Portfolio Optimization and Optimal Martingale Measures in Markets with Jumps

and

Optimal Martingale Measures

in

Markets with Jumps

Dissertation

zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.)

an der

Universit¨at Konstanz

Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik

vorgelegt von

Christina R. Niethammer

Tag der m¨undlichen Pr¨ufung: 05.05.2008

Referent: Prof. Dr. Michael Kohlmann, Universit¨at Konstanz

Referent: Prof. Dr. Ludger Overbeck, Justus-Liebig-Universit¨at Gießen

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5438/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-54386

Abstract

We discuss optimal portfolio selection with respect to utility functions of type −e^−αx, α > 0 (exponential problem) and −|1− ^αx_p |^p (p-th problem).

We considerN risky assets and a risk-free bond. Risky assets are modeled by continuous semimartingales or exponential L´evy processes.

These dynamic expected utility maximization problems are solved by transforming the model into a constrained static version and applying con- vex duality. The connection between the static and the dynamic problems is drawn as follows: firstly we construct explicit portfolios (dynamic so- lution) attaining the optimal static values for the p-th problem, secondly we establish uniform convergence in probability of these portfolios and the corresponding wealth processes to the dynamic solution of the exponential problem. Moreover, convergence of the optimal wealth processes in a supre- mum norm and convergence of the terminal values in L^v, v ≥ 1 follows under the assumption of upward bounded jumps. By construction these results yield an explicit portfolio for the exponential problem. To establish our results, we need to prove several properties on the solutions of the dual problems, i.e. the q-optimal martingale measure and the minimal entropy martingale measure.

In fact, in the presence of unbounded jumps the q-optimal martingale measure (the dual solution of the staticp-th problem) may fail to be equiv- alent. Depending on the specific formulation of the portfolio selection prob- lem, i.e. whether or not consumption is allowed, we have to consider the signed or the absolutely continuous version of theq-optimal martingale mea- sure. However, techniques usually applied in order to characterize the equiv- alent case are not suitable. An explicit form of the q-optimal signed mar- tingale measure is therefore established by a new verification procedure via a hedging argument reversing the above duality. Admitting consumption, a superhedging argument yields an explicit form of the absolutely continuous martingale measure. The convergence of both versions of theq-optimal mar- tingale measures to the minimal entropy martingale measure, whenq tends to 1 is proved, implying the convergence of the optimal strategies (portfo- lio and eventually consumption) of thep-th problem to the exponential one.

We close the thesis by a comparison of the achieved results in the continuous and discontinuous case.

iii

Zusammenfassung

In der folgenden Zusammenfassung werden die Problemstellungen der vor- liegenden Dissertation motiviert und die Hauptresultate nach einem veran- schaulichenden Beispiel n¨aher erl¨autert. Abschließend werden die neuen Ergebnisse der Dissertation stichpunktartig und nach Kapiteln geordnet zusammengefasst.

Die Problematik wird folgendermaßen veranschaulicht. Alleinige In- vestitionen in eine einzige Firma bedeuten ein hohes Risiko, welches sich aus einem gemeinsamen die Unternehmen ¨ubergreifenden und dem idio- synkratischen Risiko der Firma zusammensetzt. Tats¨achlich kann dieses Risiko aber durch eine strategische Streuung verschiedener Einzelnamen, so genannte Diversifikation, gemindert werden. Dies geschieht im All- gemeinen durch geschickte Anordnung der zur Verf¨ugung stehenden In- vestitionsm¨oglichkeiten an Hand ihrer Abh¨angigkeitsstruktur. Je besser die Diversifikation umso kleiner ist das idiosynkratische Risiko. Im Idealfall bleibt nur ein nicht diversifizierbares gemeinsames Risiko, z.B. schlechte all- gemeine Konjunktur, bestehen. Je h¨oher nun die Risikoaversion des In- vestors ist umso kleiner wird der Anteil sein, der in die risikobehafteten Anlagen (Aktien, Bonds mit Ausfallrisiko, Derivate auf solche etc.) in- vestiert wird. Potentiell zu viel investiertes Kapital wird durch Kredit fi- nanziert; ¨ubersch¨ussiges Kapital wird in risikolose Anlagem¨oglichkeiten (z.B.

risikolose Rentenpapiere) angelegt.

