Recent advances in financial engineering 2009

May 3, 2010 13:23 Proceedings Trim Size: 9in x 6in prefacevii Program August 3 Monday Chair: Masaaki Kijima 10:00–10:10 Yasuyuki Kato, Nomura Securities/Kyoto University Opening Address

Trang 2

RECENT ADVANCES IN FINANCIAL ENGINEERING

2009

Proceedings of the

KIER-TMU International Workshop

on Financial Engineering 2009

Trang 5

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4299-89-3

ISBN-10 981-4299-89-8

All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

Printed in Singapore.

RECENT ADVANCES IN FINANCIAL ENGINEERING 2009

Proceedings of the KIER-TMU International Workshop on Financial Engineering 2009

Trang 6

May 3, 2010 13:23 Proceedings Trim Size: 9in x 6in preface

PREFACE

This book is the Proceedings of the KIER-TMU International Workshop on Financial Engineering 2009 held in Summer 2009 The workshop is the succes-

sor of “Daiwa International Workshop on Financial Engineering” that was held

in Tokyo every year since 2004 in order to exchange new ideas in financial gineering among workshop participants Every year, various interesting and highquality studies were presented by many researchers from various countries, fromboth academia and industry As such, this workshop served as a bridge betweenacademic researchers in the field of financial engineering and practitioners

en-We would like to mention that the workshop is jointly organized by the tute of Economic Research, Kyoto University (KIER) and the Graduate School ofSocial Sciences, Tokyo Metropolitan University (TMU) Financial support fromthe Public Management Program, the Program for Enhancing Systematic Edu-cation in Graduate Schools, the Japan Society for Promotion of Science’s Pro-gram for Grants-in Aid for Scientific Research (A) #21241040, the Selective Re-search Fund of Tokyo Metropolitan University, and Credit Pricing Corporation aregreatly appreciated

Insti-We invited leading scholars including four keynote speakers, and various kinds

of fruitful and active discussions were held during the KIER-TMU workshop.This book consists of eleven papers related to the topics presented at the work-shop These papers address state-of-the-art techniques and concepts in financialengineering, and have been selected through appropriate referees’ evaluation fol-lowed by the editors’ final decision in order to make this book a high quality one.The reader will be convinced of the contributions made by this research

We would like to express our deep gratitude to those who submitted their pers to this proceedings and those who helped us kindly by refereeing these pa-pers We would also thank Mr Satoshi Kanai for editing the manuscripts, and Ms.Kakarlapudi Shalini Raju and Ms Grace Lu Huiru of World Scientific Publishing

pa-Co for their kind assistance in publishing this book

February, 2010

Yukio Muromachi, Tokyo Metropolitan University

v

Trang 7

KIER-TMU International Workshop

Selective Research Fund of Tokyo Metropolitan UniversityCredit Pricing Corporation

vi

Trang 8

vii

Program

August 3 (Monday)

Chair: Masaaki Kijima

10:00–10:10 Yasuyuki Kato, Nomura Securities/Kyoto University

Opening Address

Chair: Chiaki Hara

10:10–10:55 Chris Rogers, University of Cambridge

Optimal and Robust Contracts for a Risk-Constrained Principal

10:55–11:25 Yumiharu Nakano, Tokyo Institute of Technology

Quantile Hedging for Defaultable Claims

11:25–12:45 Lunch

Chair: Yukio Muromachi

12:45–13:30 Michael Gordy, Federal Reserve Board

Constant Proportion Debt Obligations: A Post-Mortem Analysis of RatingModels (with Soren Willemann)

13:30–14:00 Kyoko Yagi, University of Tokyo

An Optimal Investment Policy in Equity-Debt Financed Firms with FiniteMaturities (with Ryuta Takashima and Katsushige Sawaki)

14:00–14:20 Afternoon Coffee I

Chair: St´ephane Cr´epey

14:20–14:50 Hidetoshi Nakagawa, Hitotsubashi University

Surrender Risk and Default Risk of Insurance Companies (with Olivier LeCourtois)

14:50–15:20 Kyo Yamamoto, University of Tokyo

Generating a Target Payoff Distribution with the Cheapest Dynamic lio: An Application to Hedge Fund Replication (with Akihiko Takahashi)

