Advances in finance and stochastics

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Advances in Finance and Stochastics Springer-Verlag Berlin Heidelberg GmbH Klaus Sandmann Philipp J Schonbucher (Eds.) Advances in Finance and Stochastics Essays in Honour of Dieter Sondermann With 32 Figures Springer Klaus Sandmann Johannes Gutenberg-Universităt Mainz Lehrstuhl fiir Bankbetriebslehre Jakob Welder-Weg 55128 Mainz Germany e-mail: Philipp J Schonbucher Rheinische Friedrich- Wilhelms-Universităt Bonn Inst f Gesellschafts- u Wirtschaftswissenschaften Statistische Abteilung Adenauerallee 24-42 53113 Bonn Germany e-mail: Catalog·in·Publication Data applied for Die Deutsche Bibliothek- CIP-Einheitsaufnahme Advances in finance and stochastics: essays in honour of Dieter Sondermann/ Klaus Sandmann; Philipp J Schonbucher (ed.) ISBN 978-3-642-07792-o ISBN 978-3-662-04790-3 (eBook) DOI 10.1007/978-3-662-04790-3 Mathematics Subject Classification (2ooo): 6o-6, 91B JEL Classification: G13 This work is subject to copyright AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH Violations are liable for prosecution under the German Copyright Law © Springer-Verlag Berlin Heidelberg 2002 Originally published by Springer-Verlag Berlin Heidelberg New York in 2002 Softcover reprint of the hardcover 1st edition 2002 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typeset by the authors using a Springer TJ3X macro package Cover production: design & production GmbH, Heidelberg SPIN 10996690 41/3142db - - Printed on acid-free paper Preface Finance and Stochastics and Dieter Sondermann are directly and inextricably linked to each other The recognition and the success of this journal would not have been possible without his untiring commitment, his sensitivity for scientific quality and originality as well as his trustworthiness when dealing with the authors One could almost say: Finance and Stochastics is Dieter Sondermann, since without him this journal would not be In the preface of the first issue of Finance and Stochastics in January 1997, Dieter Sondermann referring to the significance of the thesis of Louis Bachelier, stated: 'Thus, the year 1900 may be considered as the birth date of both Finance and Stochastics' Further on he wrote: 'The journal Finance and Stochastics is devoted to the fruitful interface of these two disciplines' What is there to add? It was important to identify and to articulate such a goal, yet to translate it into action and to make it possible was crucial It is due to Dieter Sondermann's initiative and constant work that the idea of Finance and Stochastics has turned into a highly reputable and successful project His unfailing commitment as founder and chief editor has made this journal an important publication forum of international renown A publication in Finance and Stochastics is a guarantee of originality and quality for scientific papers Thus, what could have been more natural than the idea of honouring Dieter Sondermann on the occasion of his 65th birthday with a collection of research papers entitled Advances in Finance and Stochastics? Those who know him would surely agree that especially Dieter Sondermann, in his modest and undemonstrative way, would never have approved of such an honour Luckily, the person to be honoured does not have a say in the matter However, if he had had one and had not been able to prevent it happening, it is likely that he would have warned us emphatically against a conception that was one-sided and looked back upon his own contributions He might even have considered the exercise quite superfluous Instead, his one and only concern would have been for the reader interested in scientific knowledge and the solution of problems 'The future has more Futures' This 'bon mot' of the financial market also holds good for Dieter Sondermann's scientific work and his involvement which has always been diverse and with a clear focus on the future Dieter VI Preface Sondermann was among the first scientists in Germany to apply themselves to the study of Mathematical Finance Influenced by the seminal work of Fisher Black, Myron Scholes and Robert C Merton and by virtue of his own profound understanding of the theory of general equilibrium, his contributions often mark the starting point for further development In 1985, with Hedging of Non-Redundant Contingent Claims, Dieter Sondermann, together with Hans Follmer, paved the way for the pricing and hedging of options in an incomplete financial market At an early stage he recognises the importance of the theory of arbitrage for the evaluation of insurance risk, which he demonstrates in Reinsurance in Arbitrage Free Markets (1991) With a similar feel for new ground, he proved the market model approach to the term structure of interest rates in his work Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates (1997, together with Kristian R Miltersen and Klaus Sandmann) Dieter Sondermann's academic career might be considered surprising and unusual, especially in its initial phase Yet, if one looks at it from today's perspective, one can see easily how each step and each stage form an integral part of a consistent whole He was born on May 10th, 1937 in Duisburg, Germany His early years not directly point to an academic career: his employment as a forwarding agent in