financial markets with asymmetric information

8/9/2019 Financial Markets With Asymmetric Information

1/21

Financial Markets with Asymmetric Information:Information Drift, Additional Utility and Entropy

Stefan Ankirchner and Peter Imkeller

Institut fur Mathematik, Humboldt-Universitat zu Berlin,

Unter den Linden 6, 10099 Berlin, Germany

We review a general mathematical link between utility and infor-mation theory appearing in a simple financial market model withtwo kinds of small investors: insiders, whose extra informationis stored in an enlargement of the less informed agents filtration.The insiders expected logarithmic utility increment is describedin terms of the information drift, i.e. the drift one has to eliminatein order to perceive the price dynamics as a martingale from hisperspective. We describe the information drift in a very generalsetting by natural quantities expressing the conditional laws of the

better informed view of the world. This on the other hand allows toidentify the additional utility by entropy related quantities knownfrom information theory.Key words: enlargement offiltration; logarithmic utility; utilitymaximization; heterogeneous information; insider model; Shannoninformation; information difference; entropy.

2000 AMS subject classifications: primary 60H30, 94A17; sec-ondary 91B16, 60G44.

1. IntroductionA simple mathematical model of two small agents on a financial mar-

ket one of which is better informed than the other has attracted muchattention in recent years. Their information is modelled by two differentfi

ltrations: the less informed agent has the field Ft, corresponding tothe natural evolution of the market up to time t at his disposal, whilethe better informed insider knows the bigger field Gt Ft. Here is ashort selection of some among many more papers dealing with this model.Investigation techniques concentrate on martingale and stochastic controltheory, and methods of enlargement offiltrations (see Yor , Jeulin , Jacod in[22]), starting with the conceptual paper by Duffie, Huang [12]. The model

1


2/21

2

is successively studied on stochastic bases with increasing complexity:e.g. Karatzas, Pikovsky [24] on Wiener space, Grorud, Pontier [15] allowPoissonian noise, Biagini and Oksendal [7] employ anticipative calculustechniques. In the same setting, Amendinger, Becherer and Schweizer [1]calculate the value of insider information from the perspective of specificutilities. Baudoin [6] introduces the concept of weak additional informa-tion, while Campi [8] considers hedging techniques for insiders in theincomplete market setting. Many of the quoted papers deal with the cal-culation of the better informed agents additional utility.

In Amendinger et al. [2], in the setting of initial enlargements, the addi-tional expected logarithmic utility is linked to information theoretic con-cepts. It is computed in terms of an energy-type integral of the informationdrift between the filtrations (see [18]), and subsequently identified withthe Shannon entropy of the additional information. Also for initial en-largements, Gasbarra, Valkeila [14] extend this link to the Kullback-Leiblerinformation of the insiders additional knowledge from the perspectiveof Bayesian modelling. In the environment of this utility-informationparadigm the papers [16], [19], [17], [18], Corcueraet al. [9], and Ankirchner

et al. [5] describe additional utility, treat arbitrage questions and their inter-pretation in information theoretic terms in increasingly complex modelsof the same base structure. Utility concepts different from the logarith-mic one correspond on the information theoretic side to the generalizedentropy concepts of fdivergences.

In this paper we review the main results about the interpretation of thebetter informed traders additional utility in information theoretic terms

mainly developed in [4], concentrating on the logarithmic case. This leadsto very basic problems of stochastic calculus in a very general setting ofenlargements offiltrations: to ensure the existence of regular conditionalprobabilities offields of the larger with respect to those of the smallerfiltration, we only eventually assume that the base space be standard Borel.In Section 2, we calculate the logarithmic utility increment in terms of theinformation drift process. Section 3 is devoted to the calculation of the in-formation drift process by the Radon-Nikodym densities of the stochastic

kernel in an integral representation of the conditional probability processand the conditional probability process itself. For convenience, before pro-ceeding to the more abstract setting of a general enlargement, the resultsare given in the initial enlargement frameworkfirst. In Section 4 we finallyprovide the identification of the utility increment in the general enlarge-ment setting with the information difference of the two filtrations in termsof Shannon entropy concepts.