Portfo-15:20–15:50 Yasuo Taniguchi, Sumitomo Mitsui Banking Corporation/Tokyo

Metropolitan UniversityLooping Default Model with Multiple Obligors

15:50–16:10 Afternoon Coffee II

Trang 9

Chair: Hidetaka Nakaoka

16:10–16:40 St´ephane Cr´epey, Evry University

Counterparty Credit Risk (with Samson Assefa, Tomasz R Bielecki,Monique Jeanblanc and Behnaz Zagari)

16:40–17:10 Kohta Takehara, University of Tokyo

Computation in an Asymptotic Expansion Method (with Akihiko Takahashiand Masashi Toda)

Trang 10

ix

August 4 (Tuesday)

Chair: Takashi Shibata

10:00–10:45 Chiaki Hara, Kyoto University

Heterogeneous Beliefs and Representative Consumer

10:45–11:15 Xue-Zhong He, University of Technology, Sydney

Boundedly Rational Equilibrium and Risk Premium (with Lei Shi)

11:15-11:45 Yuan Tian, Kyoto University/Tokyo Metropolitan University

Financial Synergy in M&A (with Michi Nishihara and Takashi Shibata)

11:45–13:15 Lunch

Chair: Andrea Macrina

13:15–14:00 Mark Davis, Imperial College London

Jump-Diffusion Risk-Sensitive Asset Management (with Sebastien Lleo)

14:00–14:30 Masahiko Egami, Kyoto University

A Game Options Approach to the Investment Problem with ConvertibleDebt Financing

15:30–16:00 Andrea Macrina, King’s College London/Kyoto University

Information-Sensitive Pricing Kernels (with Lane Hughston)

16:00–16:30 Hiroki Masuda, Kyushu University

Explicit Estimators of a Skewed Stable Model Based on High-FrequencyData

16:30–17:00 Takayuki Morimoto, Kwansei Gakuin University

A Note on a Statistical Hypothesis Testing for Removing Noise by TheRandom Matrix Theory, and its Application to Co-Volatility Matrices (withKanta Tachibana)

Chair: Keiichi Tanaka

17:00–17:10 Kohtaro Kuwada, Tokyo Metropolitan University

Closing Address

Trang 12

May 3, 2010 10:39 Proceedings Trim Size: 9in x 6in contents

CONTENTS

Preface vProgram viiRisk Sensitive Investment Management with Affine Processes: A ViscosityApproach M Davis and S Lleo 1Small-Sample Estimation of Models of Portfolio Credit Risk

M B Gordy and E Heitfield 43Heterogeneous Beliefs with Mortal Agents

A A Brown and L C G Rogers 65Counterparty Risk on a CDS in a Markov Chain Copula Model with JointDefaults S Cr´epey, M Jeanblanc and B Zargari 91Portfolio Efficiency Under Heterogeneous Beliefs

X.-Z He and L Shi 127Security Pricing with Information-Sensitive Discounting

A Macrina and P A Parbhoo 157

On Statistical Aspects in Calibrating a Geometric Skewed Stable AssetPrice Model H Masuda 181

A Note on a Statistical Hypothesis Testing for Removing Noise by theRandom Matrix Theory and Its Application to Co-Volatility Matrices T Morimoto and K Tachibana 203Quantile Hedging for Defaultable Claims Y Nakano 219New Unified Computational Algorithm in a High-Order AsymptoticExpansion Scheme K Takehara, A Takahashi and M Toda 231Can Financial Synergy Motivate M&A?

Y Tian, M Nishihara and T Shibata 253

xi

Trang 13

Risk Sensitive Investment Management with A ffine

Processes: A Viscosity Approach∗

Mark Davis and S´ebastien Lleo

Department of Mathematics, Imperial College London, London SW7 2AZ, EnglandE-mail: mark.davis@imperial.ac.uk and sebastien.lleo@imperial.ac.uk

In this paper, we extend the jump-diffusion model proposed by Davis andLleo to include jumps in asset prices as well as valuation factors Thecriterion, following earlier work by Bielecki, Pliska, Nagai and others, isrisk-sensitive optimization (equivalent to maximizing the expected growthrate subject to a constraint on variance) In this setting, the Hamilton-Jacobi-Bellman equation is a partial integro-differential PDE The mainresult of the paper is to show that the value function of the control problem

is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation

Keywords: Asset management, risk-sensitive stochastic control, jump

diffusion processes, Poisson point processes, L´evy processes, HJB PDE,policy improvement