the 'Rhenania Allgemeine Speditions AG' in 1953, his examination in 1956 as a business assistant and finally his activity until 1958 as an expedient in the shipping company 'Vereinigte Stinnes Reederei GmbH' in Duisburg Ruhrort During those years, two notions must have become rooted in his mind, his love of the Rhine and of shipping and his love of pursuing promising ideas After his Abitur in 1960, Dieter Sondermann embarked on his studies of Mathematics, Physics and Economics at the University of Bonn Little did he know (or even hope), when leaving Bonn in 1962 with a Vordiplom and heading for Hamburg, that he was to return as a professor of Economics and Statistics not quite 17 years later Many stages and formative encounters awaited him still After his Diplom in Mathematics in 1966, Prof Dr Heinz Bauer who had noticed this promising young mathematician from Hamburg, invited him to the University of Erlangen Here, after only two years, Dieter Sondermann obtained his Ph.D in 1968 There was no respite for the young academic with such diverse interests No sooner had he obtained his doctoral degree than his interest in Economics was kindled - and this with lasting effect, not least because of Theory of Value, the 'magic' book by Gerard Debreu It fascinated him and filled him with enthusiasm With his innate capacity for sound judgement he clearly grasped the opportunities and perspectives contained in the work - and he used them In 1970 Dieter Sondermann was appointed lecturer in Mathematical Economics at the University of Saarbriicken Yet, curiosity deriving from fascination requires scientific discussion Thus his path led to the Center of Operations Research and Econometrics, CORE, in Louvain, Belgium, where from 1970 to 1972 he was a visiting research professor Here at CORE a Preface VII great number of committed young scientists met up, and it was here that the foundations were laid for many scientific and personal friendships which were to last until the present day In 1972 Dieter Sondermann returned to Bonn, this time as Visiting Associate Professor, in 1973 he was in Berkeley, USA, and a year later he accepted a full professorship in Economics at the University of Hamburg, Germany The same year he joined the editorial board of the Journal of Mathematical Economics, founded by Werner Hildenbrand, on which he served until 1985 At the same time, from 1973 until 1980, he was a member of the editorial board of the Journal of Economic Theory, and from 1983 until 1992 of that of the Applicandae Mathematicae (Acta) In 1979 he became Fellow of the lAS at the Hebrew University, Jerusalem This is also the year in which he accepted a chair in Economics and Statistics at the University of Bonn Dieter Sondermann, Bonn, the Rheinische Friedrich-Wilhelms University and the Rhine are intertwined in so many different ways His house by the Rhine serves as a refuge for him, his wife, his family and their friends Even the perennial threat of high water cannot mar his lifelong attachment to the Rhine and Rhine shipping Instead, with a calmness that is so typical of him, he will contemplate such a phenomenon of nature in statistical terms With the same calmness, full of determination, and most successfully, Dieter Sondermann manages, from 1985 until 1999, Stochastics of Financial Markets, the subproject B of the Sonderforschungsbereich 303 During these 15 years, this research team, under his leadership, gains recognition at home and abroad and makes a lasting contribution towards the development and importance of Mathematical Finance His open and problem-oriented style of discussion deeply influences work methods and fosters an atmosphere of curiosity To bring into accord both research and teaching has always been for him - and still is - a constant matter of concern In a personal and human manner that is so characteristic of him, Dieter Sondermann has, throughout the years, supported and influenced the career of his numerous members of staff Many of his students, themselves now in responsible positions at universities or in industry, remain deeply indebted to him There are as many reasons for showing our gratitude to Dieter Sondermann as there are possibilities for expressing this With Advances in Finance and Stochastics we simply want to say: Thank you! The future has more Futures, Dieter! February 2002, Bonn Klaus Sandmann Philipp Schonbucher Introduction In many areas of finance and stochastics, significant advances have been made since this field of research was opened by Black, Scholes and Merton in 1973 The collection of contributions in Advances in Finance and Stochastics reflects this variety Necessarily, a selection of topics had to be made, and we endeavoured to choose those that are currently in the focus of active research and will remain so in future This selection spans risk management, portfolio theory and multi-asset derivatives, market imperfections, interest-rate modelling and exotic options Since Follmer and Sondermann (1986) published one of the first mathematical finance papers on risk management in incomplete markets, quantitative research has developed rapidly in this area The first three papers of this volume represent the recent developments in this area In the first paper on risk management, Delbaen extends the fundamental notion of a