0

(Xtni+1− Xtni

)(Ytni+1− Ytni )

converges to 〈X, Y 〉t in probability relative to P. Hence 〈X, Y 〉t ϕ is the limit in probabilityof the sums

X0Y0 +2n∑i=0

(Xtni+1−Xtni

)(Ytni+1− Ytni )

relative to P. But the limit is also equal to 〈X,Y 〉t, and hence we have

〈X, Y 〉t ϕ = 〈X,Y 〉t

Since both processes are continuous, they coincide up indistinguishability relative to P.

Let M be a continuous (Q, FQ)-semimartingale and M = M ϕ. By Assumption 4.2.1,M is also a continuous (P, FP)-semimartingale. Moreover, the process 〈M, Z〉 relative to Q isP-indistinguishable from the one relative to P. Similarly, Lemma 4.2.2 implies that the process〈M,Z〉 coincides with 〈M, Z〉 ϕ.

Theorem 4.2.3. If M is a continuous local (P,FP)-martingale with M0 = 0, then

M − 1Z· 〈M,Z〉 (4.2)

is a continuous local (P,GP)-martingale.

Proof. Let M be a continuous local (P,FP)-martingale with M0 = 0. Then by Lemma 4.1.5, theprocess defined by M(ω, ω′) = M(ω) is a continuous local (Q, FQ)-martingale. Then Girsanov’stheorem yields that the process

M − 1Z· 〈M, Z〉


is a continuous local (P, FP)-martingale. By Lemma 4.2.2, the process 〈M, Z〉 ϕ is P-indistinguishable from the process 〈M,Z〉, whence we have

(1Z· 〈M, Z〉) ϕ =

1Z· 〈M,Z〉

up to P-indistinguishability. Finally with Lemma 4.1.3 we have that

M − 1Z· 〈M,Z〉 (4.3)

is a continuous local (P,GP)-martingale.

4.3 Conclusion

In the preceding chapters the filtration is supposed to be enlarged by some random variable Gtaking values in a Polish space (U,U). As a consequence, for every t ∈ [0, T ] regular conditionaldistributions of G given Ft exist. The development in chapter 1 is based on the paper of Jacod [9]who does not use Girsanov’s theorem, but he assumed the condition (A’) to be satisfied. And hepointed out that his result could be obtained by applying Girsanov’s theorem to the conditionalmeasures Pl = P[·|G = l], l ∈ U . Condition (A’) implies that the conditional measures Pl areabsolutely continuous with respect to P. Hence, by Girsanov, for a given local (P,Ft)-martingaleM there is a process Al such that M − Al is a local (P,Ft)-martingale. By combining theprocesses Al we obtain that M −AG is a local (P,Gt)-martingale. But combining the processesAl in a meaningful way has never worked out rigorously.

In the approach of Ankirchner [4], every local martingale is embedded into the product spaceΩ. He applies Girsanov’s theorem on the product space and then translate the results back intothe original space. One of the advantages is that there is no need to assume that the regularconditional distributions do exist, and there is also no need to show how the processes can becombined. Instead he shows how one can translate objects from Ω to Ω and vice versa. Moreoverthere is no restriction to the initial enlargement, also dynamic enlargements of the kind:

Gt = ∩s>t(Fs ∨ Ks), t ≥ 0

can be treated.

Appendix A

Some Important Theorems andLemma’s

In this appendix we give the most important theorems in abstract probability theory and thetheory of stochastic processes that we have used in the proves of some results.

A.1 Appendix Chapter 1

The following theorem and corollary assure the existence of the regular conditional probabilities.

Theorem A.1.1 (Shiryaev [15], Theorem 5).Let X be a random variable in (Ω,A,P) with values in a Borel space (E, E). Then there is aregular conditional probability of X with respect to mathcalA.

Corollary A.1.2 (Shiryaev [15], Corollary that follows Theorem 5).Let X be a random variable in (Ω,A,P) with values in a complete separable metric space (E, E).Then there is a regular conditional distribution of X with respect to A. In particular such adistribution exists for the spaces (Rn,B(Rn)) and (R∞,B(R∞)).

Theorem A.1.3 (Theoreme de Doob, Dellacherie [5]).Let x 7→ Px be a Markovian Kernel from (X,X ) into (Y,Y) and let x 7→ Qx be a bounded kernelfrom (X,X ) into (Y,Y), such that Qx is absolutely continuous with respect to Px for all x ∈ X.Then there exists a non-negative X ⊗ Y-measurable function g, such that

Q(x, dy) = g(x, y)P(x, dy), for every x ∈ X.

furthermore, we can assume that g(x, ·) = g(ξ, ·) for every couple (x, ξ) of elements of X suchthat Qx = Qξ and Qx = Qξ.