In this paper, we extend the jump diffusion risk-sensitive asset managementmodel proposed by Davis and Lleo [19] to allow jumps in both asset prices andfactor levels

Risk-sensitive control generalizes classical stochastic control by parametrizingexplicitly the degree of risk aversion or risk tolerance of the optimizing agent Inrisk-sensitive control, the decision maker’s objective is to select a control policy

h(t) to maximize the criterion

Trang 14

May 3, 2010 13:34 Proceedings Trim Size: 9in x 6in 001

2

where t is the time, x is the state variable, F is a given reward function, and the risk

sensitivity θ ∈] − 1, 0[∪]0, ∞) is an exogenous parameter representing the decisionmaker’s degree of risk aversion A Taylor expansion of this criterion around θ = 0yields

J(t, x, h; θ) = E [F(t, x, h)] −θ

which shows that the risk-sensitive criterion amounts to maximizing E [F(t, x, h)]

subject to a penalty for variance Jacobson [28], Whittle [35], Bensoussan andVan Schuppen [9] led the theoretical development of risk sensitive control whileLefebvre and Montulet [32], Fleming [25] and Bielecki and Pliska [11] pio-neered the financial application of risk-sensitive control In particular, Bieleckiand Pliska proposed the logarithm of the investor’s wealth as a reward func-tion, so that the investor’s objective is to maximize the risk-sensitive (log) re-turn of his/her portfolio or alternatively to maximize a function of the powerutility (HARA) of terminal wealth Bielecki and Pliska brought an enormouscontribution to the field by studying the economic properties of the risk-sensitiveasset management criterion (see [13]), extending the asset management modelinto an intertemporal CAPM ([14]), working on transaction costs ([12]), nu-merical methods ([10]) and considering factors driven by a CIR model ([15]).Other main contributors include Kuroda and Nagai [31] who introduced an ele-gant solution method based on a change of measure argument Davis and Lleoapplied this change of measure technique to solve a benchmarked investmentproblem in which an investor selects an asset allocation to outperform a givenfinancial benchmark (see [18]) and analyzed the link between optimal portfoliosand fractional Kelly strategies (see [20]) More recently, Davis and Lleo [19]extended the risk-sensitive asset management model by allowing jumps in assetprices

In this chapter, our contribution is to allow not only jumps in asset prices

intro-duce jumps in the factors, the Bellman equation becomes a nonlinear

Par-tial Integro-DifferenPar-tial equation and an analytical or classical C1,2 solutions

value function and the risk sensitive Hamilton-Jacobi-Bellman Partial gro Differential Equation (RS HJB PIDE), we consider a class of weak so-lutions called viscosity solutions, which have gained a widespread acceptance

theo-rem and the proof that the value function of the control problem under sideration is the unique continuous viscosity solution of the associated RS HJBPIDE In particular, the proof of the comparison results uses non-standard ar-guments to circumvent difficulties linked to the highly nonlinear nature of the

con-RS HJB PIDE and to the unboundedness of the instantaneous reward

func-tion g.

Trang 15

This chapter is organized as follows Section 2 introduces the general setting

of the model and defines the class of random Poisson measures which will beused to model the jump component of the asset and factor dynamics In Section

3 we formulate the control problem and apply a change of measure to obtain asimpler auxiliary criterion Section 4 outlines the properties of the value function

In Section 5 we show that the value function is a viscosity solution of the RS HJBPIDE before proving a comparison result in Section 6 which provides uniqueness

Our analytical setting is based on that of [19] The notable difference is that

we allow the factor processes to experience jumps

The growth rates of the assets are assumed to depend on n valuation factors

X1(t), , X n (t) which follow the dynamics given in equation (4) below The assets market comprises m risky securities S i , i = 1, , m Let M := n + m Let

(Ω, {Ft} , F , P) be the underlying probability space On this space is defined an

RM-valued (Ft )-Brownian motion W(t) with components W k (t), k = 1, , M.

