Stochastic processes and applications to mathematical finance

December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+The series of the Ritsumeikan conferences has been aimed to hold blies of those interested in the applications of theor

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STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE

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Proceedings of the 6the Ritsumeikan International Symposium

World Scientific

NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

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STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Proceedings of the 6th Ritsumeikan International Symposium

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December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+

The series of the Ritsumeikan conferences has been aimed to hold blies of those interested in the applications of theory of stochastic processes and stochastic analysis to financial problems The Conference, counted as the 6th

assem-one, was also organized in this line: there several eminent specialists aswell as active young researchers were jointly invited to give their lectures(see the program cited below) and as a whole we had about hundred par-ticipants The present volume is the proceedings of this conference based

on those invited lectures

We, members of the editorial committee listed below, would expressour deep gratitude to those who contributed their works in this proceed-ings and to those who kindly helped us in refereeing them We wouldexpress our cordial thanks to Professors Toshio Yamada, Keisuke Haraand Kenji Yasutomi at the Department of Mathematical Sciences, of Rit-sumeikan University, for their kind assistance in our editing this volume

We would thank also Mr Satoshi Kanai for his works in editing TeX filesand Ms Chelsea Chin of World Scientific Publishing Co for her kind andgenerous assistance in publishing this proceedings

December, 2006, Ritsumeikan University (BKC)Jir ˆo Akahori

Shigeyoshi Ogawa

Shinzo Watanabe

v

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vi

STOCHASTIC PROCESSES AND APPLICATIONS TO

MATHEMATICAL FINANCE

Date March 6–10, 2006

Place Rohm Memorial Hall/Epoch21, in BKC, Ritsumeikan University

1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan

Program

March, 6 (Monday): at Rohm Memorial Hall

10:00–10:10 Opening Speech, by Shigeyoshi Ogawa (Ritsumeikan

Uni-versity)

10:10–11:00 T Lyons (Oxford University)

Recombination and cubature on Wiener space

11:10–12:00 S Ninomiya (Tokyo Institute of Technology)

Kusuoka approximation and its application to finance

12:00–13:30 Lunch time

13:30–14:20 T Fujita (Hitotsubashi University, Tokyo)

Some results of local time, excursion in random walk and Brownianmotion

14:30–15:20 K Hara (Ritsumeikan University, Shiga)

Smooth rough paths and the applications

15:20–15:50 Break

15:50–16:40 X-Y Zhou (Chinese University of Hong-Kong)

Behavioral portfolio selection in continuous time

17:30– Welcome party

March, 7 (Tuesday): at Rohm Memorial Hall

10:00–10:50 M Schweizer (ETH, Zurich)

Aspects of large investor models

11:10–12:00 J Imai (Tohoku University, Sendai)

A numerical approach for real option values and equilibrium gies in duopoly

strate-12:00–13:30 Lunch time

The 6th Ritsumeikan International Conference on

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vii

13:30–14:20 H Pham (Univ Paris VII)

An optimal consumption model with random trading times and uidity risk and its coupled system of integrodifferential equations

liq-14:30–15:20 K Hori (Ritsumeikan University, Shiga)

Promoting competition with open access under uncertainty

15:20–15:50 Break

15:50–16:40 K Nishioka (Chuo University, Tokyo)

Stochastic growth models of an isolated economyMarch, 8 (Wednesday): at Rohm Memorial Hall

10:00–10:50 H Kunita (Nanzan University, Nagoya)

Perpetual game options for jump diffusion processes

11:10–11:50 E Gobet (Univ Grenoble)

A robust Monte Carlo approach for the simulation of generalizedbackward stochastic differential equations

12:00– Excursion

March, 9 (Thursday): at Epoch21

10:00–10:50 P Imkeller (Humbold University, Berlin)

Financial markets with asymmetric information: utility and entropy

11:00–12:00 M Pontier (Univ Toulouse III)

Risky debt and optimal coupon policy

12:00–13:30 Lunch time

13:30–14:20 H Nagai (Osaka University)

Risk-sensitive quasi-variational inequalities for optimal investmentwith general transaction costs

14:30–15:20 W Runggaldier (Univ Padova)