Um diese Beziehungen quantifizieren zu k¨onnen, ben¨otigen wir ein stochastisches Modell f¨ur die risikobehafteten und risikolosen Anlagem¨og- lichkeiten im Markt. Zun¨achst betrachten wir stetige Semimartingale und spezielle Prozesse mit Spr¨ungen (L´evy Prozesse). Des Weiteren m¨ussen wir die Risikoeinstellung des Investors in dieses Modell integrieren, zum Beispiel an Hand von isoelastischen und exponentiellen Nutzenfunktionen. In diesem Rahmen werden wir nun versuchen, eine optimale, dynamische Anlagestra- tegie zu finden, die den erwarteten Nutzen des Verm¨ogens des Investors zu einem gegeben Endzeitpunkt maximiert. Es zeigt sich, dass dieses optimale Portefeuille von der oben erw¨ahnten Abh¨angigkeitsstruktur entscheidend beeinflusst wird. In vielen Modellen geschieht dies durch die Kovarianzma- trix der Anlagem¨oglichkeiten. Des Weiteren m¨ochten wir untersuchen, ob der Einsatz von Modellen mit Spr¨ungen tats¨achlich der Realit¨at entspricht.

v

Am Ende der Dissertation werden wir daher kurz auf eine Methode zur Sch¨atzung von Kovarianzen eingehen, die gleichzeitig einen Test auf die Existenz von Spr¨ungen liefert.

Im Folgenden soll nun ein Beispiel diese Vorgehensweise erl¨autern. Wir betrachten eine konkave Nutzenfunktion U und bezeichnen das gegebene Anfangsverm¨ogen des Investors mit ˜x. Hiermit soll nun die folgende Maximierungsaufgabe gel¨ost werden. Es wird ein Verm¨ogensprozess Y gesucht, der den erwarteten Nutzen des Endverm¨ogens Y_T maximiert:

V(˜x)_C ≡ sup

Y∈WC(˜x)

E[U(Y_T)], x˜∈R (1) mit

WC(˜x) ={Y|Y_t= ˜x+ Z _t

0

ϑ^′_udS_u−C_t, ϑ∈ A, C∈C},

wobei S ein N-dimensionales Semimartingal als Modell f¨ur die risikobe- hafteten Anlagen,ϑ^′ = (ϑ¹, ..., ϑ^N) eine zul¨assige Anlagestrategie aus einer MengeA, sowieC ∈C, die Menge der zul¨assigen Konsumprozesse ist. Die risikolose Anlagem¨oglichkeit wird ohne große Einschr¨ankungen an die All- gemeinheit konstant auf 1 gesetzt.

Die L¨osung wird wie folgt erzielt: Das gegebene dynamische Problem

¨

uber Verm¨ogensprozesse aus WC(˜x) wird in ein statisches Problem ¨uber nicht zeitabh¨angige Zufallsvariablen ¨uberf¨uhrt. Um dieses statische Pro- blem zu l¨osen, bedienen wir uns dann Methoden aus der konvexen Analysis.

Im Falle einer exponentiellen Nutzenfunktion (−e^−αx, α > 0) und im ein- fachsten Modell einer geometrisch Brownschen Bewegung

S_t=S₀+ Z t

0

S_uµ_udu+ Z t

0

S_uσ_udW_u, S=diag(S⁽¹⁾, ..., S^(N)), mit deterministischen zeitabh¨angigen Koeffizienten µ (Drift) und σ (Ko- varianzmatrix) ergibt sich dann die folgende optimale Anlagestrategie und der folgende optimale Konsum:

ϑ(t) =ˆ S⁻¹_t (σ_tσ_t^′)⁻¹µ_t, C ≡0. (2) Im Beispiel beobachten wir, dass das optimale Portefeuille entscheidend durch die unterliegende Kovarianzstrukturσdes Anlageprozesses beeinflusst wird. Es wird nicht konsumiert, da die betrachtete Nutzenfunktion strikt ansteigend ist und Konsum von der Nutzenfunktion nicht honoriert wird.

Das wird sich ¨andern, wenn wir nicht immer ansteigende Nutzenfunktionen betrachten. Konsum setzt eine konkave Nutzenfunktion im fallenden Teil quasi konstant auf das Maximum der Funktion.