3/21

3

2. Additional Logarithmic Utility and Information DriftLet us first fix notations for our simple financial market model. First of

all, to simplify the exposition, we assume that the trading horizon is givenby T = 1. Let (, F, P) be a probability space with a filtration (Ft)0t1.We consider a financial market with one non-risky asset of interest ratenormalized to 0, and one risky asset with price Xt at time t [0, 1]. Weassume that X is a continuous (Ft)semimartingale with values in R andwrite A for the set of all Xintegrable and (Ft)predictable processes such that 0 = 0. If A, then we denote by ( S) the usual stochasticintegral process. For all x > 0 we interpret

x + ( X)t, 0 t 1,as the wealth process of a trader possessing an initial wealth x and choos-ing the investment strategy on the basis of his knowledge horizon corre-sponding to the filtration (Ft).Throughout this paper we will suppose the preferences of the agents to bedescribed by the logarithmic utility function.

Therefore it is natural to suppose that the traders total wealth hasalways to be strictly positive, i.e. for all t

[0, 1]

(1) Vt(x) = x + ( X)t > 0 a.s.Strategies satisfying Eq. (1) will be called xsuperadmissible. The agentswant to maximize their expected logarithmic utility from terminal wealth.So we are interested in the exact value of

u(x) = sup{E log(V1(x)) : A, x superadmissible}.

Sometimes we will write uF(x), in order to stress the underlying filtration.The expected logarithmic utility of the agent can be calculated easily, if onehas a semimartingale decomposition of the form

(2) Xt = Mt +

t0

s dM,Ms,

where is a predictable process. Such a decomposition has to be expected

in a market in which the agent trading on the knowledgefl

ow (Ft) has noarbitrage opportunities. In fact, ifX satisfies the property (NFLVR), then itmay be decomposed as in Eq. (2) (see [10]). It is shown in [3] thatfinitenessof u(x) already implies the validity of such a decomposition. Hence adecomposition as in (2) may be given even in cases where arbitrage exists.We state Theorem 2.9 of [5], in which the basic relationship between optimallogarithmic utility and information related quantities becomes visible.


4/21

4

Proposition 2.1. Suppose X can be decomposed into X = M+ M,M. Thenfor any x > 0 the following equation holds

(3) u(x) = log(x) +1

2E

1

0

2s dM,Ms.

Letusgivethecoreargumentsprovingthisstatementinaparticularsetting,and for initial wealth x = 1. Suppose that X is given by the linear sde

dXtXt

= tdt + dWt,

with a one-dimensional Wiener process W , and assume that the smalltraders filtration (Ft) is the (augmented) natural filtration of W. Here is a progressively measurable mean rate of return process which satisfies1

0|t|dt < , Pa.s. Let us denote investment strategies per unit by , so

that the wealth process V(x) is given by the simple linear sde

dVt(x)

Vt(x)

= t

dXt

Xt.

It is obviously solved by the formula

Vt(x) = exp[

t0

s dWs 12

t0

2s ds +

t0

s s ds].

Due to the local martingale property of

t

0s dWs, t [0, 1], the expected

logarithmic utility of the regular trader is deduced from the maximizationproblem

(4) uF(1) = max

E[

10

s s ds 12

10

2s ds].

The maximization of

1

0 s s ds 1

21

0 2

s ds

for given processes is just a more complex version of the one-dimensionalmaximization problem for the function

12

2


5/21

5

with R. Its solution is obtained by the critical value = and thus

(5) uF(1) =1

2 E[

1

0 2

s ds].

This confirms the claim of Proposition 2.1.This proposition motivates the following definition.

Definition 2.1. A filtration (Gt) is called finite utility filtration for X, ifX isa (Gt)semimartingale with decomposition dX = dM + dM,M, where is (Gt)predictable and belongs to L2(M), i.e. E

10

2 dM,M < . Wewrite

F = {(Ht) (Ft)(Ht) is a finite utility filtration for X}.We now compare two traders who take their portfolio decisions not on the

basis of the same filtration, but on the basis of different information flowsrepresented by thefiltrations (Gt) and (Ht) respectively. Suppose that bothfiltrations (Gt) and (Ht) are finite utility filtrations. We denote by(6) X = M + M,Mthe semimartingale decomposition with respect to (

Gt) and by

(7) X = N+ N, Nthe decomposition with respect to (Ht). Obviously,

M,M = X, X = N, Nand therefore the utility difference is equal to

uH

(x)

uG

(x) =1

2E

1

0

(2

2) d

M,M

.