Moreover, let (Z, B Z) be a Borel space1 Let p be an (Ft)-adapted σ-finite Poisson

point process on Z whose underlying point functions are maps from a countable set D p ⊂ (0, ∞) into Z Define

Consider Np(dt, dz), the Poisson random measure on (0, ∞)×Z induced by p

Fol-lowing Davis and Lleo [19], we concentrate on stationary Poisson point processes

of class (QL) with associated Poisson random measure Np(dt, dx) The class (QL)

is defined in [27] (Definition II.3.1, p 59) as

(QL) with respect to (Ft) if it is σ-finite and there exists ˆNp= ˆNp(t, U)

such that

(i) for U ∈ Z p , t 7→ ˆ Np(t, U) is a continuous (F t)-adapted increasing process;

(ii) for each t and a.a ω ∈ Ω, U 7→ ˆ Np(t, U) is a σ-finite measure on (Z, B(Z));

(iii) for U ∈ Z p , t 7→ ˜ Np(t, U) = Np(t, U) − ˆ Np(t, U) is an (F t)-martingale;The random measuren ˆNp(t, U)o

is called the compensator of the point process p.

1Z is a standard measurable (metric or topological) space and B Zis the Borel σ-field endowed to

Z.

Trang 16

4

Since the Poisson point processes we consider are stationary, then their sators are of the form ˆNp(t, U) = ν(U)t, where ν is the σ-finite characteristic

compen-measure of the Poisson point process p For notational convenience, we define the

Poisson random measure ¯Np(dt, dz) as

(See for example Definition II.4.1 in Ikeda and Watanabe [27] where F P and F2,locP

are given in equations II(3.2) and II(3.5) respectively.)

Let S0denote the wealth invested in the money market account with dynamicsgiven by the equation:

asset Let S i (t) denote the price at time t of the ith security, with i = 1, , m The dynamics of risky security i can be expressed as:

Trang 17

i < 0 < γmax i < +∞, i = 1, , m for i = 1, , m Furthermore, define

i ] is the smallest closed hypercube containing ˜S.

In addition, the vector-valued function γ(z) satisfies:

Define the set J as

For a given z, the equation h0γ(z) = −1 describes a hyperplane in R m Under sumption 2.1 J is a convex subset of Rm

We will assume that:

and

Assumption 2.3 The systematic (factor-driven) and idiosyncratic (asset-driven) jump risks are uncorrelated, i.e ∀z ∈ Z and i = 1, , m, γi (z)ξ0(z) = 0.

Trang 18

6

The second assumption implies that there cannot be simultaneous jumps in thefactor process and any asset price process This assumption, which will provesufficient to show the existence of a unique optimal investment policy, may appearsomewhat restrictive as it does not enable us to model a jump correlation structureacross factors and assets, although we can model a jump correlation structurewithin the factors and within the assets

Remark 2.1 Assumption (2.3) is automatically satisfied when jumps are only

allowed in the security prices and the state variable X(t) is modelled using a

diffu-sion process (see [19] for a full treatment of this case)

Let Gt := σ((S (s), X(s)), 0 ≤ s ≤ t) be the sigma-field generated by the rity and factor processes up to time t.

secu-An investment strategy or control process is an R m-valued process with the

interpretation that h i (t) is the fraction of current portfolio value invested in the ith asset, i = 1, , m The fraction invested in the money market account is then

h0(t) = 1 −Pm

i=1 hi (t).

conditions are satisfied:

1 h(t) is progressively measurable with respect to {B([0, t]) ⊗ G t}t≥0 and isc`adl`ag;

Proof The proof of this result is immediate.

Definition 2.3 A control process h(t) is in class A(T ) if the following conditions

are satisfied:

1 h(t) ∈ H ∀t ∈ [0, T ];

Trang 19

G(z, h; θ) = 1 − 1 + h0γ(z)−θ

(10)

Definition 2.4 We say that a control process h(t) is admissible if h(t) ∈ A(T ).

The proportion invested in the money market account is h0(t) = 1 −Pm

i=1 hi (t) Taking this budget equation into consideration, the wealth V(t, x, h), or V(t), of the investor in response to an investment strategy h(t) ∈ H , follows the dynamics

dV(t) V(t−) =

a0+ A00X(t)dt + h0(t)a − a01 +A − 1A00X(t)dt +h0(t)ΣdW t+

Z

h0(t)γ(z) ¯ Np(dt, dz)

Defining ˆa := a − a01 and ˆA := A − 1A00, we can express the portfolio dynamics as

as-sensitive management problem is to find h∗(t) ∈ A(T ) that maximizes the control

Trang 20

Z

( 1θ

Trang 21

is a standard Brownian motion under the measure Pθhand we define the Pθh pensated Poisson measure as

"

exp(θ

t g(X s , h(s); θ)ds − θ ln v

)#(20)

where Eh,θ t,x[·] denotes the expectation taken with respect to the measure Pθh

and with initial conditions (t, x).

