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Multidimensional Wick-Itˆo Formula for Gaussian Processes.. Multidimensional Wick-Itˆ o Formula for Gaussian Processes 5where The mapping 1i[0,t]7→ Xtican be extended to a linear isometr

Stochastic Analysis, Stochastic Systems, and Applications to Finance

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ISBN-13 978-981-4355-70-4

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STOCHASTIC ANALYSIS, STOCHASTIC SYSTEMS, AND APPLICATIONS

TO FINANCE

Contents

Preface viiContributors and Addresses ix

Part I Stochastic Analysis and Systems

1 Multidimensional Wick-Itˆo Formula for Gaussian Processes 3

D Nualart and S Ortiz-Latorre

2 Fractional White Noise Multiplication 27

A H Tsoi

3 Invariance Principle of Regime-Switching Diffusions 43

C Zhu and G Yin

Part II Finance and Stochastics

4 Real Options and Competition 63

A Bensoussan, J D Diltz, and S R Hoe

5 Finding Expectations of Monotone Functions of Binary RandomVariables by Simulation, with Applications to Reliability,

Finance, and Round Robin Tournaments 101

M Brown, E A Pek¨oz, and S M Ross

6 Filtering with Counting Process Observations and Other

Factors: Applications to Bond Price Tick Data 115

X Hu, D R Kuipers, and Y Zeng

vi Contents

7 Jump Bond Markets Some Steps towards General Models

in Applications to Hedging and Utility Problems 145

M Kohlmann and D Xiong

8 Recombining Tree for Regime-Switching Model: Algorithm

and Weak Convergence 193

Preface

This volume contains 11 chapters It is an expanded version of the paperspresented at the first Kansas–Missouri Winter School of Applied Probabil-ity, which was organized by Allanus Tsoi and was held at the University ofMissouri, February 14 and 15, 2008 It brought together researchers fromdifferent parts of the country to review and to update the recent advances,and to identify future directions in the areas of applied probability, stochas-tic processes, and their applications

After the successful conference was over, there was a strong support

of publishing the papers delivered in the conference as an archival volume.Based on the support, we began the preparation on this project In addition

to papers reported at the conference, we have invited a number of colleagues

to contribute additional papers

As an archive, this volume presents some of the highlights of the ference, as well as some of most recent developments in stochastic systemsand applications This book is naturally divided into two parts The firstpart contains some recent results in stochastic analysis, stochastic processesand related fields It explores the Itˆo formula for multidimensional Gaussianprocesses using the Wick integral, introduces the notion of fractional whitenoise multiplication, and discusses the LaSalle type of invariance principlesfor hybrid switching diffusions The second part of the book is devoted to fi-nancial mathematics, insurance models, and applications Included here areoptimal investment policies for irreversible capital investment projects un-der uncertainty in monopoly and Stackelberg leader-follower environments,

con-viii Preface

finding expectations of monotone functions of binary random variables bysimulation, with applications to reliability, finance, and round robin tour-naments, jump bond markets with general models in applications to hedg-ing and utility problems, algorithm and weak convergence for recombiningtree in a regime-switching model, applications of counting processes andmartingales in survival analysis, extended filtering micro-movement modelwith counting process observations and applications to bond price tick data,optimal reinsurance for a jump diffusion model, recursive algorithms andnumerical studies for mean-reverting asset trading

Without the encouragement and assistance of many colleagues, this ume would have never come into being We thank all the authors of thisvolume, and all of the speakers of the conference for their contributions Thefinancial support provided by the University of Missouri for this conference

vol-is also greatly acknowledged

Allanus TsoiColumbia, MissouriDavid NualartLawrence, KansasGeorge YinDetroit, Michigan

Contributors and Addresses

• Alain Bensoussan, School of Management, University of Texas atDallas, Richardson, TX 75083-0688, USA & The Hong Kong Poly-technic University, Hong Kong Email: alain.bensoussan@utdallas.edu

• Mark Brown, Department of Mathematics, City College, CUNY,New York, NY, USA Email: cybergarf@aol.com

• J David Diltz, Department of Finance and Real Estate, The versity of Texas at Arlington, Arlington, TX 76019, USA Email:diltz@uta.edu

Uni-• SingRu Hoe, School of Management, University of Texas at Dallas,Richardson, TX 75083-0688, USA Email: celinehoe@utdallas.edu

• Xing Hu, Department of Economics, Princeton University, ton, 08544, USA Email: xinghu@princeton.edu

Prince-• Michael Kohlmann, Department of Mathematics and Statistics,University of Konstanz, D-78457, Konstanz, Germany Email:michael.kohlmann@uni-konstanz.de

• David R Kuipers, Department of Finance, Henry W BlochSchool of Business and Public Administration, University of Mis-souri at Kansas City, Kansas City, MO 64110, USA Email:kuipersd@umkc.edu