Furthermore, Eqs. (6) and (7) imply

(8) M = N ( ) M,M a.s.IfGt Ht for all t 0, Eq. (8) can be interpreted as the semimartingaledecomposition ofM with respect to (Ht). In this case one can show thatthe utility difference depends only on the process = . In fact,

uH(x) uG(x) = 12 E1

0(2 2) dM,M

=1

2E(

10

2 dM,M) E(1

0

dM,M)

=1

2E(

10

2 dM,M).


6/21

6

The last equation is due to the fact that NM =

dM,M is a martingalewith respect to (Ht), and is adapted to thisfiltration. It is therefore naturalto relate to a transfer of information.

Definition 2.2. Let (Gt) be a finite utility filtration and X = M + M,Mthe Doob-Meyer decomposition of X with respect to (Gt). Suppose that(Ht) is a filtration such that Gt Ht for all t [0, 1]. The (Ht)predictableprocess satisfying

M

0

t dM,Mt is a (Ht) local martingale

is called information drift (see [18]) of (Ht) with respect to (Gt).

The following proposition summarizes the findings just explained, andrelates the information drift to the expected logarithmic utility increment.

Proposition 2.2. Let (Gt) and (Ht) be two finite utility filtrations such thatGt Ht for all t [0, 1]. If is the information drift of(Ht) w.r.t. (Gt), then wehave

uH(x) uG(x) = 12 E1

02 dM,M.

3. The Information Drift and the Law of Additional InformationIn this section we aim at giving a description of the information drift

between two filtrations in terms of the laws of the information incrementbetween two filtrations. This is done in two steps. First, we shall consider

the simplest possible enlargement of filtrations, the well known initialenlargement. In a second step, we shall generalize the results available inthe initial enlargement framework. In fact, we consider general pairs offiltrations, and only require the state space to be standard Borel in order tohave conditional probabilities available.3.1 Initial enlargement, Jacods condition

In this setting, the additional information in the larger filtrations is atall times during the trading interval given by the knowledge of a random

variable which, from the perspective of the smallerfi

ltration, is knownonly at the end of the trading interval. To establish the concepts in fairsimplicity, we again assume that the smaller underlying filtration (Ft) isthe augmentedfiltration of a one-dimensional Wiener process W. Let G bean F1measurable random variable, and let

Gt = Ft (G), t [0, 1].


7/21

7

Suppose that (Gt) is small enough so that W is still a semimartingale withrespect to this filtration. More precisely, suppose that there is an informa-tion drift G such that

10

|Gs | ds < P-a.s.,

and such that

(9) W= W+

.0

Gs ds

with a (Gt) Brownian motion W. To clarify the relationship between theadditional information G and the information drift G , we shall workunder a condition concerning the laws of the additional information Gwhich has been used as a standing assumption in many papers dealingwith grossissement de filtrations. See Yor [27], [26], [28], Jeulin [21]. Thecondition was essentially used in the seminal paper by Jacod [20], andin several equivalent forms in Follmer and Imkeller [13]. To state andexploit it, let us first mention that all stochastic quantities appearing in the

sequel, often depending on several parameters, can always be shown topossess measurable versions in all variables, and progressively measurableversions in the time parameter (see Jacod [20]).

Denote by PG the law ofG , and for t [0, 1], , by PGt (, dl) theregular conditional law ofG given Ft at . Then the condition, whichwe will call Jacods condition, states that

(10) PGt (, dg) is absolutely continuous with respect to PG(dg) for P a.e. .

Also its reinforcement

(11) PGt (, dg) is equivalent to PG(dg) for P a.e. ,

will be of relevance. Denote the Radon-Nikodym density process of theconditional laws with respect to the law by

pt(, g) =dPGt (, )

dPG(g), g R, .

By the very definition, t Pt(, dg) is a local martingale with values in thespace of probability measures on the Borel sets ofR. This is inheritedto t pt(, g) for (almost) all g R. Let the representations of thesemartingales with respect to the (Ft)Wiener process Wbe given by

pt(, g) = p0(, g) +t

0

kgu dWu, t [0, 1]


8/21

8

with measurable kernels k. To calculate the information drift in terms ofthese kernels, take s, t [0, 1], s t, and let A Fs and a Borel set B on thereal line determine the typical set A

G

1[B] in a generator of

Gs. Then we

may write

E([Wt Ws] 1A 1B(G)) = E(

B

1A [Wt Ws] PGt (, dg))

=

B

E(1A [Wt Ws] [pt ps](, g)) PG(dg)

= B

E(1A t

s

kgu du) P

G(dg)

=

B

E(1A

ts

kgu

pu(, g) pu(, g) du) PG(dg)

=

B

E(1A

ts

kgu

pu(, g) du pt(, g)) PG(dg)

= E(

B

1Ak

gu

pu(, g) PGt (, dg))

= E(1A 1B(G)

t

s

kgupu(, g) |g=G du).