On filtering in a model for credit risk

15:20–15:50 Break

15:50–16:40 D A To (Univ Natural Sciences, HCM city)

A mixed-stable process and applications to option pricing

16:50– Short Communications

1 Y Miyahara (Nagoya City University)

2 T Tsuchiya (Ritsumeikan University, Shiga)

3 K Yasutomi (Ritsumeikan University, Shiga)

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viii

March, 10 (Friday): Epoch21

10:00–10:50 R Cont (Ecole Polytechnique, France)

Parameter selection in option pricing models: a statistical approach

11:10–12:00 T V Nguyen (Hanoi Institute of Mathematics)

Multivariate Bessel processes and stochastic integrals

12:00–13:30 Lunch time

13:30–14:20 J-A, Yan (Academia Sinica, China)

A functional approach to interest rate modelling

14:30–15:20 M Arisawa (Tohoku University, Sendai)

A localization of the L´evy operators arising in mathematical finances

15:20–15:50 Break

15:50–16:40 A N Shiryaev (Steklov Mathem Institute, Moscow)

Some explicit stochastic integral representation for Brownian tionals

func-18:30– Reception at Kusatsu Estopia Hotel

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January 17, 2007 19:18 Proceedings Trim Size: 9in x 6in contents

CONTENTS

Preface vProgram viFinancial Markets with Asymmetric Information: Information Drift,Additional Utility and Entropy S Ankirchner and P Imkeller 1

A Localization of the L´evy Operators Arising in Mathematical

Finances M Arisawa 23Model-free Representation of Pricing Rules as Conditional

Expectations S Biagini and R Cont 53

A Class of Financial Products and Models Where Super-replication

Risky Debt and Optimal Coupon Policy and Other Optimal

Strategies D Dorobantu and M Pontier 85Aﬃne Credit Risk Models under Incomplete Information

R Frey, C Prosdocimi, and W J Runggaldier 97Smooth Rough Paths and the Applications

K Hara and T Lyons 115From Access to Bypass: A Real Options Approach

K Hori and K Mizuno 127The Investment Game under Uncertainty: An Analysis of

Equilibrium Values in the Presence of First or Second Mover

Advantage J Imai and T Watanabe 151Asian Strike Options of American Type and Game Type

M Ishihara and H Kunita 173Minimal Variance Martingale Measures for Geometric L´evy

xi

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January 17, 2007 19:18 Proceedings Trim Size: 9in x 6in contents

xii

A Remark on Impulse Control Problems with Risk-sensitive

Criteria H Nagai 219

A Convolution Approach to Multivariate Bessel Proceses

T V Nguyen, S Ogawa, and M Yamazato 233Spectral Representation of Multiply Self-decomposable Stochastic

Processes and Applications

N V Thu, T A Dung, D T Dam, and N H Thai 245

Numerical Approximation by Quantization for Optimization

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December 26, 2006 10:43 Proceedings Trim Size: 9in x 6in 01Ankirchner˙Imkeller

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

Stefan Ankirchner and Peter Imkeller

Institut f ¨ur Mathematik, Humboldt-Universit¨at zu Berlin,Unter den Linden 6, 10099 Berlin, Germany

We review a general mathematical link between utility and mation theory appearing in a simple financial market model withtwo kinds of small investors: insiders, whose extra information

infor-is stored in an enlargement of the less informed agents’ filtration.The insider’s expected logarithmic utility increment is described

in terms of the information drift, i.e the drift one has to eliminate

in 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 thebetter 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 of filtration; logarithmic utility; utility

maximization; heterogeneous information; insider model; Shannoninformation; information diﬀerence; entropy

2000 AMS subject classifications: primary 60H30, 94A17;

sec-ondary 91B16, 60G44

1 Introduction

A simple mathematical model of two small agents on a financial ket one of which is better informed than the other has attracted muchattention in recent years Their information is modelled by two diﬀerent

the natural evolution of the market up to time t at his disposal, while

short selection of some among many more papers dealing with this model.Investigation techniques concentrate on martingale and stochastic controltheory, and methods of enlargement of filtrations (see Yor , Jeulin , Jacod in[22]), starting with the conceptual paper by Duﬃe, Huang [12] The model