coherent risk measure in two directions from the original definition in Artzner (1999): the underlying probability space is now be a general probability space (and not finite) and the class of risks that are measured is extended to encompass all random variables on this space Using methods from the theory of convex games he is able to prove the analogies of the results of Artzner (1999) in this much more general setup But not everything carries through identically from the discrete setup: Delbaen shows that now a coherent risk measure has to be allowed to assume infinite values, representing completely unacceptable risks The following contribution by Follmer and Schied also treats coherent risk measures, but only as a special case of a more general class of risk measures: the convex risk measures The authors show that convex risk measures can be represented as a supremum of expectations under different measures, corrected by a penalty function that depends on the probability measure alone They also connect these risk measures to utilitybased risk measurement The third article on risk management is authored by Embrechts and Novak who give a survey of recent developments in the modelling and measurement of extremal events While the first two articles are concerned with the question of a consistent allocation of risk capital to a given set of risks, this article gives asymptotic answers to the question of the probability with which this level of risk capital will be exceeded X Introduction The part on portfolio theory opens with a paper by Werner in which he develops a multi-period extension to the CAPM, the APT and similar factor pricing models By measuring the risk of the assets in terms of the risk of the underlying dividend streams (instead of the one-period returns), the author is able to give conditions under which exact factor pricing relationships hold In contrast to this portfolio-selection problem, Duan and Pliska consider the pricing of options on multiple co-integrated assets Apart from providing necessary conditions for cointegration of a set of assets with GARCH-stochastic volatilities, they also study the effect that cointegrating relationships under the physical measure have on the dynamics of the assets under the equilibrium pricing measure and on the dynamics of risk premia In the following paper, Madan, Milne and Elliott study the effects that arise when several investors use different, individual factor pricing models, and these models are aggregated While Werner took the factor structure as given in his model, Madan want to understand where economy-wide risk factors and riskpremia arise from, they shift the focus from asset-returns to identifying and explaining investor-specific risk exposures Market imperfections are the theme of the next three contributions Kabanov and Stricker consider super-hedging strategies under transaction costs They characterise the hedging-set (the set of initial endowments that allow a self-financing super-replication) of a contingent claim in a general setup with non-constant transaction costs In the following paper, Frey and Patie address the problem of hedging options in illiquid markets In a simulation study they show that a hedging strategy based upon a nonlinear partial differential equation that includes liquidity effects can significantly improve the performance of the hedge In Frey and Patie's contribution illiquidity takes the form of market impact, Le the transactions of a large trader move prices, but he is able to trade at any time he chooses Rogers and Zane consider a different kind of illiquidity in the third paper of this group: Here, traders are only allowed to trade at Poisson arrival times which they cannot influence The traders' objective is a consumption/investment problem similar to Merton (1969) Rogers and Zane establish that Merton's investment rule (investing a fixed proportion of wealth in the risky asset) is still optimal, and characterize the modification of the optimal consumption process Using an asymptotic expansion, they assess the cost of illiquidity to the investor The two contributions on interest-rate modelling both build upon the market-modelling approach for observed effective interest rates by Miltersen, Sandmann and Sondermann (1997) Bhar provide an estimation methodology for a short-rate model which explicitly recognizes the fact that the short term interest-rate is unobservable Their approach aims to connect the stochastic models for the continuously compounded short rate with the observed effective, discretely compounded rates Introduction XI Schlogl analyses this connection in the other direction and shows that every market model implies a model for the continuously compounded short rate that is uniquely determined by the interpolation method used for rates maturing between tenor dates He provides an interpolation method which preserves the Markovian properties of discrete-tenor models but allows for continuous stochastic dynamics of the short rate The final set of contributions has its focus on specific pricing problems that arise in the pricing of exotic options, in particular the connection between insurance and financial markets, optimal stopping, and barrier features which all affect the payoff of the option in a nonlinear way The connection between the