• Ruihua Liu, Department of Mathematics, University of Dayton,

300 College Park, Dayton, OH 45469-2316, USA Email: hua.liu@notes.udayton.edu

rui-x Contributors and Addresses

• Shangzhen Luo, Department of Mathematics, University of ern Iowa, Cedar Falls, Iowa, 50614-0506, USA Email: luos@uni.edu

North-• David Nualart, Department of Mathematics, University of Kansas,Lawrence, KS 66045, USA Email: nualart@math.ku.edu

• Salvador Ortiz-Latorre, Departament de Probabilitat, L`ogica i tad´ıstica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona,Spain Email: sortiz@ub.edu

Es-• Erol A Pekoz, School of Management, Boston University,

595 Commonwealth Avenue, Boston, MA 02215, USA Email:pekoz@bu.edu

• Sheldon M Ross, Department of Industrial and Systems ing, University of Southern California, Los Angeles, CA 90089,USA Email: smross@usc.edu

Engineer-• Jianguo Sun, Department of Statistics, University of Missouri,USA Email: sunj@missouri.edu

• Allanus Hak-Man Tsoi, Department of Mathematics, University ofMissouri, Columbia, MO 65211, USA Email: tsoia@missouri.edu

• Dewen Xiong, Department of Mathematics, Shanghai JiaotongUniversity, Shanghai 200240, People’s Republic of China Email:xiongdewen@sjtu.edu.cn

• George Yin, Department of Mathematics, Wayne State University,Detroit, MI 48202, USA Email: gyin@math.wayne.edu

• Yong Zeng, Department of Mathematics and Statistics, University

of Missouri at Kansas City, Kansas City, MO 64110, USA Email:zengy@umkc.edu

• Qing Zhang, Department of Mathematics, University of Georgia,Athens, GA 30602, USA Email: qingz@math.uga.edu

• Chao Zhu, Department of Mathematical Sciences, University

of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA Email:zhu@uwm.edu

• Chao Zhuang, Marshall School of Business, University of SouthernCalifornia, Los Angeles, CA 90089, USA Email: czhuang@usc.edu

S Ortiz-Latorre Departament de Probabilitat, L` ogica i Estad´ ıstica, Universitat de Barcelona

Gran Via 585, 08007 Barcelona, Spain Email: sortiz@ub.edu

An Itˆ o formula for multidimensional Gaussian processes using the Wick gral is obtained The conditions allow us to consider processes with infinite quadratic variation As an example we consider a correlated heterogenous frac- tional Brownian motion We also use this Itˆ o formula to compute the price of

inte-an exchinte-ange option in a Wick-fractional Black-Scholes model.

Keywords : Wick-Itˆ o formula; Gaussian processes; Malliavin calculus.

1 Introduction

The classical stochastic calculus and Itˆo’s formula can be extended to martingales There has been a recent interest in developing a stochasticcalculus for Gaussian processes which are not semimartingales such as thefractional Brownian motion (fBm for short) These developments are mo-tivated by the fact that fBm and other related processes are suitable inputnoises in practical problems arising in a variety of fields including finance,telecommunications and hydrology (see, for instance, Mandelbrot and VanNess7 and Sottinen13)

semi-A possible definition of the stochastic integral with respect to the fBm

is based on the divergence operator appearing in the stochastic calculus

of variations This approach to define stochastic integrals started from thework by Decreusefond and ¨Ust¨unel3and was further developed by Carmona

4 D Nualart and S Ortiz-Latorre

and Coutin2 and Duncan, Hu and Pasik-Duncan4 (see also Hu5 and alart9 for a general survey papers on the stochastic calculus for the fBm).The divergence integral can be approximated by Riemman sums definedusing the Wick product, and it has the important property of having zeroexpectation

Nu-Nualart and Taqqu11,12 have proved a Wick-Itˆo formula for generalGaussian processes In 11 they have considered Gaussian processes withfinite quadratic variation, which includes the fBm with Hurst parameter

H > 1/2 The paper 12 deals with the change-of-variable formula for sian processes with infinite quadratic variation, in particular the fBm withHurst parameterH ∈ (1/4, 1/2) The lower bound for H is a natural one,see Al`os, Mazet and Nualart.1

Gaus-The aim of this paper is to generalize the results of Nualart and Taqqu12

to the multidimensional case We introduce the multidimensional Wick-Itˆointegral as a limit of forward Riemann sums and prove a Wick-Itˆo formulaunder conditions similar to those in Nualart and Taqqu,12allowing infinitequadratic variation processes

The paper is organized as follows In Section 2, we introduce the ditions that the multidimensional Gaussian process must satify and statethe Itˆo formula Section 3 contains some definitions in order to introducethe Wick integral In Section 4 we prove some technical lemmas using ex-tensively the integration by parts formula for the derivative operator Theconvergence results used in the proof of the main theorem are proved inSection 5 Section 6 is devoted to study two examples related to the multi-dimensional fBm with parameterH > 1/4 Finally, in section 7 we use ourItˆo formula to compute the price of an exchange option on a Wick-fractionalBlack-Scholes market

con-2 Preliminaries

LetX = {Xt, t ∈ [0, T ]} be a d-dimensional centered Gaussian process withcontinuous covariance function matrixR(s, t), that is,