1

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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 agent’s 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 information drift between the filtrations (see [18]), and subsequently identified with

the Shannon entropy of the additional information Also for initial largements, Gasbarra, Valkeila [14] extend this link to the Kullback-Leiblerinformation of the insider’s additional knowledge from the perspective

en-of Bayesian modelling In the environment en-of this utility-information

paradigm the papers [16], [19], [17], [18], Corcuera et al [9], and Ankirchner

et al [5] describe additional utility, treat arbitrage questions and their

inter-pretation in information theoretic terms in increasingly complex models

of the same base structure Utility concepts diﬀerent from the mic one correspond on the information theoretic side to the generalized

In this paper we review the main results about the interpretation of thebetter informed trader’s additional utility in information theoretic termsmainly developed in [4], concentrating on the logarithmic case This leads

to very basic problems of stochastic calculus in a very general setting ofenlargements of filtrations: to ensure the existence of regular conditional

filtration, 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 stochastickernel 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 framework first 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 terms

of Shannon entropy concepts

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3

2 Additional Logarithmic Utility and Information Drift

Let us first fix notations for our simple financial market model First ofall, to simplify the exposition, we assume that the trading horizon is given

by T = 1 Let (Ω, F , P) be a probability space with a filtration (F t)0≤t≤1

We consider a financial market with one non-risky asset of interest rate

integral process For all x> 0 we interpret

x + (θ · X) t , 0 ≤ t ≤ 1,

as the wealth process of a trader possessing an initial wealth x and

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 has

always to be strictly positive, i.e for all t∈ [0, 1]

want to maximize their expected logarithmic utility from terminal wealth

So we are interested in the exact value of

The expected logarithmic utility of the agent can be calculated easily, if onehas a semimartingale decomposition of the form

0 ηs d M, M s,

arbitrage opportunities In fact, if X satisfies the property (NFLVR), then it

may be decomposed as in Eq (2) (see [10]) It is shown in [3] that finiteness

of u(x) already implies the validity of such a decomposition Hence a

decomposition 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

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4

Let us give the core arguments proving this statement in a particular setting,

dX t

X t = αt dt + dW t,

with a one-dimensional Wiener process W, and assume that the small

is a progressively measurable mean rate of return process which satisfies

maximization problem for the function

2π2

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This proposition motivates the following definition

ζ is (Gt)−predictable and belongs to L2(M), i.e E1

0 ζ2d M, M < ∞ We

write

F = {(Ht)⊃ (Ft)(H

t) is a finite utility filtration for X}

We now compare two traders who take their portfolio decisions not on thebasis of the same filtration, but on the basis of diﬀerent information flowsrepresented by the filtrations (Gt) and (Ht) respectively Suppose that bothfiltrations (Gt) and (Ht) are finite utility filtrations We denote by

the utility diﬀerence depends only on the process µ = ζ − β In fact,

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6

with respect to (Ht), andζ is adapted to this filtration It is therefore natural

to relateµ to a transfer of information

(Ht) is a filtration such thatGt ⊂ Ht for all t∈ [0, 1] The (Ht)−predictable

is called information drift (see [18]) of (H t) with respect to (Gt)

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

Gt⊂ Ht for all t ∈ [0, 1] If µ is the information drift of (H t ) w.r.t (G t ), then we have

3 The Information Drift and the Law of Additional Information

In this section we aim at giving a description of the information driftbetween 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 initial enlargement In a second step, we shall generalize the results available in

the 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, Jacod’s condition

In this setting, the additional information in the larger filtrations is atall times during the trading interval given by the knowledge of a randomvariable which, from the perspective of the smaller filtration, is knownonly at the end of the trading interval To establish the concepts in fair

the augmented filtration of a one-dimensional Wiener process W Let G be

Gt= Ft ∨ σ(G), t ∈ [0, 1].