markets for insurance and financial risks has been a long-standing area of interest to Dieter Sondermann Nielsen and Sandmann analyse in their contribution one example where this connection is particularly evident: equity-linked life and pension insurance contracts The authors give results for the existence of a fair periodic premium and provide approximate and numerical results for their magnitude Optimal stopping is the theme of the contributions by Schweizer; Shepp, Shiryaev and Sulem; and Peskir and Shiryaev Schweizer analyses the optimal stopping problems posed by Bermudan options As Bermuda options can only be exercised in a subset of the lifetime of the option, the early exercise strategies are subject to this additional restriction Schweizer shows under which conditions the problem can be reduced to a modified American (unrestricted) optimal stopping problem, and how super-replication strategies can be derived in this setup Shepp, Shiryaev and Sulem consider an option that combines American early exercise, a knockout barrier and lookback-features: the barrier version of the Russian option Here, the early exercise strategies are restricted by the knockout barrier of the option Despite the complicated structure of the option, they are able to provide the optimal exercise strategy and the value function of this derivative The following contribution by Schiirger contains an analysis of the distribution, moments and Laplace transforms of the suprema of several stochastic processes - a problem with immediate applications for the pricing of barrier and lookback options Schiirger gives explicit formulae for these quantities for Bessel processes as well as for strictly stable Levy processes with no positive jumps For this he uses an elegant transformation from the maximum of a stochastic process to its first hitting time The final contribution again addresses the question of optimal stopping Peskir and Shiryaev analyse the Poisson disorder problem, the problem of detecting a change in the intensity of a Poisson process In this context they show that the smooth-pasting condition is not always valid for the optimal value function if the state vector can be discontinuous 298 Goran Peskir and Albert N Shiryaev (d [2], page 713 or [4], page 307) It follows that (o respectively Thus, the infimum in (7) may indeed be viewed as taken over all stopping times T of (7rt}t?:o, and the optimal stopping problem (7) falls into the class of optimal stopping problems for Markov processes We thus proceed by finding the infinitesimal operator of the Markov process (7rt)t?:o By Ito's formula, upon making use of the fact easily verified (see (15) below) that the innovation process is a martingale under P1T with respect to (Ff)t>o, it follows from (12) that the infinitesimal operator of (7rt)t?:o acts on E e l [O,l] according to the following rule: (Lf)(7r) =( A - (Al - AO)7r )(1-7r)1'(7r) +(Al7r+AO(l-7r))(/(Al7r+;O~l_7rJ -/(7r)) (13) It may be noted that the equations (10)-(13) for A = reduce to the analogous equations in [6] We may assume that for each r ~ a probability measure Qr is defined on (.a, F) such that Qr«() r) Thus, under Qr the observed process X = (Xdt?:o is given by = = Xt for all t tion: ~ where r = ~ 1t 1(s :::; r)dN;O + o It follows that P 1t 1T /(8) r)dN;l (14) admits the following decomposi- (15) which appears to be an elegant tool, for instance, to check that the innovation process (Xt)t>o defined above is a martingale under P1T • Moreover,-using (15) it is straightforwardly verified that the following facts are valid: 1f H V (1f) is concave (continuous) and decreasing on [0,1]; The stopping time T* = inf {t ~ 011ft ~ B.} is optimal in the problem (5)+(7), where B* is the smallest 1f from [0,1] satisfying V(1f) = 1- 1f The map (16) (17) Solving the Poisson Disorder Problem 299 Thus V(7r) < 1- 7r for all7r E [0, B*) and V(7r) = 1- 7r for all7r E [B*, 1] It should be noted in (17) that 7rt = ., c as well as to compute the value V (7r) for 7r E [0, B.) (especially for 7r = 0) We tackle these questions by forming a free-boundary problem A Free-Boundary Problem Being aided by the general (optimal stopping) theory of Markov processes (see e.g [8]), and making use of the preceding facts, we are naturally led to formulate the following free-boundary problem for 7r I-t V(7r) and B defined above: = -C7r = 1- 7r V(B.-) = 1- B (lLV)(7r) V(7r) (0 < 7r < B.) (18) (19) (B* ::; 7r ::; 1) (continuous fit) (20) In some cases (specified below) the following condition will be satisfied as well: V'(B.) = -1 (smooth fit) (21) However, we will also see below that this condition may fail Finally, it is easily verified by passing to the limit for 7r that each continuous solution 7r I-t V(7r) of the system (18)+(19) must necessarily satisfy: + V'(O+) = ° (normal entrance) (22) whenever V (0+) is finite This condition proves useful in the case when >'1 < >'0 For a similar free-boundary differential-difference problem corresponding to the case> = above we refer to [6] Solving the free-boundary problem (18) It turns out that the case >'1 < >'0 is much different from the case >'1 > >'0 Thus assume first that >'1 > >'0 and consider the equation (18) on (0, B] for some < B < given and fixed Introduce the 'step' function (23) 300 Goran Peskir and Albert N Shiryaev for 7r ::; B