InNiethammer [103] haben wir uns schon mit exponentiellen Nutzen- funktionen in einem stetigen Semimartingalmodell besch¨aftigt. Die Arbeit enth¨alt allerdings nur die Herleitung des optimalen Endverm¨ogens (L¨osung

des statischen Problems) f¨ur exponentielle und eine Transformation von iso- elastischen (−(1− _2m^x )^2m) Nutzenfunktionen sowie deren Zusammenhang.

Ein explizites Portefeuille und die Verbindung der dynamischen L¨osungen ist nicht gegeben. In der Zwischenzeit ist es aber gelungen, diese L¨ucke zu schließen, siehe Kohlmann and Niethammer [76] und Kapitel 2, in welchem die Ergebnisse weiter auf den Fall −|1−^x_p|^p verallgemeinert wer- den.

Stetige Semimartingale reichen jedoch h¨aufig nicht aus, um Aktien- kurse ad¨aquat zu modellieren. Extrem negative Entwicklungen des Aktien- kurses sind zum Beispiel h¨aufig durch Spr¨unge gekennzeichnet, w¨ahrend Aufw¨artsbewegungen des Aktienkurses meist stetig erfolgen. Die Beobach- tung von Spr¨ungen zeigt sich vor allem bei starkem Kursverfall. F¨ur den Gegenstand der Portefeuilleoptimierung ist es daher nicht unrealistisch anzunehmen, dass Spr¨unge nach oben beschr¨ankt sind. Diese Annahme reicht dann im Wesentlichen aus, die oben genannten Resultate aus [76]

f¨ur gewisse Sprungprozesse (L´evy Prozesse) zu beweisen. Dies ist sogar f¨ur mehrdimensionale L´evy Prozesse gelungen, siehe Niethammer [106] bzw.

Kapitel 3.

Jedoch gibt es eine Reihe h¨aufig verwendeter Modelle mit unbeschr¨ank- ten Spr¨ungen, zum Beispiel ein Varianz-Gamma-Modell. In Kapitel 4 wer- den daher Verallgemeinerungen diskutiert. Es stellt sich heraus, dass im Fall von−|1−^x_p|^p ein ¨Ubergang zu signierten bzw. absolutstetigen Markt- maßen zwingend notwendig ist. Diese Maße k¨onnen nun aber nicht mehr mit ¨ublichen Methoden behandelt werden. Man k¨onnte daher denken, dass sich das Thema Portefeuilleoptimierung nun auf den ersten Fall mit beschr¨ankten Spr¨ungen begrenzt. Fassen wir die Maße aber als L¨osung der dualen Seite auf, kann mit ihrer Hilfe das Portefeuilleoptimierungsproblem gel¨ost werden. Dar¨uber hinaus ist es sogar m¨oglich, mittels der L¨osung des Portefeuilleproblems interessante Resultate ¨uber signierte und absolut- stetige Maße zu erfahren. Wir kehren die Dualit¨at also um und k¨onnen damit unter nicht sehr restriktiven Annahmen die explizite Form des q-optimalen signierten Martingalmaßes und des q-optimalen absolutstetigen Martingal- maßes pr¨asentieren. Weiter zeigen wir deren Konvergenz zum Entropie- minimalen Maß und kl¨aren letztendlich, wie restriktiv eine Beschr¨ankung auf ¨aquivalente Martingalmaße sein kann. Mit Hilfe des q-optimalen sig- nierten Martingalmaßes k¨onnen wir das dynamische Optimierungsproblem l¨osen, wenn es uns nicht erlaubt ist zu konsumieren. Erlauben wir jedoch Konsum, was bei einer nicht ansteigenden Nutzenfunktion durchaus sinnvoll erscheint, bedienen wir uns dann des q-optimalen absolutstetigen Martin- galmaßes. Die Konvergenz beider Portefeuilleprobleme zum exponentiellen Problem folgt aus der Konvergenz der Maße. Die gestellten Annahmen sind hierbei ¨außerst schwach, und einfache Charakterisierungen werden bewiesen.

Die Ergebnisse pr¨asentieren somit eine vollst¨andige Darstellungq-optimaler Maße in Modellen mit exponentiellen L´evy-Prozessen. Die Resultate in

Kapitel 4 sind Teil der ArbeitBender and Niethammer [14].