Ri,j(s, t) = E[XsiXtj],fori, j = 1, , d For s = t, we have the covariance matrix Vt=R(t, t)

We denote by H be the space obtained as the completion of the set ofstep functions inA = [0, T ] × {1, , d} with respect the scalar product

Multidimensional Wick-Itˆ o Formula for Gaussian Processes 5

where

The mapping 1i[0,t]7→ Xtican be extended to a linear isometry between thespace H and the Gaussian Hilbert space generated by the process X Wedenote by letI1:h 7→ X (h) , h ∈ H this isometry

Let H⊗mdenote themth tensor power of H, equipped with the followingscalar product

Now consider the set of smooth random variables S A random variable

F ∈ S has the form

F = f (X (h1), , X (hn)), (1)withh1, , hn∈ H, n ≥ 1, and f ∈ Cb∞(Rn) (f and all its partial deriva-tives are bounded) In S one can define the derivative operator D as

(Ω; H) By iteration one obtains

Definition 2.1 For m ≥ 1, the space Dm,2

is the completion of S withrespect to the norm kF km,2 defined by

kF k2m,2= E[F2] +

m

X

E[ DiF 2H⊗i]

6 D Nualart and S Ortiz-Latorre

The Wick product F  X (h) between a random variable F ∈ D1,2 andthe Gaussian random variableX (h) is defined as follows

Definition 2.2 LetF ∈ D1,2andh ∈ H Then the Wick product F X (h)

is defined by

F  X (h) = F X (h) − hDF, hiH.Actually, the Wick product coincides with the divergence (or the Sko-rohod integral) ofF h, and by the properties of the divergence operator wecan write

E[F X (h)] = E [hDF, hiH] (2)The Wick integral of a stochastic processu with respect to X is defined

as the limit of Riemann sums constructed using the Wick product For this

we need some notation Denote by D the set of all partitions of [0, T ]

π = {0 = t0< t1< · · · < tn =T }such that

Multidimensional Wick-Itˆ o Formula for Gaussian Processes 7

s∆iXk] 2→ 0, as |π| → 0,

where ∆iXj=Xtji+1− Xtji, and π runs over all partitions of [0, T ]

in the class D

Our purpose is to derive a change-of-variable formula for the process

f (Xt), wheref : Rd→ R if a function satisfying the following condition.(A4) For every multi-indexα = (α1, , αd) ∈ Nd with |α| := α1+ · · · +

αd≤ 7, the iterated derivatives

8 D Nualart and S Ortiz-Latorre

Besides the multi-index notation for the derivatives, we will also use thefollowing notation for iterated derivatives Letf (x1, , xd) be a sufficientlysmooth function, then

The next theorem is the main result of the paper

Theorem 3.1 Suppose that the Gaussian process X and the function fsatisfy the preceding assumptions (A1) to (A4) Then the forward integrals(see Definition 2.3)

Z t

0

∂jf (Xs)  dXsj, 0 ≤ t ≤ T, j = 1, , dexist and the following Wick-Itˆo formula holds:

3 (i) + 14!Tπ

4 (i) ,where

Multidimensional Wick-Itˆ o Formula for Gaussian Processes 9

whereδi= (ti, ti+1] Taking into account that

ϕj,ki = EhXtji+1− Xtji Xk

i.This gives

10 D Nualart and S Ortiz-Latorre

Multidimensional Wick-Itˆ o Formula for Gaussian Processes 11

Remark 3.1 We can also consider a function f (t, x) depending on timesuch that the partial derivative ∂f∂t(t, x) exists and is continuous In thiscase we obtain the additional termRt

Lemma 4.2 Let F ∈ D2,2 andh, g ∈ H Then

E[F X (h) X (g)] = E[ 2F, h g

H ⊗2] + E [F ] hh, giH.Proof See Nualart and Taqqu12, Lemma 6

Lemma 4.3 Let F ∈ D2,2, h, g ∈ H, ξ = X (h) X (g) − hh, giH Then

E[F ξ1ξ2] = E[ 2(F ξ1), h2 g2H⊗2]

Now, by the Leibniz rule for the derivative operator,

D2(F ξ ) = D2F
ξ + 2DF Dξ +F D2ξ ,

12 D Nualart and S Ortiz-Latorre

where

Dξ1=h1X (g1) +X (h1)g1,

D2ξ1= 2 (h1 g1),and thus

D2(F ξ1) = D2F
ξ1+ 2X (g1) (DF h1) + 2X (h1) (DF g1)

+2F (h1 g1) =A1+ 2A2+ 2A3+ 2A4.Then,

E[hA1, h2 g2iH⊗2] = E[ξ1 2F, h2 ... (Springer-Verlag,Berlin, 2006)

11 D Nualart and M.S Taqqu, Wick-Itˆo formula for regular processes and cations to the Black and Scholes formula, Stochastics and Stochastics Reports,