Observe that S(7r) > 7r for all < 7r < and find points < B2 < B1 < Bo := B such that S(Bn) = B n - for n 2: It is easily verified that (n = 0,1, ) Denote In = (Bn, Bn-d (24) for n 2: 1, and introduce the 'distance' function for 7r ::; B, where [x] denotes the integer part of x Observe that d is defined to satisfy: 7r E In {=} d(7r, B) =n (26) for all < 7r ::; B Now consider the equation (18) first on h upon setting V(7r) = 1- 7r for 7r E (B, S(B)] This is then a first-order linear differential equation which can be solved explicitly Imposing a continuity condition at B (which is in agreement with (20) above) we obtain a unique solution 7r H V(7rj B) on 11 lt is possible to verify that the following formula holds: (27) where 7r H Vp ,l (7rj B) is a (bounded) particular solution of the non-homogeneous equation in (18): (28) and 7r H Vg (7r) is a general solution of the homogeneous equation in (18): Vg (7r) (1 - 7r )'Y1 = IA - (A1 - ' Ao )7rI 'Yo = (1 - 7r ) exp ( (A1 if A =j: A1 - AO A1 - 7r) ) , _ Ao)(1 if A = A1 - Ao (29) where 1'1 = Ad (A1 - Ao - A) and 'Yo = (AO + A) / (A1 - AO - A), and the constant is determined by the continuity condition V(B-j B) = - B leading to CI (B) cI(B) = 1_ (AlA + AoC B _ A(A1 - c) ) Vg (B) Al (AO + A) Al (AO + A) (30) where Vg(B) is obtained by replacing 7r in (29) by B [We see from (28)-(30) however that the continuity condition at B cannot be met when B equals jj Solving the Poisson Disorder Problem 301 from (33) below unless B equals A(AI - C)/(AAI + do) from (40) below (the latter is equivalent to C = Al -Ao-A) Thus, if B = B :/= A(AI -C)/(AAI +do) then there is no solution 7f f-t V(7f; B) on It that satisfies V(7f; B) = - 7f for 7f E (B, S(B)] and is continuous at B It turns out, however, that this analytic fact has no significant implication for the solution of (5)+(7).] Next consider the equation (18) on 12 upon using the solution found on It and setting V(7f) = CI (B)Vg(7f) + Vp,1 (7f; B) for 7f E (Bl, S(BI)] This is then again a first-order linear differential equation which can be solved explicitly Imposing a continuity condition over I UIt at BI (which is in agreement with (16) above) we obtain a unique solution 7f f-t V(7f; B) on 12 It turns out, however, that the general solution of this equation cannot be expressed in terms of elementary functions (unless A = as shown in [6]) but one needs, for instance, the Gauss hypergeometric function As these expressions are increasingly complex to record, we omit the explicit formulas in the sequel Continuing the preceding procedure by induction as long as possible (considering the equation (18) on In upon using the solution found on I n- I and imposing a continuity condition over In U I n- I at B n - I ) we obtain a unique solution 7f f-t V(7f; B) on In given as (31) where 7f f-t Vp,n (7f; B) is a (bounded) particular solution, 7f f-t Vg (7f) is a general solution given by (29), and B f-t cn(B) is a function of B (and the four parameters) [We will see however in Theorem 4.1 below that in the case B > B > with B from (33) below the solution (31) exists for 7f E (B, B] but explodes at B unless B = B* ] The key difference in the case Al < AO is that S(7f) < 7f for all < 7f < so that we need to deal with points B := Bo < BI < B2 < such that S(Bn) = B n- I for n 2: Then the facts (24)-(26) remain preserved provided that we set In = [B n- I , Bn) for n 2: In order to prescribe the initial condition when considering the equation (18) on It, we can take B = c: > small and make use of (22) upon setting V(7f) = v for all7f E [S(B),B) where v E (0,1) is a given number satisfying V(B) = v Proceeding by induction as earlier (considering the equation (18) on In upon using the solution found on I n- I and imposing a continuity condition over I n- I U In at B n- I ) we obtain a unique solution 7f f-t V(7f;C:,v) on In given as (32) where 7f f-t Vp,n(7f;C:,v) is a particular solution, 7f f-t Vg(7f) is a general solution given by (29), and c: f-t cn(c:) is a function of c: (and the four parameters) We shall see in Theorem 4.1 below how these solutions can be used to determine the optimal 7f f-t V(7f) and B* Two key facts about the solution Both of these facts hold only in the case when Al > AO The first fact to be observed is that 302 Goran Peskir and Albert N Shiryaev E= A Al - Ao (33) is a singularity point of the equation (18) whenever A < Al - Ao This is clearly seen from (29) where Vg (1f) -+ 00 for 1f -+ E The second fact of interest is that B=_A_ (34) A+c is a smooth-fit point of the system (18)-(20) whenever Al > AO and C # Al - AO - A, i.e V'(B-; B) = -1 in the notation of (31) above This can be verified by (27) using (28)-(30) It means that B is the unique point which in addition to (18)-(20) has the power of satisfying the smooth-fit condition (21) It may also be noted in the verification above that the equation V'(B-; B) = -1 has no solution when c = Al - Ao - A as the only candidate 13 := B = E satisfies: V'(13-;13) = _ Ao Al (35) This identity follows readily from (27)-(30) upon noticing that CI (B) = O Thus, when C runs from +00 to Al - AO - A, the smooth-fit point B runs from o to the singularity point E, and once B has reached E for C = Al - AO - A, the smooth-fit condition (21) breaks down and gets replaced by the condition (35) above We will soon attest below that