Abschließend sei ein Wort zur Bedeutung der Ergebnisse in der vor- liegenden Dissertation gestattet. Ausgangspunkt waren vor Jahren Porte- feuilleoptimierungs- und Hedgingprobleme, die mit Methoden der stochasti- schen Dualit¨at gel¨ost wurden. Dann verlegte man sich auf die zum Aus- gangsproblem duale Seite und untersuchte extensiv Eigenschaften optimaler Martingalmaße, insbesondere ihre Existenz und Konvergenz. Da man hoffte, diese auch als Risiko- und Preismaße benutzen zu k¨onnen, geriet das Aus- gangsproblem mehr und mehr aus dem Blickwinkel der Forschung. Die vor- liegende Doktorarbeit ist, neben der vollst¨andigen Besprechungq-optimaler Maße in exponentiellen L´evy-Modellen, der erste Ansatz, die simultan auf der dualen Seite erzielten Resultate zu benutzen, um das “eigentliche” Pro- blem weiter zu erforschen. Unsere Idee optimale (signierte) Martingalmaße mit Hilfe des Portefeuilleoptimierungsproblems zu finden, erscheint f¨ur wei- tere Forschungsvorhaben weiterhin ¨außerst viel versprechend, da sie in all- gemeinen Semimartingalmodellen einsetzbar ist.

Wir fassen die Ergebnisse zusammen:

Kapitel 1

Werkzeuge und einige grundlegende Resultate:

1. Einf¨uhrung in L´evy Prozesse

2. Darstellung des allgemeinen Portefeuilleproblems, vor allem des Zusammenhangs zwischen dynamischer, statischer und dualer L¨osung:

• Herleitung der optimalen Endwerte f¨ur das so genannte “p-te Pro- blem” (f¨ur Nutzenfunktionen vom Typ−|1−^αx_p |^p)

• Konvergenz der optimalen Endwerte f¨ur das p-te Problem zum exponentiellen Problem (−e^−αx) unter allgemeinen Bedingungen

• Literatur¨uberblick Kapitel 2

Portefeuilleoptimierung im stetigen Semimartingalmodell

• Konvergenz der optimalen Endwerte f¨ur das p-te Problem zum expo- nentiellen Problem (−e^−αx) unter der Inversen H¨older-Bedingung

• Herleitung des optimalen Portefeuilles ϑ^(p)f¨ur dasp-te Problem unter der Annahme eines deterministischen “Mean-Variance-Tradeoff”- Prozesses

• Konvergenz der Portefeuilles ϑ^(p) zum optimalen Portefeuille f¨ur die exponentielle Nutzenfunktion −e^−αx

Kapitel 3

Portefeuilleoptimierung f¨ur Sprungprozesse (exponentielle L´evy Prozesse)

• Hilfsresultate zu q-optimalen ¨aquivalenten Martingalmaßen und dem Entropie-minimalen Martingalmaß unter Annahme Cq aus Jean- blanc et al. [69] (dort zur Gewinnung desq-optimalen ¨aquivalenten Maßes)

• Konvergenz der optimalen Endwerte f¨ur dasp-te Problem zum expo- nentiellen Problem unter AnnahmeC_q

• Herleitung des optimalen Portefeuilles ϑ^(p) f¨ur Nutzenfunktionen

−|1−^αx_p |^p unter C_q

• Konvergenz der Portefeuilles ϑ^(p) zum optimalen Portefeuille f¨ur die exponentielle Nutzenfunktion−e^−αx

Kapitel 4

Von der Portefeuilleoptimierung zu optimalen Martingalmaßen

• Diskussion der Einschr¨ankungen f¨ur die Resultate in Kapitel 3 durch AnnahmeCq, insbesondere f¨ur unbeschr¨ankte Spr¨unge

• Pr¨asentation hinreichender Bedingungen f¨ur die Existenz einer ex- pliziten Form von q-optimalen absolutstetigen und signierten Maßen (die Bedingungen werden vonCq impliziert)

• Ein Verifikationsansatz basierend auf einem Hedgingargument f¨ur das q-optimale signierte Martingalmaß

• Ein Verifikationsansatz basierend auf einem Superhedgingargument f¨ur dasq-optimale absolutstetige Martingalmaß

• Aus den Verifikationsans¨atzen entwickeln wir dann eine explizite Darstellung desq-optimalen signierten Maßes und desq-optimalen ab- solutstetigen Maßes (ohneC_q vorauszusetzen)