• the exponentially transformed criterion

˜I(v, x, h; t, T ; θ) := E h,θ

t,x

"

exp(θ

t g(Xs , h(s); θ)ds − θ ln v

)#

(21)which we will find convenient to use in our derivations

We have completed our reformulation of the problem under the measure Pθh Thestate dynamics (18) is a jump-diffusion process and our objective is to maximizethe criterion (20) or alternatively minimize (21)

In this section we derive the risk-sensitive Hamilton-Jacobi-Bellman partialintegro differential equation (RS HJB PIDE) associated with the optimal controlproblem Since we do not anticipate that a classical solution generally exists, wewill not attempt to derive a verification theorem Instead, we will show that the

Trang 22

10

value function Φ is a solution of the RS HJB PIDE in the viscosity sense In fact,

we will show that the value function is the unique continuous viscosity solution

of the RS HJB PIDE This result will in turn justify the association of the RS HJBPIDE with the control problem and replace the verification theorem we wouldderive if a classical solution existed

Let Φ be the value function for the auxiliary criterion function I(v, x; h; t, T )

defined in (20) Then Φ is defined as

Similarly, let ˜Φ be the value function for the auxiliary criterion function

˜I(v, x; h; t, T ) Then ˜Φ is defined as

Trang 23

Under Assumption 2.2 the term

Z

nh

1 + h0γ(z)−θ

− 1ioν(dz)

which is also concave in h ∀z ∈ Z a.s dν Therefore, the supremum is reached

for a unique optimal control h∗, which is an interior point of the set J defined in

equation (7), and the supremum, evaluated at h∗, is finite

Trang 24

12

As in [19], we will use “zero beta” (0β) policies (initially introduced byBlack [16]))

Definition 4.1 1.20β-policy]By reference to the definition of the function g in

equation (15), a ‘zero beta’ (0β) control policy ˇh(t) is an admissible control policy for which the function g is independent from the state variable x.

In our problem, the set Z of 0β-policies is the set of admissible policies ˇh

which satisfy the equation

ˇh0A = −Aˆ 0

As m > n, there is potentially an infinite number of 0β-policies as long as the

following assumption is satisfied

Without loss of generality, we fix a 0β control ˇh as a constant function of time

so that

g(x, ˇh; θ) = ˇg where ˇg is a constant.

Proposition 4.1 The value function Φ(t, x) is convex in x.

Proof See the proof of Proposition 6.2 in [19].

Trang 25

4.3 Boundedness

bounded, i.e there exists M > 0 such that

t g(Xs , h(s); θ)ds − θ ln v

Z

( 1θ

Trang 26

14

Remark 4.1 Key to this assumption is the condition (35) which imposes a

spe-cific constraint on one element of each of the 2n vectors β k (t) To clarify the structure of this constraint, define Mβ−as the square n × n matrix whose i-th column (with i = 1, , n) is the n-element column vector β i (t) Then all the elements

m−

j j , j = 1, , m on the diagonal of M−

β are such that

m−j j=βj j (t) < 0 Similarly, define M+

β as the square n × n matrix whose i-th column (with i =

1, , n) is the n-element column vector β n+i (t) Then all the elements m+

j j , j =

1, , m on the diagonal of M+

β are such that

m+j j=βn+ j j (t) > 0 Note that there is no requirement for either M−β or M+

β to have full rank

It would in fact be perfectly acceptable to have rank 1 as a result of columnduplication

Moreover, the existence of 2n constant controls ¯h k , k = 1, , 2n such that (33) satisfies (35) is only guaranteed when J = R n However, since finding the controls

is equivalent to solving a system of at most n inequalities with m variables and

m > n, it is likely that one could find constant controls after some adjustments to the elements of the matrices A0, A, B or to the maximum jump size allowed.

Proposition 4.3 Suppose Assumption 4.2 holds and consider the 2n constant

controls ¯h k , k = 1, , 2n parameterizing the 4n functions

αk : [0, T ] → R, k = 1, , 2n

βk : [0, T ] → R n , k = 1, , 2n such that for i = 1, , n,

βi i (t) < 0

βn+i i (t) > 0

following upper bounds:

˜

Φ(t, x) ≤ eαk (t)+β k0 (t)x

in each element xi , i = 1, , n of x.