in all these cases the smooth-fit point B is actually equal to the optimal-stopping point B* from (17) above Observe that the equation (18) has no singularity points when Al < Ao This analytic fact reveals a key difference between the two cases Conclusions In parallel to the two analytic properties displayed above we begin this section by stating the relevant probabilistic properties of the a posteriori probability process Sample-path properties of (1ft)t>o First consider the case Al > Ao Then from (12) we see that (1ft)t>o call only jump towards (at times of the jumps of the process X ) Moreover, the sign of the drift term A(1 - 1f) (AI - AO)1f(1 - 1f) = (AI - Ao)(E - 1f)(1 - 1f) is determined by the sign of E - 1f Hence we see that (1ftk~o has a positive drift in [0, E), a negative drift in (E, 1], and a zero drift at E Thus, if (1ftk~o starts or ends up at E, it is trapped there until the first jump of the process X occurs At that time (1ftk::o finally leaves E by jumping towards This also shows that after Solving the Poisson Disorder Problem 303 (i) o o (ii) o Fig Sample-path properties of the a posteriori probability process (7rt)t>o from (6)+(12) The point Ii is a singularity point (33) of the free-boundary equation (18) once (7Tt)t::::o leaves [0, B) it never comes back The sample-path behaviour of (7Tt}t>O when Al > Ao is depicted in figure (Part i) below Next consider the case Al < AO' Then from (12) we see that (7Tdt>o can only jump towards (at times of the jumps of the process X ) Moreover, the sign of the drift term A(1- 7T) - (AI - AO)7T(1- 7T) = (A + (AO - Ad7T)(1-7T) is always positive Thus (7Tt)t>o always moves continuously towards and can only jump towards O The sample-path behaviour of (7Tt)t>o when Al < AO is depicted in Figure (Part ii) Sample-path behaviour and the principles of smooth and continuous fit With a view to (17), and taking < B < given and fixed, we shall now examine the manner in which the process (7Ttk:o enters [B, 1] if starting at B - d7T where d7T is infinitesimally small Our previous analysis then shows the following (see Figure 1) If >'1 > >'0 and B < B, or >'1 < >'0, then (7Ttk:::O enters [B,1] by passing through B continuously If, however, >'1 > >'0 and B > B then the only way for (7Ttk:::o to enter [B, 1] is by jumping over B (Jumping exactly at B happens with probability zero.) 304 Goran Peskir and Albert N Shiryaev The case Al > Ao and B = B is special If starting outside [B,I] then (7rdt>o travels towards B by either moving continuously or by jumping However,-the closer (7rth>o gets to B the smaller the drift to the right becomes, and if there were no jump over B eventually, the process (7rt)t>o would never reach B as the drift to the right tends to zero together with the distance of (7rdt>o to B This fact can be formally verified by analysing the explicit representation of ( are given and fixed Then there exists B E (0,1) such that the stopping time (36) is optimal in (5) and (7) Moreover, the optimal cost function 7r t-* V(7r) from (5)+(7) solves the free-boundary problem (18)-(20), and the optimal threshold B is determined as follows (i) If Al > Ao and c > Al - Ao - A, then the smooth-fit condition (21) holds at B., and the following explicit formula is valid (cf [2) and [1]): (37) In this case B* < B where B is a singularity point of the free-boundary equation (J8) given in (33) above (see Figure 2) (ii) If Al > Ao and c = Al - Ao - A, then the smooth-fit condition breaks down at B and gets replaced by the condition (35) above (V' (B -) = -Ao/At) The optimal threshold B is still given by (37), and in this case B* = B (see figure 3) (iii) If Al > Ao and c < Al - Ao - A, then the smooth-fit condition does not hold at B*, and the optimal threshold B is determined as a unique solution in (B, 1) of the following equation: (38) Solving the Poisson Disorder Problem 305 where the map B 1-+ deB, B) is defined in (25), and the map B 1-+ cn(B) is defined by (30) and (31) above (see Figure below) In particular, when c satisfies: AlAO(Al - AO - A) < AlAo + (AI - AO)(A + AO) - C < \ Al - \ AO - \ A (39) then the following explicit formula is valid: B* = A(AI - c) AAI + CAo (40) which in the case c = Al - AO - A reduces again to (37) above In the cases (i)-(iii) the optimal cost function 11" 1-+ V(1I") from (5)+(7) is given by (31) with B* in place of B for all < 11" ~ B* (with V(O) = V(O+)) and V(1I") = 1-11" for B* ~ 11" ~ (iv) If Al < AO then the smooth-fit condition holds at B*, and the optimal threshold B* can be determined using the normal entrance condition (22) as follows (see Figure 5) For e > small let Ve denote a unique number in (0,1) for which the map 11" 1-+ V(1I"je,ve ) from (32) hits the map 11" 1-+ - 11" smoothly at some B; from (0,1) Then we have: B = limBe g.j.O * V(1I") = lim V(1I"je,ve ) (41) (42) giO for all < 11" B ~ 11" ~ < B (with V(O) = V(O+)) and V(1I") - 11" for Proof We have already established in (17) above that T* from (36) is optimal in (5) and (7) for some B* E [0,1] to be found It thus follows by the strong Markov property of the process (1I"dt~0 together with (16) above that the optimal cost function 11" 1-+ V(1I") from (5)+(7) solves the free-boundary problem (18)-(20) Some of these facts will also be reproved below First consider the case Al > A.Q In Section 3.2 above it