The bottom line of this chain of arguments shows that

W= W

0

klupu(, g) |g=G du

i s a (G

t)

martingale, hence a (G

t)

Brownian motion provided that10

| kgupu(,g) |g=G| du < Pa.s.. This completes the deduction of an explicitformula for the information drift ofG in terms of quantities related to thelaw ofG in which we use the common oblique bracket notation to denotethe covariation of two martingales (for more details see Jacod [20]).

Theorem 3.1. Suppose that Jacods condition (10) is satisfied, and furthermorethat

(12) Gt = kg

t

pt(, g) |g=G =ddt p(, g), Wt

pt(, g) |g=G, t [0, 1],

satisfies

(13)

10

|Gu | du < Pa.s..


9/21

9

Then

W= W+

0

Gs ds

is a Gsemimartingale with a GBrownian motion W.To see how restrictive condition (10) may be, let us illustrate it by

looking at two possible additional information variables G.

Example 1:Let > 0 and suppose that the stock price process is a regular diffusiongiven by a stochastic differential equation with bounded volatility and

drift , t = (Xt), t [0, 1], where is a smooth function without ze-roes. Let G = X1+. Then in particular X is a time homogeneous Markovprocess with transition probabilities Pt(x, dy), x R+, t [0, 1], which areequivalent with Lebesgue measure onR+. For t [0, 1], the regular condi-tional law ofG given Ft is then given by P1+t(Xt, dy), which is equivalentwith the law ofG. Hence in this case, even the strong version of Jacodshypothesis (11) is verified.

Example 2:

LetG = sup

t[0,1]Wt.

To abbreviate, denote for t [0, 1]

Gt = sup0st

Ws, G1t = supts1

(Ws Wt).

Finally, let p1

t denote the density function ofG1

t. Then we may write forevery t [0, 1]

(14) G = Gt [Wt + G1t].

Now Gt is Ftmeasurable, independent of G1t , and therefore for Borelsets A on the real line we have

(15) PGt (

, A) =

GtWt

p1

t(y)dy

Gt (A) +

A[GtWt,[p1

t(y)dy.

Note now that the family of Dirac measures in the first term of (15) issupported on the random points Gt, and that the law ofGt is absolutelycontinuous with respect to Lebesgue measure on R+. Hence there cannot

be any common reference measure equivalent with Gt Pa.s. Therefore inthis example Jacods condition is violated.


10/21

10

It can be seen that there is an extension of Jacods framework intowhich example 2 still fits. This is explained in [18], [19], and resides ona version of Malliavins calculus for measure valued random elements. Ityields a description of the information drift in terms of traces of logarithmicMalliavin gradients of conditional laws ofG. We shall not give details here,since we will go a considerable step ahead of this setting. In fact, in thefollowing subsection we shall further generalize the framework beyondthe Wiener space setting.

3.2 General enlargementAssume again that the price process X is a semimartingale of the form

X = M + M,Mwith respect to afinite utility filtration (Ft). Moreover, let (Gt) b e afiltrationsuch that Ft Gt, and let be the information drift of (Gt) relative to (Ft).We shall explain how the description of by basic quantities related tothe conditional probabilities of the larger algebras Gt with respect tothe smaller ones Ft, t 0 generalizes from the setting of the previoussubsection. Roughly, the relationship is as follows. Suppose for all t 0there is a regular conditional probability Pt(, ) ofF given Ft, which canbe decomposed into a martingale component orthogonal to M , plus acomponent possessing a stochastic integral representation with respect to

M with a kernel function kt(, ). Then, provided is square integrable withrespect to dM,M P, the kernel function at t will be a signed measure inits set variable. This measure is absolutely continuous with respect to theconditional probability itself, if restricted to Gt, and coincides with theirRadon-Nikodym density.

As a remarkable fact, this relationship also makes sense in the reversedirection. Roughly, if absolute continuity of the stochastic integral ker-nel with respect to the conditional probabilities holds, and the Radon-Nikodym density is square integrable, the latter turns out to provide aninformation drift in a Doob-Meyer decomposition of X in the largerfiltration.