• Konvergenz derq-optimalen Maße zum Entropie-minimalen Maß

• Herleitung des optimalen Portefeuilles ϑ^(p) f¨ur das p-te Porte- feuilleproblem mit unbeschr¨ankten Spr¨ungen, aber ohne Konsum

• Herleitung des optimalen Portefeuilles und des optimalen Konsums (ϑ^(p,C), C^(p,C)) f¨ur das p-te Portefeuilleproblem mit unbeschr¨ankten Spr¨ungen und Konsum

• Konvergenz der Portefeuilles ϑ^(p) und ϑ^(p,C) zum optimalen Porte- feuille f¨ur die exponentielle Nutzenfunktion −e^−αx

• Konvergenz des Konsums C^(p,C) im dynamischen Problem mit Kon- sum gegen 0 f¨urp gegen unendlich

• Nachweis der neuen Bedingungen f¨ur viele praktisch relevante Modelle wie das Merton, Kou oder Variance-Gamma-Modell, die unterC_qnicht verwendet werden k¨onnen

Kapitel 5 Anwendung

• Kurze Einf¨uhrung in eine Sch¨atzmethode f¨ur Spr¨unge und Kovarianzen

• Illustration der Ergebnisse und Vergleich der optimalen Portefeuilles

• Eine Anwendung im Kreditrisikomanagement

Introduction

In this thesis, we are interested in finding an optimal investment decision at a fixed time horizon T. In particular, we are going to derive an optimal portfolio for an investor depending on his preferences and on market pa- rameters. This is related to an auxiliary dual problem of finding an optimal martingale measure. We solve this dual problem and explain its relation to the portfolio optimization problem. We close the thesis by discussing the be- havior of the obtained portfolio in an explicitly parameterized model. That means we estimate unknown parameters of the model, e.g. the volatility and correlation structure of the stocks in order to explicitly obtain an optimal portfolio.

The following introduction is supposed to give the reader a first glance at the importance of portfolio optimization. Portfolio optimization is viewed from different perspectives while focusing on an intuitive treatment.

Motivation

Not least since the default of Enron when a large number of American in- vestors lost their whole savings and pension funds it is widely known that a well diversified portfolio is the key to a successful risk management. An investment in a single stock will barely be optimal, if other investment op- portunities are available. So apart from some odd cases a first principle in portfolio optimization dictates: never invest in a single stock, always diver- sify your portfolio to reduce your overall risk. In fact, this means by applying a skillful investment strategy in more than one stock, the investor faces less risk, but nonetheless gets the same or even higher expected (portfolio) re- turn. Equivalently, we may look for a strategy that takes highest possible expected return given a certain fixed level of risk that is not allowed to be surpassed. When shifting this fixed level of risk, we get a strategy depending on this level of risk, the efficient frontier (known as Markowitz’s problem).

A solution can be obtained by a quadratic programming procedure. This approach further leads to a well-known performance measure, the so-called Sharpe ratio (µ−r)/σ. Here, µ denotes the expected return of the port- folio, σ the standard deviation of the portfolio return, and r the riskless return. These concepts go back to Markowitz[91, 92] andSharpe[119]

xi

or see alsoMarkowitz[93]. In 1972, an analytical solution of Markowitz’s problem was derived byMerton [96].

An applicable concept of steering a portfolio was born. That was not the end of the research in this field. Problems appeared. Firstly, risk is mea- sured by the standard deviation which is a quite questionable risk measure (see Bluhm et al. [17]; F¨ollmer and Schied [49]). Overshooting the expected return is punished in the same manner as a potential undershoot- ing of the expected return. Consequently, new measures were developed.

One of them is the Value at Risk at a fixed level α, i.e. the α-quantile of the considered loss distribution. It has become a regularly applied measure in industry. However, it shares with the standard deviation the problem of not sufficiently taking into account potential high losses coming from heavy tailed distributions. Moreover, as indicated byArtzner et al. [7; 8] both do not necessarily satisfy conditions for a “good” risk measure. Diversified portfolios might be punished by the VaR. As an alternative the expected shortfall (in main cases equivalent to the tail condition expectation) is con- sidered. It satisfies the proposed conditions and is applied in practice.