was shown that for each given and fixed B E (0, B) the problem (18)-(20) with B in place of B has a unique continuous solution given by the formula (31) Moreover, this solution is (at least) C l everywhere but possibly at B where it is (at least) Co As explained following (30) above, these facts also hold for B = B when B equals A(AI -C)/(AAI +CAo) from (40) above We will now show how the optimal threshold B is determined among all these candidates B when c> Al - AO - A - (i)+(ii): Since the innovation process Xt = X t - J~ (Al11"s- +AO(1-1I"s- ))ds is a martingale under P1f with respect to (:Ff)t~o, it follows by (12) that 1I"t=1I"+A!ot(1-1I"s_)dS+Mt (43) 306 Goran Peskir and Albert N Shiryaev (i) ·· t · ·· 1t o-t 1-1t 1to-t V(1t) 1t B (ii) 1t 1to-tV(1t;B) Fig A computer drawing of the maps 7r f-t V(7rj B) from (31) for different B from (0,1) in the case Al = 4, Ao = 2, A = 1, c = The singularity point fj from (33) equals 1/2, and the smooth-fit point E from (34) equals 1/3 The optimal threshold B coincides with the smooth-fit point E The optimal cost function 7r f-t V(7r) from (5)+(7) equals 7r f-t V(7rjB.) for ::; 7r ::; B and - 7r for B ::; 7r ::; (This is presented in part (i) above.) The solutions 7r f-t V(7rj B) for B > B are ruled out since they fail to satisfy ::; V(7r) ::; - 7r for all 7r E [0,1] (This is shown in part (ii) above.) The general case Al > Ao with c > Al - Ao - A looks very much the same Solving the Poisson Disorder Problem 307 1t smooth lit (C>t 1't O·t ) u" B continuous lit (C E, and all solutions 71' t-t V(7I'j B) for B < B hit - 71' for some 71' > E (This is shown in part (i) above.) A simple numerical method based on the preceding fact suggests the following estimates 0.750 < B < 0.752 The optimal cost function 71' t-t V(7I') from (5)+(7) equals 71' t-t V(7I'j B.) for ::;: 71' ::;: B and 1- 71' for B ::;: 71' ::;: The solutions 71' t-t V(7I'j B) for B ::;: E are ruled out since they fail to be concave (This is shown in part (ii) above.) The general case Al > AO with c < Al - AO - A looks very much the same Solving the Poisson Disorder Problem 309 X-+V(X;E,V) X-+ l-x x Fig A computer drawing of the maps 7r I -t V(7rj c, v) from (32) for different v from (0,1) with c = 0.001 in the case >'1 = 2, >'0 = 4, > = 1, c = For each c > there is a unique number Ve E (0,1) such that the map 7r I -t V(7rjc,v e ) hits the map 7r I -t - 7r smoothly at some B! E (0,1) Letting c j we obtain B; -t B and V(7rj c, v e ) -t V(7r) for all 7r E [0, I] where B is the optimal threshold from (17) and 7r I -t V(7r) is the optimal cost function from (5)+(7) never optimal to stop (7rtk~o in [0, B), as well as that (7rdt~o must be stopped immediately after entering [B,l] as it will never return to the 'favourable' region [0, B) again This proves that B equals the optimal threshold B., i.e that T from (36) with B from (37) is optimal in (5) and (7) The claim about the breakdown of the smooth-fit condition (21) when C = Al - AD - A has been already established in the paragraph containing (35) above (cf Figure 3) The general answer (37) has been obtained in [1] (iii): It was shown in Section 3.2 above that for each given and fixed B E (i3,1) the problem (18)-(20) with B in place of B has a unique continuous solution on (i3,1] given by the formula (31) We will now show that there exists a unique point B E (i3,1) such that lim 1r.(.B V(7r; B) = ±oo if B E (i3, B.) U (B., 1) and lim1r.(.B V(7r; B.) is finite This point is the optimal threshold, i.e the stopping time T from (36) is optimal in (5) and (7) Moreover, the point B can be characterized as a unique solution of the equation (38) in (i3, 1) In order to verify the preceding claims we will first state the following observation which proves useful Setting g(7r) = 1-7r for < 7r < we have: 310 Goran Peskir and Albert N Shiryaev (Lg)(1r) ~ -C1r (45) where iJ is given in (34) This is verified straightforwardly using (13) Now since B is a singularity point of the equation (18) (recall our discussion in Section 3.3 above), and moreover 1r I-t V(1r) from (5)+(7) solves (18)(20), we see that the optimal threshold B* from (17) must satisfy (38) This is due to the fact that a particular solution 1r I-t Vp ,n(1rj B*) for n = d(B, B*) in (31) above is taken bounded The key remaining fact to be established is that there cannot be two (or more) points in (B, 1) satisfying (38) Assume on the contrary that there are two such points Bl and B2 We may however assume that both Bl and B2 are larger than iJ since for BE (B, iJ) the solution 1r I-t V(1rj B) is ruled out by the fact that V(1rj B) > - 1r for 1r E (B - €, B) with € > small This fact is verified directly using (27)-(30) Thus, each map 1r I-t V(1rj B i ) solves (18)-(20) on (0, B i ] and is continuous (bounded) at B for i = 1,2 Since S(1r) > 1r for all < 1r < when Al > Ao, it follows easily from (13) that each solution 1r I-t V(1rj B i ) of (18)-(20) must also satisfy -00 < V(O+j B i ) < +00 for i = 1,2 In order to make use of the preceding fact we shall set h/3(1r) = (1 + (fJI)B) - fJ1r for ~ 1r ~ Band h/3(1r) = l-1r for B ~ 1r ~ Since both maps 1r I-t V(1rj B i ) are bounded on (0, B) we can fix fJ > large enough so that V(1rj B i ) ~ h/3(1r) for all < 1r ~ Band i = 1,2 Consider then the auxiliary optimal stopping problem: W(1r) := i~f E1r (h/3(1rT) + C iT 