To provide some details of this fundamental relationship, we need towork with conditional probabilities. We therefore assume that (, F, P) isstandard Borel (see [23]). Unfortunately, since we have to apply standardtechniques of stochastic analysis, the underlying filtrations have to be as-sumed completed as a rule. On the other hand, for handling conditionalprobabilities it is important to have countably generated conditioning fields. For this reason we shall use small versions (F0t ), (G0t ) which arecountably generated, and big versions (Ft), (Gt) that are obtained as thesmallest right-continuous and completed filtrations containing the small


11/21

11

ones, and thus satisfy the usual conditions of stochastic calculus. We fur-ther suppose that F0 is trivial and that every (Ft)local martingale has acontinuous modification, and of course

F0

t G0

tfor all t

0. We assume

that M a (F0t )local martingale. The regular conditional probabilities rel-ative to the algebras F0t are denoted by Pt. For any set A F theprocess

(t, ) Pt(, A)is an (F0t )martingale with a continuous modification adapted to (Ft) (seee.g. Theorem 4, Chapter VI in [11]). We may assume that the processesPt(, A) are modified in such a way that Pt(, ) is a measure on F forPMalmost all (, t), where PM is given on

[0, 1] de

fi

ned by PM(

)=

E

01(, t)dM,Mt, F B+. It is known that each of these martingales

may be described in the unique representation (see e.g. [25], Chapter V)

(16) Pt(, A) = P(A) +t

0ks(, A)dMs + LAt ,

where k(, A) is (Ft)predictable and LA satisfies LA,M = 0.Note that trivially each

field in the left-continuous filtration (

G0t

) is

also generated by a countable number of sets.We claim that the existence of an information drift of (Gt) relative to

(Ft) for the process M depends on the validity of the following condition,which is the generalization of Jacods condition (10) to arbitrary stochastic

bases on standard Borel spaces.

Condition 3.1. kt(, )G0t

is a signed measure and satisfies

kt(, )G0t Pt(, )G0t

for PMa.a (, t).If (3.1) is satisfied, one can show (see [4]) that there exists an (Ft

Gt)predictable process such that for PMa.a. (, t)

(17) t(, ) =

dkt(, )

dPt(, ) G0t().

It is also immediate from the definition that

(18) t(, ) Pt(, d) dM,Mt = t(, ) dM,Mt.

On the basis of these simple facts it is possible to identify the informationdrift, provided (3.1) is guaranteed.


12/21

12

Theorem 3.1. Suppose Condition 3.1 is satisfied and is as in (17). Then

t() = t(, )

is the information drift of(Gt) relative to (Ft).

Proof. We give the arguments in case M is a martingale. For 0 s < t andA G0s we have to show

E [1A(Mt Ms)] = E1A

ts

u(, ) dM,Mu

.

Observe

E [1A(Mt Ms)] = E [Pt(, A)(Mt Ms)]

= E

(Mt Ms)

t0

ku(, A) dMu+ E[(Mt Ms)LAt ]

= E t

s

ku(

, A) d

M,M

u

= E

ts

A

u(, ) dPu(, d) dM,Mu

= E

1A()

ts

u(, ) dM,Mu

,

where we used (18) in the last equation.

We now look at the problem from the reverse direction. As an immedi-ate consequence of (18) and Proposition 2.2 note that (Gt) is a finite utilityfiltration if and only if

2t (,

) Pt(, d) dM,Mt dP() < .

Starting with the assumption that (Gt) is a finite utility filtration, whichthus amounts to E

10 2 dM,M < , we derive the validity of Condition

3.1.In the sequel, (Gt) denotes a finite utility filtration and its predictable

information drift, i.e.

(19) M = M

0

t dM,Mt


13/21

13

is a (Gt)local martingale. To prove absolute continuity, we first define ap-proximate Radon-Nikodym densities. This will be done along a sequenceof partitions of the state space which generate the respective fields of the

bigger filtration. So let tni= i2n for all n 0 and 0 i 2n. We denote by

T the set of all tni

. It is possible to choose a family offinite partitions (Pi,n)such that

for all t T we have G0t = (Pi,n : i, n 0 s.t. tni = t),

Pi,n

Pi+1,n

,

ifi < j, n < m and i 2n = j 2m, then Pi,n Pj,m.

We define for all n 0 the following approximate Radon-Nikodym densi-ties

nt (, ) =

2n

1

i=0

APi,n

1]tni

,tni+1

](t)1A() kt(, A)

Pt(, A).