Trang 27

Proof Setting Z = Rn− {0} and recalling that the dynamics of the state variable

X(t) under the Pθh-measure is given by

We will now limit ourselves the class Hcof constant controls By the

opti-mality principle, for an arbitrary admissible constant control policy ¯h, we have

˜

Φ(t, x) ≤ ˜I(x; ¯h; t, T ) ≤ E t,x

"

exp(θ

t g(Xs , ¯h)ds − θ ln v

where

α : t ∈ [0, T ] → R

β : t ∈ [0, T ] → R n

are functions solving two ODEs

Indeed, applying the Feynman-Kac formula, we find that the function W(t, x)

satisfies the integro-differential PDE:

Trang 28

Z

( 1θ

Trang 29

Equations (41) and (42) are respectively equations (33) and (34) from

As-sumption 4.2 By AsAs-sumption 4.2, there exists 2n constant controls ¯h k , k =

1, , 2n such that for i = 1, , n,

βi i (t) < 0

βn+i

i (t) > 0

where βi

j (t) denotes the jth component of the vector β i (t) We can now conclude

that we have the following upper bounds

˜

Φ(t, x) ≤ eαk (t)+β k0 (t)x for each element x , i = 1, , n of x.

Trang 30

18

Remark 4.3 To obtain the upper bounds and the asymptotic behaviour, we do

not need the 2n constant controls to be pairwise different In fact, we need at least

2 different controls and at most 2n different controls Moreover, we could consider

wider classes of controls extending beyond constant controls This would requiresome modifications to the proof but would also alleviate the assumptions requiredfor the result to hold

Remark 4.4 For a given constant control ¯h, equation (39) is a linear

n-dimensional ODE However, if in the dynamics of the state variable X(t), Λ and Ξ depended on X, the ODE would be nonlinear Once ODE (39) is solved, obtaining α(t) from equation (40) is a simple matter of integration.

solution of ODE (39) is the same whether the dynamics of S (t) and X(t) is the

jump diffusion considered here or the corresponding pure diffusion model The

converse is, however, not true since in the pure diffusion setting h ∈ R m, while in

the jump diffusion case h ∈ J ⊂ R m

In recent years, viscosity solutions have gained a widespread acceptance as aneffective technique to obtain a weak sense solution for HJB PDEs when no classi-

cal (i.e C1,2) solution can be shown to exist, which is the case for many stochasticcontrol problems Viscosity solutions also have a very practical interest Indeed,once a solution has been interpreted in the viscosity sense and the uniqueness ofthis solution has been proved via a comparison result, the fundamental ‘stability’result of Barles and Souganidis [8] opens the way to a numerical resolution ofthe problem through a wide range of schemes Readers interested in an overview

of viscosity solutions should refer to the classic article by Crandall, Ishii and ons [17], the book by Fleming and Soner [26] and Øksendal and Sulem [30], aswell as the notes by Barles [5] and Touzi [34]

Li-While the use of viscosity solutions to solve classical diffusion-type stochasticcontrol problems has been extensively studied and surveyed (see Fleming andSoner [26] and Touzi [34]), this introduction of a jump-related measure makes thejump-diffusion framework more complex As a result, so far no general theoryhas been developed to solve jump-diffusion problems Instead, the assumptionsmade to derive a comparison result are closely related to what the specific problemallows Broadly speaking, the literature can be split along two lines of analysis,depending on whether the measure associated with the jumps is assumed to befinite

In the case when the jump measure is finite, Alvarez and Tourin [1] consider

a fairly general setting in which the jump term does not need to be linear in the

function u which solves the integro-differential PDE In this setting, Alvarez and

Trang 31

Tourin develop a comparison theorem that they apply to a stochastic differentialutility problem Amadori [3] extends Alvarez and Tourin’s analysis to price Eu-ropean options Barles, Buckdahn and Pardoux [6] study the viscosity solution ofintegro-differential equations associated with backward SDEs (BSDEs).