1rtdt) (46) where the supremum is taken over all stopping times T of (1rt}t>o Extend the map 1r I-t V(1rj B i ) on [Bi' 1] by setting V(1rj B i ) = - 1r for Bi ~ 1r ~ and denote the resulting (continuous) map on [0,1] by 1r I-t Vi(1r) for i = 1,2 Then 1r I-t Vi(1r) satisfies (18)-(20), and since Bi ~ iJ, we see by means of (45) that the following condition is also satisfied: (LVi)(1r) -C1r (47) for 1r E [Bi' 1] and i = 1,2 We will now show that the preceding two facts have the power of implying that Vi(1r) = W(1r) for all1r E [0,1] with either i E {I, 2} given and fixed It follows by Ito formula that Vi(1rt) = Vi(1r) where M = (Mth~o Mt + i t (LVi)(1rs_)ds + M t (48) is a martingale (under P1r) given by = i t (Vi (1rs- + Ll1rs) - Vi(1rs-»)dXs (49) Solving the Poisson Disorder Problem 311 and Xt = X t - J~(>\l7rs- + Ao(1- 7r s_))ds is the innovation process By the optional sampling theorem it follows from (48) using (47) and the fact that Vi(7r) ~ h{3(7r) for all7r E [0,1) that Vi(7r) ~ W(7r) for all7r E [0,1) Moreover, defining Ti = inf{t ~ 017rt ~ B i } it is easily seen by (43) for instance that E 7r (Ti) < 00 Using then that 7r t-+ Vi(7r) is bounded on [0,1)' it follows easily by the optional sampling theorem that E7r(Mr.) = Since moreover Vi(7r r ) = h{3(7rr J and (LVi)(7rs -) = -C7rs_ for all s ~ Ti, we see from (48) that the inequality Vi (7r) ~ W (7r) derived above is actually equality for all 7r E [0,1) This proves that V(7rj Bd = V(7rj B ) for all 7r E [0,1]' or in other words, that there cannot be more than one point B* in (fJ, 1) satisfying (38) Thus, there is only one solution 7r t-+ V(7r) of (18)-(20) which is finite at fJ (see Figure below), and the proof of the claim is complete (i v): It was shown in Section 3.2 above that the map 7r t-+ V (7r j e, v) from (32) is a unique continuous solution of the equation (LV)(7r) = -C7r for e < 7r < satisfying V(7r) = v for all 7r E [S(e),e] It can be checked using (29) that (50) (51) for 7r E 11 = [e,e1) where S(e1) = c Moreover, it may be noted directly from (13) above that L(f + c) = L(f) for every constant c, and thus V(7rj e, v) = V(7rj e, 0) +v for all7r E [S(e), 1) Consequently, the two maps 7r t-+ V(7rj e, v') and 7r t-+ V(7rj e, v") not intersect in [S(e), 1) when v' and v" are different Each map 7r t-+ V (7r j e, v) is concave on [S (c), 1) This fact can be proved by a probabilistic argument using (15) upon considering the auxiliary optimal stopping problem (46) where the map 7r t-+ h{3 (7r) is replaced by the concave map hu(7r) = v /\ (1 - 7r) [It is a matter of fact that 7r t-+ W(7r) from (46) is concave on [0,1] whenever 7r t-+ h{3(7r) is so.) Moreover, using (29)+(50)+(51) in (32) with n = it is possible to see that for v close to we have V(7rj e, v) < for some 7r > e, and for v close to we have V(7rje, v) > 1- 7r for some 7r > e (see Figure below) Thus a simple concavity argument implies the existence of a unique point B~ E (0,1) at which 7r t-+ V (7rj e, ve ) for some Ve E (0, 1) hits 7r t-+ - 7r smoothly The key nontrivial point in the verification that V(7rj e, ve ) equals the value function W(7r) of the optimal stopping problem (46) with 7r t-+ hUe (7r) in place of 7r t-+ h{3 (7r) is to establish that (L(V (.j e, v e ))) (7r) ~ -C7r for all7r E (B~, S-l (B;)) Since B; is a smooth-fit point, however, this can be done using the same method which we applied in part of the proof of Theorem 2.1 in [6] Moreover, when e I- then clearly (41) and (42) are valid (recall (16) and (22) above), and the proof of the theorem is complete ° ° ° o 312 Goran Peskir and Albert N Shiryaev Concluding the paper we would like to mention that the fixed false-alarm formulation of the Poisson disorder problem (cf [(8), page 205)) raises some new interesting questions not present in the Wiener process version of the same problem References Davis, M H A (1976) A note on the Poisson disorder problem Banach Center Publ (65-72) Gal'chuk, L I and Rozovskii, B L (1971) The "disorder" problem for a Poisson process Theory Probab Appl 16 (712-716) Jacod, J and Shiryaev, A N (1987) Limit Theorems for Stochastic Processes Springer-Verlag, Berlin Heidelberg Lipster, R S and Shiryaev, A N (2nd ed 2001) Statistics of Random Processes II Springer-Verlag, New York Marcellus, R L (1990) A Markov renewal approach to the Poisson disorder problem Comm Statist Stochastic Models (213-228) Peskir, G and Shiryaev, A N (1998) Sequential testing problems for Poisson processes Research Report No 400, Dept Theoret Statist Aarhus (20 pp) Ann Statist 28, 2000, (837-859) Shiryaev, A N (1967) Two problems of sequential analysis Cybernetics (63-69) Shiryaev, A N (1978) Optimal Stopping Rules Springer-Verlag, New York ...Springer-Verlag Berlin Heidelberg GmbH Klaus Sandmann Philipp J Schonbucher (Eds.) Advances in Finance and Stochastics Essays in Honour of Dieter Sondermann With 32 Figures Springer Klaus Sandmann... Catalog in Publication Data applied for Die Deutsche Bibliothek- CIP-Einheitsaufnahme Advances in finance and stochastics: essays in honour of Dieter Sondermann/ Klaus Sandmann;... Sandmann Philipp Schonbucher Introduction In many areas of finance and stochastics, significant advances have been made since this field of research was opened by Black, Scholes and Merton in
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