Note that kt(,A)Pt(,A) is (Ft)predictable and 1]tni ,tni+1](t)1A() is (Gt)predictable.Hence the product of both functions, defined as a function on 2 [0, 1], ispredictable with respect to (Ft Gt). By the very definition, for PMalmostall(, t)

[0, 1]thediscreteprocess(mt (,

))m

1 is a martingale. To have

a chance to see this martingale converge as m , we will prove uniformintegrability which will follow from the boundedness of the sequence inL2(Pt(, )). This again is a consequence of the following key inequality (formore details see [4]).

Lemma 3.1. Let 0 s < t 1 and P = {A1, . . . , An} be a finite partition ofinto G0s measurable sets. Then

E

ts

nk=1

kuPu

2(, Ak) 1Ak dM,Mu 4E

ts

2u dM,Mu

< .

Proof. An application of Itos formula, in conjunction with (16) and (19),


14/21

14

yields

nk=1

1Ak log Ps(, Ak) 1Ak log Pt(, Ak)

=

nk=1

t

s

1

Pu(, Ak) 1Ak dPu(, Ak)

+1

2

ts

1

Pu(, Ak)2 1Ak dP(, Ak), P(, Ak)u

=

nk=1

t

s

kuPu (, Ak) 1Ak d Mu

ts

kuPu (, Ak) 1Ak u dM,Mu

t

s

1

Pu(, Ak) 1Ak dLAku +

1

2

ts

kuPu

2(, Ak) 1Ak dM,Mu

+1

2

ts

1

Pu(, Ak)2 1Ak dLAk , LAku

(20)

Note that Pt(, A

k)log Pt(

, A

k) is a submartingale bounded from below for

all k. Hence the expectation of the left hand side in the previous equation isat most 0. One readily sees that the stochastic integral process with respectto M in this expression is a martingale and hence has vanishing expectation,while a similar statement holds for the stochastic integral with respect tothe singular parts LAk . Consequently we may deduce from Eq. (20) and theKunita-Watanabe inequality

E

nk=1

1

2t

s

kuPu

2(, Ak) 1Ak dM,Mu

En

k=1

ts

kuPu

(, Ak) 1Ak u dM,Mu

E

ts

nk=1

kuPu

2(, Ak) 1Ak dM,Mu

12

E

ts

2u dM,Mu 1

2

,

which implies

E

ts

nk=1

kuPu

2(, Ak) 1Ak dM,Mu 4E

ts

2u dM,Mu

.

This completes the proof.


15/21

15

Lemma 3.1 will now allow us to obtain a Radon-Nikodym density

process provided the given information drift satisfies E 1

02 dM,M 0,


16/21

16

H0t = ( t+ ) and G0t = F0t H0t . Again let (Ft) and (Gt) be the smallestrespective extensions of (F0t ) and (G0t ) satisfying the usual conditions. Aninvestor having access to the information represented by (Gt) knows atany time whether within the next time units the Wiener process will hitthe level a, provided the level has not yet been hit. In this example, theinformation drift of (Gt) is already completely determined as the densityprocess ofkt(, ) relative to Pt(, ) along the algebras H0t (this followsfrom a slight modification of the proof of Theorem 3.1).

Let St = sup0rt Wr, F(a, x, u) = P((ax) u) and recall that F(a, x, u) =

u

0

y2y3

exp( (ax)22y )dy, for all x < a (see Ch.III, p.107 in [25]). Note thatfor all r u 1 we have Pr(, {(a) u}) = 1{Sra} + 1{Sr


17/21

17

assume (PRP), then we would obtain that the additional utility is onlybounded by this entropy (see [4]).

Recall that the entropy of a measure relative to a measure on a-algebra S is defined by

HS() =

log

dd

S

dP, if on S and the integral exists,, else.

We first fix a time s [0, 1] and try to measure the entropy of theinformation contained in G0s relative to the filtration (F0u ), conditional tothe

algebra

Fs. To this end we introduce an auxiliary filtration obtained

by enlarging (Fu) with G0s at time s,

Ku =Fu if 0 u < s

r>u Fr G0s , ifu [s, 1]

and we denote by s the information drift of (Ku) relative to M. Theconditional entropy of the algebra G0s relative to the filration (F0u ) on thetime interval [s, t], t (s, 1], will be defined by

H(s, t) =

HG0s (Pt(, )Ps(, ))dP().