We now come to a second problem related to the above approaches. The solution given in [96] was still designed for a single period. So apart from the research on different risk measures motivated by Markowitz’s seminal work, there was the strong desire to handle multiperiod models, see e.g. [26; 51; 62;

102; 113], and continuous-time models, see e.g. [40; 74; 86; 95]. However, the work on such dynamic models has been dominated by considering expected utility as an objective function (sometimes also called the Merton problem going back to Merton [95]). Apart from the quadratic utility function (see [40]) this does not exactly meet the original idea of Markowitz and rather offers a second approach of optimizing investor’s aims with respect to his preferences. Although utility functions are rarely applied in practice, utility functions are a smart concept to describe investor’s preference. An exact form of an investor’s utility function is rarely known, nevertheless a qualitative strategy can be obtained from typical behavior explained by utility functions. Finally, it is a frequently discussed topic in mathematical finance and serves as a good interpretation for optimal pricing measures (see Section 1.3). We will therefore remain with the utility approach in the sequel.

Before we continue with utility maximization, we recommend some intro- ductory references concerning portfolio optimization. A good introductory textbook isMeucci[97] also containing aspects of the estimation of market invariants and several risk measures. Another book concerned with practi- cal, though mathematical, principles also containing portfolio optimization on VaR and expected shortfall is Deutsch [38]. A book looking at an optimal capital allocation is Matten [94]. In contrast to these books, a textbook focusing on a precise mathematical foundation is F¨ollmer and Schied [49]. A further reference following the original idea of Markowitz

but also drawing a connection to utility functions is given inBowden [20].

In this reference, constrained problems with respect to the tail conditional expectation and the VaR are discussed and compared. Effective utility func- tions corresponding to the constrained problems are derived, i.e. functions describing investor’s preferences. Some further reading about risk measures is given in [1; 42]. A more detailed discussion of the related literature on maximization of expected utility is presented in Section 1.3. Moreover, an exact statement of our theoretical results concerning portfolio optimization and its relation to the current research in the area of utility functions is stated there. We will next focus on explaining the intuitive idea of our results.

When applying the concept of utility functions, satisfaction of an investor is measured by the expected utility of the terminal wealth at time T, i.e.

E(U(Y_T)) with terminal wealth Y_T where Y represents the wealth process andU denotes the utility function of the investor. If an investor gets greater terminal wealth, the satisfaction will be higher. U is therefore expected to be increasing. Moreover, U should be designed of diminishing marginal utility, i.e. the more terminal wealth the investor already has the less valuable is a further unit of wealth. For example consider the first raindrop in a desert in comparison to a whole thunderstorm. Summarizing, U should be concave and increasing to meet the qualitative preference structure of an investor.

Whereas the qualitative shape of a utility function is quite obvious, we unfortunately have to say that in practice an explicit form of a function describing preferences is quite difficult to find.

We thus observe that a quadratic loss function, corresponding to the above mean-variance approach, is not really appropriate to quantify this qualitative shape of U as it also punishes overshooting, i.e. the explained problem of the standard deviation. Still the mean-variance example shows that a not everywhere increasing but concave utility function can make sense, in particular for pricing issues (to punish under- and overshooting of the price) or in the sense of Markowitz’s original idea. Allowing for consump- tion in principle leads to an equivalent problem where the utility function becomes constant in the decreasing part of the original utility function; we can interpret this effect as saturation for the investor. Except from consid- ering the standard devition, it further makes sense to take higher moments into account. We therefore treat two types of utility functions, exponential utility functions −e^−αx, α >0 (henceforth called exponential problem) and isoelastic type functions−|1−^αx_p |^p (henceforth called p-th problem). Both are strictly concave but only the exponential function is strictly increasing.

(As mentioned, if we allow for consumption in thep-th problem, effectively we make the utility function constant when x is greater thanp/α). More- over, the connection of both types of utility functions is obvious, we have pointwise convergence if p gets large. The question arises if this still holds for the solutions of the two problems. In order to prove such results, we first

solve both problems and then show convergence forpgoing to infinity. Both problems are solved via a dual problem. Its optimizer can be interpreted as an optimal pricing measure. Their convergence then yields the asserted convergence by applying duality in the limit. Note, we do not treat so-called power utilities, i.e. isoelastic functions ^x_p^p, p ∈ (0,1), for x > 0, and −∞

for x ≤ 0. For a quite intuitive treatment also containing an analysis of stochastic correlations see Buraschi et al. [25]. The authors emphasize the importance of correlation risk. In Chapter 5 we therefore recall some literature on correlation and jump estimation.