Theorem A.1.4 (Williams [16], Monotone Class Theorem for Functions).Let H be a class of bounded functions from Ω into R satisfying the following conditions:

(i) H is a vector space over R;

(ii) the constant function 1 is an element of H;

38


(iii) if (fn)n is a sequence of non negative functions in H such that fn ↑ f where f is a boundedfunction on Ω, Then f ∈ H.

Then if H contains the indicator function of every set in some π-system I, then H containsevery bounded σ-measurable function on S.


Theorem A.2.1 (Jacod [8], Thereme 7).Let Y be a semimartingale relative to F and G. Every F-predictable process ϑ ∈ Lsm(Y,QG,F)is in Lsm(Y,QG,G), and the integral stochastic

∫ϑdY relative to F is the same relative to G.

Corollary A.2.2 (Jacod [7], Corollaire 9.21).Let F and G be two filtrations on the probability space (Ω,A,P) such that F is a subfiltration ofG and let X ∈Mloc(G) ∩Mloc(F) and q ∈ [1,∞). Then

Lqloc(X,G) = P(G) ∩ Lqloc(X,F).

If H ∈ P(G)∩Lqloc(X,F), there exists then a common version∫HdX of the processes F-

∫HdX

and G-∫HdX.

Theorem A.2.3 (Malliavin [11], Theorem 3.5.1).Let G and F be two sub-σ-algebras of the probability space (Ω,A,P) and let H denote the σ-algebra they generate. Let V be the vector subspace of L∞(A) defined by

V =h ∈ L∞(A) : h =

n∑i=1

figi, with fi ∈ L∞(F), gi ∈ L∞(G).

Then V ⊂ L2(H) and V is dense in L2(H).

Theorem A.2.4 (He, Wang and Yan [6], Theorem 10.5).The collection of all bounded martingales denoted by M∞ is dense in H1.

Theorem A.2.5 (He, Wang and Yan [6], Theorem 13.4).Assume M ∈Mloc,0. Then the following statements are equivalent:

(1) L(M) = Mloc,0, i.e., M has the strong property of predictable representation,

(2) L1(M) = M10,

(3) M∞0 ⊂ L(M).


(Ω,F ,F,P) is a filtered probability space. Let H be the vector space of bounded predictableprocesses H of the form

H =n−1∑i=1

hi1]ti,ti+1],


where 0 < t1 < t2 < . . . < tn and hi is bounded Fti-measurable.If X is a real process, we associate to every H ∈ H the process:

J(X,H)t =n−1∑i=1

hi(Xt∧ti+1 −Xt∧ti).

The following theorem is the characterization of semimartingales:

Theorem A.3.1 (Jacod [7], Theorem 9.3, p.279).An F-adapted, cadlag process X is a semimartingale, if and only if for every sequence (Hn)n≥0

of elements of H converging uniformly to H ∈ H and for every t ≥ 0, the variables J(X,Hn)tconverges in probability to J(X,H)t.

Index

(Ω,F ,P) 1F 1G 1(U,U) 1G 1S 1H, H0, H 2Ω 2O(H0) 2P(H0) 3pl 4pG, 1/pG 4P 4M(R,H) 10Hp(R,H) 10H∞(R,H) 10Lp(M,R,H) 10Lsm(S,R,H) 10H2loc(Q

F,F) 10QF 10ZF 10ZG 11QG 11V (x, π) 25AH(x, T ) 25HF

(P|Q

)29

41

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[2] Amendinger, J., Becherer, D., Schweizer, M., (2003): A monetary value for initial informa-tion in portfolio optimization. Finance and Stochastics 7, 29-46.

[3] Amendinger, J., Imkeller, P., Schweizer, M., (1998): Additional logarithmic utility of aninsider. Stochastic Processes and their Applications 75, 263-286.

[4] Ankirchner, S., Dereich, S., Imkeller, P., (2004): Enlargement of filtrations and continuousGirsanov-type embeddings.

[5] Dellacherie, C., (1979/1980): Sur les noyeaux σ-finis. Seminaire de Probabilites XV, Lec-tures Notes in Mathematics. Vol. 850. Springer-Verlag. pp. 371-387.

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[9] Jacod, J., (1985): Grossissement Initial, Hypothese (H’) et Theoreme de Girsanov. In:Jeulin, Th., Yor, M. (eds.), Grossissement de filtration: exemples et applications, LectureNotes in Mathematics, 1118. Springer-Verlag.

[10] Jeulin, Th., (1980): Semi-Martingales et Grossissement d’une Filtration. Lecture Notes inMathematics, 833. Springer-Verlag.

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