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7

respect to this filtration More precisely, suppose that there is an tion driftµGsuch that

under a condition concerning the laws of the additional information G

which has been used as a standing assumption in many papers dealing

with grossissement de filtrations See Yor [27], [26], [28], Jeulin [21] The

condition was essentially used in the seminal paper by Jacod [20], and

in several equivalent forms in F¨ollmer and Imkeller [13] To state andexploit it, let us first mention that all stochastic quantities appearing in thesequel, 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])

t(ω, dl) the

we will call Jacod’s condition, states that

(10) P G

t(ω, dg) is absolutely continuous with respect to PG (dg) for P− a.e ω ∈ Ω.

Also its reinforcement

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

p t(ω, g) = dP

G

t(ω, ·)

By the very definition, t → P t(·, dg) is a local martingale with values in the

p t(·, g) = p0(·, g) +

0

k u g dW u , t ∈ [0, 1]

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8

with measurable kernels k To calculate the information drift in terms of these kernels, take s, t ∈ [0, 1], s ≤ t, and let A ∈ F s and a Borel set B on the real line determine the typical set A ∩ G−1[B] in a generator ofGs Then wemay write

the covariation of two martingales (for more details see Jacod [20])

Theorem 3.1 Suppose that Jacod’s condition (10) is satisfied, and furthermore

that

g t

p t(·, g)|g =G=

d

dt p(·, g), W t

p t(·, g) |g =G , t ∈ [0, 1], satisfies

(13)

0 |µG

u | du < ∞ P−a.s

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To see how restrictive condition (10) may be, let us illustrate it by

looking at two possible additional information variables G.

Example 1:

given by a stochastic diﬀerential equation with bounded volatility σ and

process with transition probabilities P t (x, dy), x ∈ R+, t ∈ [0, 1], which are

condi-tional law of G givenFt is then given by P1+ −t (X t , dy), which is equivalent with the law of G Hence in this case, even the strong version of Jacod’s

sets A on the real line we have

Note now that the family of Dirac measures in the first term of (15) is

this example Jacod’s condition is violated

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Malliavin gradients of conditional laws of G 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

there is a regular conditional probability P t(·, ·) of F given Ft, which can

be decomposed into a martingale component orthogonal to M, plus a

component possessing a stochastic integral representation with respect to

respect to dM, M ⊗ P, the kernel function at t will be a signed measure in

its set variable This measure is absolutely continuous with respect to theconditional probability itself, if restricted toGt, andα 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 an

filtration

To provide some details of this fundamental relationship, we need towork with conditional probabilities We therefore assume that (Ω, 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 conditional

t ), (G0

t) which are

smallest right-continuous and completed filtrations containing the small

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11

ones, and thus satisfy the usual conditions of stochastic calculus We

t ⊂ G0

process

(t, ω) → P t(ω, A)

is an (F0

e.g Theorem 4, Chapter VI in [11]) We may assume that the processes

where k(·, A) is (F t)−predictable and LAsatisfiesL A , M = 0.

t−) isalso generated by a countable number of sets

which is the generalization of Jacod’s condition (10) to arbitrary stochasticbases on standard Borel spaces

Gt)−predictable process γ such that for PM −a.a (ω, t)

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12

αt(ω) = γt(ω, ω)

is the information drift of (Gt ) relative to (F t ).

k u(·, A) dM, Mu

= E

t s

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

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13

ap-proximate Radon-Nikodym densities This will be done along a sequence

bigger filtration So let t n

i = i

T the set of all t n

i It is possible to choose a family of finite partitions (Pi ,n)such that

P t(ω, A).

Note that k t(ω,A)

P t(ω,A)is (Ft)−predictable and 1]t n

i ,t n i+1](t)1 A(ω) is (Gt)−predictable

predictable with respect to (Ft⊗ Gt ) By the very definition, for P M−almostall (ω, t) ∈ Ω×[0, 1] the discrete process (γm

t(ω, ·))m≥1is a martingale To have

integrability which will follow from the boundedness of the sequence in

L2(P t(ω, ·)) This again is a consequence of the following key inequality (formore details see [4])

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Note that P t(·, Ak ) log P t(·, Ak) is a submartingale bounded from below for

all k Hence the expectation of the left hand side in the previous equation is

at most 0 One readily sees that the stochastic integral process with respect

while a similar statement holds for the stochastic integral with respect to

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15

Lemma 3.1 will now allow us to obtain a Radon-Nikodym densityprocess provided the given information driftα satisfies E01α2d M, M <