We will now show that 2 H(s, t) is equal to the square-integral ofs on [s, t]. To this end let (Pm)m0 be an increasing sequence offinite partitionssuch that (Pm : m 0) = G0s . Then

H(s, t) = HG0s (Pt(, )Ps(, ))dP()

= E

APm

1A log Ps(, A) 1A log Pt(, A)

= E

APm

t

s

kuPu

(, A) 1A d Mu t

s

kuPu

(, A) 1Asu dM,Mu

+1

2

ts

kuPu

2(, A) 1A dM,Mu,

where the last equation follows from (20). Since M is a local martingale,we obtain by stopping and taking limits if necessary

H(s, t) = E

APm

t

s

kuPu

(, A) 1Au dM,Mu 12

ts

kuPu

2(, A) 1A dM,Mu

.


18/21

18

Lemma 3.1 implies that

APm

kuPu

2(, A) 1A() is an L2(Pu(, ))-bounded

martingale for PM

a.a. (, u), and therefore, by Theorem 3.1

limm

E

ts

APm

kuPu

2(, A) 1A dM,Mu = E

ts

(su)2 dM,Mu.

Similarly we have

limm

E

ts

APm

kuPu

(, A) 1A su dM,Mu = Et

s

(su)2 dM,Mu.

and hence

H(s, t) = 12

E

ts

(su)2 dM,Mu.(22)

We are now in a position to introduce a notion of conditional entropybetween our filtrations (G0t ) and (F0t ). For any partition : 0 = t0 t1 . . . tk = 1 we will use the abbreviations

=

ki=1 and

=

ki=1

Definition 4.1. Let (n

) be a sequence of partitions of [0, 1] with mesh |n

|converging to 0 as n . The limit of the sums n H(ti1, ti) as n is called conditional entropy of (G0t ) relative to (F0t ) and will be denoted byHG0 |F0 .Theorem 4.1. The conditional entropy HG0|F0 is well defined and it satisfies

HG0 |F0 = 12 E1

0

2udM,Mu.

Proof. Let(n)beasequenceofpartitionsof[0, 1]withmesh || convergingto 0 as n . For all n we define auxiliary filtrations

Dnt =s>t

(F0s G0ti ) ift [ti, ti+1[.

Since all (Dnt ) are subfitrations of (G0t ), the respective information drifts nofM exist. It follows immediately from Eq. (22) that

n

H(ti1, ti) = 12

E

t

s

(nu)2 dM,Mu.

As it is shown in Theorem 4.4 in [4], the information drifts n converge inL2(M) to the information drift . Consequently, the conditional entropy of

(G0t ) relative to (F0t ) is well defined and equals 12 E1

02uM,Mu.


19/21

19

The conditional entropy HG0 |F0 can be interpreted as a multiplicative inte-gral along the filtration (G0t ). More precisely, if for any s t 1 we define

d(s, t, , ) =Pt(,

)

Ps(,) G0s (), and if is a partition of [0, 1], then

H(ti1, ti) =

log

Pti (, )Pti1 (, )

G0ti1 ()Pti (, d

)

dP()

=

log d(ti1, ti, , )dP()

= log

d(ti

1, ti, , )dP()

In the special case where (G0t ) is obtained by an initial enlargment with arandom variable G, we have Pt(,)Ps(,)

G0s =Pt(,)Ps(,)

(G)

and hence

HG0|F0 =

logP1(, d)

P(d)

(G)

()P1(, d)

dP()

=

HF1

(G)(P1(, d

)P(d)

P

P).

The image of the measure P1(, d)P(d) under the mapping (, ) (M(), G()) is the joint distribution of M = (Mt)0t1 and G. Conse-quently, in the initial enlargement case, HG0|F0 is equal to the entropy ofthe joint distribution ofM and G relative to the product of the respectivedistributions, which is also known as the mutual information between Mand G. To sum up, we obtain a very simple formula for the additionallogarithmic utility under initial enlargements.

Theorem 4.2. Let G be a random variable and Gt = s>t Fs (G). ThenuG(x) uF(x) coincides with the mutual information between M and G.

References1. J. Amendinger, D. Becherer, and M. Schweizer. A monetary value for initial

information in portfolio optimization. Finance Stoch., 7(1):2946, 2003.2. J. Amendinger, P. Imkeller, and M. Schweizer. Additional logarithmic utility of

an insider. Stochastic Process. Appl., 75(2):263286, 1998.3. S. Ankirchner. Information and Semimartingales. Ph.D. thesis, Humboldt Uni-

versitat Berlin, 2005.4. S. Ankirchner, S. Dereich, and P. Imkeller. The shannon information offiltra-

tions and the additional logarithmic utility of insiders. Annals of Probability,34:743778, 2006.