The Lévy measure is the most extensively studied measure with singularities.Pham [33] derives a comparison result for the variational inequality associatedwith an optimal stopping problem Jakobsen and Karlsen [29] analyse in detailthe impact of the Lévy measure’s singularity and propose a maximum principle.Amadori, Karlsen and La Chioma [4] focus on geometric Lévy processes and thepartial integro differential equations they generate before applying their results

to BSDEs and to the pricing of European and American derivatives A recentarticle by Barles and Imbert [7] takes a broader view of PDEs and their non-local operators However, the authors assume that the nonlocal operator is broadlyspeaking linear in the solution which may prove overly restrictive in some cases,including our present problem

As far as our jump diffusion risk-sensitive control problem is concerned, wewill promote a general treatment and avoid restricting the class of the compen-sator ν At some point, we will however need ν to be finite This assumption willonly be made for a purely technical reason arising in the proof of the comparisonresult (in Section 6) Since the rest of the story is still valid if ν is not finite, and inaccordance with our goal of keeping the discussion as broad as possible, we willwrite the rest of the article in the spirit of a general compensator ν

Before proceeding further, we will introduce the following definition:

defined as

u∗(x) = lim sup

y→x u(y) and the lower semicontinuous envelope u∗(x) of u(x) is defined as

Trang 32

Definition 5.2 A functional H(x, r, p, A) is said to be proper if it satisfies the

following two properties:

plays a similar role to the functional H in the general equation (43), and we note

that it is indeed “proper” As a result, we can develop a viscosity approach toshow that the value function Φ is the unique solution of the associated RS HJBPIDE

We now give two equivalent definitions of viscosity solutions adapted fromAlvarez and Tourin [1]:

• a definition based on the notion of semijets;

• a definition based on the notion of test function

Before introducing these two definitions, we need to define parabolic semijet ofupper semicontinuous and lower semicontinuous functions and to add two addi-tional conditions

Trang 33

• the Parabolic superjet P2,+u as

k→∞ (t k , x k , u(t k , x k )) = (t, x, u(t, x))

Let u ∈ LS C([0, T ] × R n ) and (t, x) ∈ [0, T ] × R n We define:

• the Parabolic subjet P2,−u as P2,−u := −P2,+u , and

• the closure of the Parabolic subjet P

2,−

u as P2,−u =−P2,+u

C(R n ), ϕ ≥ 1 and R > 0 such that for

Trang 34

22

(6) and (7) in [1]) In our setting, we note that since the value function Φ and the

function x 7→ e xare locally bounded, these two conditions are satisfied

Remark 5.1 Note that the jump-related integral term

Z

(u(s, y + ξ(z)) − u(s, y)) −θ

2(u(s, y + ξ(z)) − u(s, y))2

Similarly, by definition of the Parabolic subjet P2,−u , for t = s, the pair (q, A)

satisfies the inequality

Trang 35

Definition 5.4 A locally bounded function u ∈ US C([0, T ] × R n) satisfying

Con-dition 5.1 is a viscosity subsolution of (23), if for all x ∈ R n , u(T, x) ≤ g0(x), and for all (t, x) ∈ [0, T ] × R n , (p, q, A) ∈ P2,+u(t, x), we have

A locally bounded function u ∈ LS C([0, T ] × R n) satisfying Condition 5.2 is

a viscosity supersolution of (23), if for all x ∈ R n , u(T, x) ≥ g0(x), and for all (t, x) ∈ [0, T ] × R n , (p, q, A) ∈ P2,−u(t, x), we have

subsolution of (23), if for all x ∈ R n , u(T, x) ≤ g0(x), and for all (t, x) ∈ [0, T ]×R n,

ψ ∈ C2([0, T ] × R n ) such that u(t, x) = ψ(t, x), u < ψ on [0, T ] × R n \ {(t, x)}, we

A locally bounded function v ∈ LS C([0, T ] × R n) is a viscosity supersolution

of (23), if for all x ∈ R n , v(T, x) ≥ g0(x), and for all (t, x) ∈ [0, T ] × R n, ψ ∈

C2([0, T ] × R n ) such that v(t, x) = ψ(t, x), v > ψ on [0, T ] × R n \ {(t, x)}, we have

2

ψ) −Z

Remark 5.2 A more classical but also more restrictive definition of viscosity

solution is as the continuous function which is both a supersolution and a solution of (23) (see Definition 5.1 in Barles [5]) The line of reasoning we willfollow will make full use of the latitude afforded by our definition and we willhave to wait until the comparison result is established in Section 6 to prove thecontinuity of the viscosity solution

Trang 36

sub-May 3, 2010 13:34 Proceedings Trim Size: 9in x 6in 001

24

To show that the value function is a (discontinuous) viscosity solution of theassociated RS HJB PIDE (23), we follow an argument by Touzi [34] which en-ables us to make a greater use of control theory in the derivation of the proof

Theorem 5.1 Φ is a (discontinuous) viscosity solution of the RS HJB PIDE (23)

on [0, T ] × R n , subject to terminal condition (25).