∞ Note that our main result implicitly contains the statement that the

t , P M−a.e

∞ Then the kernel k is absolutely continuous with respect to P t(ω, ·)|G0

t− , for

P M −a.a (ω, t) ∈ Ω × [0, 1] This means that Condition 3.1 is satisfied Moreover,

to (Ft ) by the formula

αt(ω) = γt(ω, ω)

t (ω, ·))m≥1is an L2(P t(ω, ·))–boundedmartingale and hence, for a.a fixed (ω, t), (γm

Observe that ˜k t(ω, ·) is absolutely continuous with respect to Pt(ω, ·) and

that we have for all A∈ Pj ,m with j2 −m ≤ t

holds for all A ∈ G0

t− Hence, by choosing k t(·, A) = ˜kt(·, A) for all A ∈ G0

t−,the proof is complete

We close this section by illustrating the method developed by means of anexample

t the

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the level a, provided the level has not yet been hit In this example, the

process of k t(ω, ·) relative to Pt(ω, ·) along the σ−algebras H0

t (this followsfrom a slight modification of the proof of Theorem 3.1)

Let S t= sup0≤r≤t W r , F(a , x, u) = P(τ(a−x) ≤ u) and recall that F(a, x, u) =

is straightforward to show that

k r(ω, {τ(a) ∧ t + δ ≤ u}) = 1[0,u](r)

So the processαt(ω) = γt(ω, ω) is the information drift of (Gt)

4 Additional Utility and Entropy of Filtrations

As in Subsection 3.2 let X = (X t)0≤t≤1be a semimartingale, (Ft) and (Gt)two finite utility filtrations such thatFt ⊂ Gt , t ∈ [0, 1], and let µ be the

information drift of (Gt) relative to (Ft) As before we assume that thereexist countably generated filtrations (F0

t ) and (G0

t) such that (Ft) and (Gt)are obtained as the the smallest respective extensions satisfying the usualconditions

analyis we will assume throughout this section that M has the predictable

that the utility diﬀerence uG(x) − uF(x) can be interpreted as a conditional

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s relative to the filration (F0

[s, t] To this end let (P m)m≥0be an increasing sequence of finite partitionssuch thatσ(Pm : m≥ 0) = G0

we obtain by stopping and taking limits if necessary

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We are now in a position to introduce a notion of conditional entropy

t) and (F0

t ) For any partition∆ : 0 = t0 ≤ t1 ≤

≤ t k= 1 we will use the abbreviations$∆=$k

i=1and%

∆=%k

i=1

converging to 0 as n→ ∞ The limit of the sums$∆n H(t i−1, t i ) as n→ ∞

is called conditional entropy of (G0

t) are subfitrations of (G0

t), the respective information driftsµn

of M exist It follows immediately from Eq (22) that

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19

inte-gral along the filtration (G0

t ) More precisely, if for any s ≤ t ≤ 1 we define d(s , t, ω, ω)= P t(ω,·)

random variable G, we have P t(ω,·)

the joint distribution of M and G relative to the product of the respective distributions, which is also known as the mutual information between M and G To sum up, we obtain a very simple formula for the additional

logarithmic utility under initial enlargements

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21

Probab., 28(4):1095–1122, 1996.

25 D Revuz and M Yor Continuous martingales and Brownian motion, volume

293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of

Mathematical Sciences] Springer-Verlag, Berlin, third edition, 1999.

26 M Yor Entropie d’une partition, et grossissement initial d’une filtration In

Grossissements de filtrations: exemples et applications T Jeulin, M.Yor (eds.),

vol-ume 1118 of Lecture Notes in Math Springer, Berlin, 1985.

27 M Yor Grossissement de filtrations et absolue continuit´e de noyaux In

Grossissements de filtrations: exemples et applications T Jeulin, M.Yor (eds.),

vol-ume 1118 of Lecture Notes in Math Springer, Berlin, 1985.