5. S. Ankirchner and P. Imkeller. Finite utility on financial markets with asym-metric information and structure properties of the price dynamics. Ann. Inst.H. Poincare Probab. Statist., 41(3):479503, 2005.


20/21

20

6. F. Baudoin. Conditioning of brownian functionals and applications to the mod-elling of anticipations on a financial market. PhD thesis, Universite Pierre etMarie Curie, 2001.

7. F. Biagini and B. Oksendal. A general stochastic calculus approach to insidertrading. Preprint, 2003.

8. L. Campi. Some results on quadratic hedging with insider trading. Stochasticsand Stochastics Reorts, 77:327248, 2003.

9. J. Corcuera, P Imkeller, A. Kohatsu-Higa, and D. Nualart. Additional utility ofinsiders with imperfect dynamical information. Preprint, September 2003.

10. F. Delbaen and W. Schachermayer. The existence of absolutely continuous localmartingale measures. Ann. Appl. Probab., 5(4):926945, 1995.

11. C. Dellacherie and P.-A. Meyer. Probabilities and potential, volume 29 ofNorth-

Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1978.12. D. Duffie and C. Huang. Multiperiod security markets with differential in-

formation: martingales and resolution times. J. Math. Econom. , 15(3):283303,1986.

13. H. Follmer and P. Imkeller. Anticipation cancelled by a Girsanov transfor-mation: a paradox on Wiener space. Ann. Inst. H. Poincare Probab. Statist.,29(4):569586, 1993.

14. D. Gasbarra and E. Valkeila. Initial enlargement: a bayesian approach. Theoryof Stochastic Processes, 9:2637, 2004.

15. A. Grorud and M. Pontier. Insider trading in a continuous time market model.International Journal of Theoretical and Applied Finance, 1:331347, 1998.

16. P. Imkeller. Enlargement of the Wiener filtration by an absolutely continu-ous random variable via Malliavins calculus. Probab. Theory Related Fields,106(1):105135, 1996.

17. P. Imkeller. Random times at which insiders can have free lunches. Stochasticsand Stochastics Reports, 74:465487, 2002.

18. P. Imkeller. Malliavins calculus in insider models: additional utility and freelunches. Math. Finance , 13(1):153169, 2003. Conference on Applications of

Malliavin Calculus in Finance (Rocquencourt, 2001).19. P. Imkeller, M. Pontier, and F. Weisz. Free lunch and arbitrage possibilities in a

financial market model with an insider. Stochastic Process. Appl., 92(1):103130,2001.

20. J. Jacod. Grossissement initial, hypothese (H), et theoreme de Girsanov. In Th.Jeulin and M. Yor, editors, Grossissements de filtrations: exemples et applications,pages 1535. Springer-Verlag, 1985.

21. Th. Jeulin. Semi-martingales et grossissement dunefiltration,volume833ofLectureNotes in Mathematics. Springer, Berlin, 1980.

22. Th. Jeulin and M. Yor, editors. Grossissements de filtrations: exemples et appli-cations, volume 1118 ofLecture Notes in Mathematics. Springer-Verlag, Berlin,1985. Papers from the seminar on stochastic calculus held at the Universite deParis VI, Paris, 1982/1983.

23. K.R. Parthasarathy. Introduction to probability and measure. Delhi etc.: MacMillanCo. of India Ltd. XII, 1977.

24. I. Pikovsky and I. Karatzas. Anticipative portfolio optimization. Adv. in Appl.


21/21

21

Probab., 28(4):10951122, 1996.25. D. Revuz and M. Yor. Continuous martingales and Brownian motion , volume

293 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of

Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999.26. M. Yor. Entropie dune partition, et grossissement initial dune filtration. In

Grossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.), vol-ume 1118 ofLecture Notes in Math. Springer, Berlin, 1985.

27. M. Yor. Grossissement de filtrations et absolue continuite de noyaux. InGrossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.), vol-ume 1118 ofLecture Notes in Math. Springer, Berlin, 1985.

28. M. Yor. Some aspects of Brownian motion. Part II. Lectures in Mathematics ETHZurich. Birkhauser Verlag, Basel, 1997. Some recent martingale problems.

financial markets with asymmetric information

Documents