Proof.

a log transformation of Φ In the next three steps, we prove that ˜Φ is a viscositysolution of the exponentially transformed RS HJB PIDE by showing that it is 1) aviscosity subsolution; 2) a viscosity supersolution; and hence 3) a viscosity solu-tion Finally, applying a change of variable result, such as Proposition 2.2 in [34],

we conclude that Φ is a viscosity solution of the RS HJB PIDE (23)

Step 1: Exponential Transformation

In order to prove that the value function Φ is a (discontinuous) viscosity lution of (23), we will start by proving that the exponentially transformed valuefunction ˜Φ is a (discontinuous) viscosity solution of (27)

so-Step 2: Viscosity Subsolution

Let (t0, x0) ∈ Q := [0, t] × R n and u ∈ C1,2(Q) satisfy

0 = ( ˜Φ∗− u)(t0, x0) = max

(t,x)∈Q( ˜Φ∗(t, x) − u(t, x)) (44)and hence

k→∞

˜

Φ(t k , x k) = ˜Φ∗(t0, x0)and define the sequence {ξ}kas ξk:= ˜Φ(t k , x k ) − u(t k , x k ) Since u is of class C1,2,

Trang 37

Fix h ∈ J and consider a constant control ˆh = h Denote by X kthe state

process with initial data X t k k = x k and, for k > 0, define the stopping time

τk:= infns > tk : (s − t k , X s k − x k) < [0, δk) × αBn

o

for a given constant α > 0 and where Bnis the unit ball in Rnand

δk:= pξk 1 − 1{0}(ξk) + k−11{0}(ξk)From the definition of τk, we see that limk→∞τk = t0

By the Dynamic Programming Principle,

˜

Φ(t k , x k) ≤ Et k ,x k

"

exp(θ

Z τk

t k g(Xs , ˆh s ; θ)ds

)

˜Φ(τk , X k k)

Z τk

t k g(Xs , ˆh s )ds

Z τk

t k g(Xs , ˆh s )ds

Trang 38

26

By the Itˆo product rule, and since dZ s · u s= 0, we get

duse Z s= u sde Z s+ e Z s dus and hence for t ∈ [t k, τk]

Noting that u(t k , x k )e Z tk = u(t k , x k) and taking the expectation with respect to the

initial data (t k , x k), we get

Trang 39

a.s by the Bounded Convergence Theorem, since the random variable

is bounded for large enough k.

Hence, we conclude that since ˆh sis arbitrary,

This argument proves that ˜Φ is a (discontinuous) viscosity subsolution of the

PDE (27) on [0, t) × R nsubject to terminal condition ˜Φ(T, x) = e g0(x;T )

Step 3: Viscosity Supersolution

This step in the proof is a slight adaptation of the proof for classical control

problems in Touzi [34] Let (t0, x0) ∈ Q and u ∈ C1,2(Q) satisfy

0 = ( ˜Φ∗− u)(t0, x0) < ( ˜Φ∗− u)(t, x) for Q\(t0, x0) (47)

We intend to prove that at (t0, x0)

Trang 40

ζ := max

Note that ζ < ∞ by boundedness of ξ(z) and thus Mδ< ∞

Now let (t k , x k) be a sequence in Nδsuch that

lim

k→∞ (t k , x k ) = (t0, x0)and

lim

k→∞

˜

Φ(t k , x k) = ˜Φ∗(t0, x0)Since ( ˜Φ− u)(t k , x k ) → 0, we can assume that the sequence (t k , x k) satisfies

for defined by (51)

Consider the -optimal control h

k, denote by ˜X

kthe controlled process defined

by the control process h

kand introduce the stopping time

( ˜Φ− u)(τ k, ˜X k(τk)) ≥ ( ˜Φ∗− u)(τ k, ˜X k(τk )) ≥ 3e −δθMδ (54)

22

(6) and (7) in [1]) In our setting, we note that since the... true since in the pure diffusion setting h ∈ R m, while in

the jump diffusion case h ∈ J ⊂ R m

In recent years, viscosity solutions have gained a...

As in [19], we will use “zero beta” (0β) policies (initially introduced byBlack [16]))

Definition 4.1 1.20β-policy]By reference to the definition of the function g in< /b>

Định dạng
Số trang	284
Dung lượng	1,88 MB