28 M Yor Some aspects of Brownian motion Part II Lectures in Mathematics ETH

Z ¨urich Birkh¨auser Verlag, Basel, 1997 Some recent martingale problems

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December 26, 2006 15:34 Proceedings Trim Size: 9in x 6in ritsuhregloc

A Localization of the L´evy Operators Arising in

where c(z)dz is a positive Radon measure, called L´evy density, defined on

R Nmin(|z|2, 1)c(z)dz<C1,(2)

C2

|z|1 +γ<|c(z)|< C3

|z|1 +γ ∀z ∈ RN∩ {|z|<1},

(3)

whereγ ∈ (0, 2), C i > 0 (1<i<3) are constants We assume that there exists

a ”uniform” constant M> 1 such that for a constant θ0∈ [0, 1],

|g(x) − g(y)|<M|x − y|θ 0 ∀x, y ∈ RN,(4)

The second-order fully nonlinear partial diﬀerential operator F is

(Degenerate ellipticity) :

F(x , p, X) ≥ F(x, p, Y) if X<Y,

23

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24

(6)

R+∪ {0} → R+∪ {0} such that limσ↓0w(σ) = 0, limσ↓0η(σ) = 0, and

|F(x, p, X) − F(y, p, X)|<w(|x − y|)|p| q + η(|x − y|)||X||

to get rid of the singularity of the L´evy measure, we shall use the following

superjet (resp subjet) and its residue Let ˆx∈ R N, and let (p, X) ∈ J2 ,+

R Nu( ˆx) (resp (p, X) ∈ J2 ,−

Then, for anyδ > 0 there exists ε > 0 such that

) holds We use this pair of numbers (ε, δ) satisfying (8) (resp (9)) for

any (p, X) ∈ J2,+R Nu( ˆx) (resp (p , X) ∈ JR2,−Nv( ˆx)) in the following definition of

viscosity solutions

is a viscosity subsolution (resp supersolution) of (1), if for any ˆx∈ R N, any

(p, X) ∈ J2,+R Nu( ˆx) (resp ∈ J2,−R Nv( ˆx)), and any pair of numbers (ε, δ) satisfying(8) (resp (9)), the following holds for any 0< ε<ε

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(II) N ≥ 2, and F satisfies the following uniform ellipticity.

(Uniform ellipticity) : There existsλ0> 0 such that

the case of (II), we claim that for any θ ∈ (0, 1), there exists Cθ > 0 suchthat (11) holds (See Theorem 3.2 in below.) (These results hold for moregeneral problem

As for the case other than (I) and (II), that is N ≥ 2 and F is not necessarily

uniformly elliptic (i.e (10) is not satisfied), we study the following two

problems in the torus T Ninstead of (1) The first one is, forλ > 0,

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nonlinear degenerate elliptic operator, satisfying the following conditions.(Periodicity) :

where X<Y(X, Y∈ SM), 0 < M<N

from R+∪ {0} → R+∪ {0} such that limσ↓0w(σ) = 0, limσ↓0η(σ) = 0, and

where Cθ> 0 is independent on λ > 0 (See Theorems 4.1 and 4.2 in below.)The method to derive the above uniform H¨older continuity (11) and theH¨older continuity (17) is based on the argument used in the proof of thecomparison result (See Ishii and Lions [21], for the similar argument inthe PDE case.)

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27

Next, we shall state the strong maximum principle for the L´evy erator In [18], for the second-order uniformly elliptic integro-diﬀerentialoperator

the strong maximum principle was given, where λ0I <(a ij)1<i,j<N<Λ0I

with-out assuming the uniform ellipticity of the partial diﬀerential operator F in(1) (see Theorem 5.1 in below, and M Arisawa and P.-L Lions [9])

Finally, we shall apply these regularity results (11), (17) and the strongmaximum principle, to study the so-called ergodic problem In the case ofthe Hamilton-Jacobi-Bellman (HJB) operator

the ergodicity of the corresponding controlled diﬀusion process, for

as follows

f (x) such that the following problem has a periodic viscosity solution u(x)

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Stochastic processes and applications to mathematical finance

The Model and Some Basic Examples

Finite Dimensional Computation of the Filter