Before we proceed we define the structure of the risky assets the investor can buy or sell. For simplicity the risk-free bond is assumed to be set con- stant to 1. For the risky assets firstly we assume that the evolution of the process is controlled by a continuous semimartingale. A geometric Brown- ian motion serves as an example. In a next step, jumps are added to this Brownian motion. If jumps are rare events and appear finitely often in a finite time interval, the exponential of a Brownian motion plus a compound Poisson process is considered. In fact, every piecewise constant L´evy process is a compound Poisson process, see Cont and Tankov [32, Prop. 3.3].

More generally, we are able to prove our results for general exponential L´evy processes (also containing processes with infinite activity), under the condi- tion of upward bounded jumps. Fortunately, this boundedness condition is actually no major disadvantage as the normal evolution of a stock is usually driven by a continuous process whereas extreme negative events are more properly explained by large negative jumps. Moreover, it is well known that empirical results hardly show large positive jumps. Nonetheless, there are popular models possessing unbounded jumps. In such settings we barely find results in the literature as techniques are quite different from the rest of thesis. Results are discussed in a separate chapter (Chapter 4).

With the specification of such processes at hand, we can now solve the ex- ponential and thep-th problem. The standard reference for the exponential problem isDelbaen et al. [36] or slightly more generalSchachermayer [116]. In both articles, theN-dimensional process of risky assets is assumed to be locally bounded, e.g. continuous on a finite time horizon. However, many L´evy processes do not possess this property. Fortunately for exponen- tial L´evy processes and an exponential utility function, an explicit solution can be found in Kallsen[72]. It remains to derive a solution of the p-th problem and its convergence to the exponential one, which is a main result of this thesis.

We optimize over wealth processes which are random and time- dependent. The first difficulty we have to tackle is thus that the optimization problem is dynamic. Fortunately, we are only interested in optimizing the terminal value. We can therefore transform the dynamic problem into a constrained static problem independent of time. Intuitively, the constraints mean that the “price” of the terminal value under every suitable pricing mea-

sure must be less or equal to the initial wealth. The constrained problem is solved by methods from convex analysis relying on the book by Luen- berger [88]. The technique to solve the static problem is developed in Niethammer[103] in the continuous case. It is based on a duality relation to optimal pricing measures as already applied in a different setting in ear- lier work e.g. byKramkov and Schachermayer[82]; Schachermayer [116]. The solution method can almost be adopted to prove the results in the jump case.

Still we need to show that the dynamic problem is actually equivalent to the static one. In the continuous case for thep-th problem, the equivalence can be shown directly, seeNiethammer[103]. However, an explicit strategy is not derived and the jump case is not considered at all. For continuous semimartingales an explicit strategy is derived under a slightly stronger assumption in this thesis. In the case of exponential L´evy processes, we are even able to construct an explicit strategy under the above boundedness condition (upward bounded jumps). Moreover, in both cases we are now able to show the convergence of the portfolios implying the convergence of the terminal values, whereas in [103] only the convergence of the terminal value in the continuous case is shown. All these new results are summarized in two articles,Kohlmann and Niethammer[76] andNiethammer[106].

The version given here in Chapter 2/3 presents the results of these articles slightly extended.

As mentioned there are popular models possessing unbounded jumps, however unbounded jumps can potentially induce zero or negative probabil- ity weights of the corresponding optimal pricing measures, i.e. the measures are absolutely continuous or signed. This might not be a good property for pricing, but for portfolio optimization this does not cause any conceptual problem. The performed optimization over these signed or absolutely con- tinuous measures is just an auxiliary problem to solve the original portfolio optimization problem. To the best of our knowledge we therefore suggest for the first time a technique to solve the dual problem that is essentially differ- ent from the existing literature because of the missing positivity condition of the “densities” of the optimal signed or absolutely continuous measures.

In fact, we use the portfolio optimization problem to solve the dual problem by a hedging (signed measures) or superhedging argument (absolutely con- tinuous measures). Notice, in the case of absolutely continuous measures a hedging strategy will not be sufficient in many cases. We therefore suggest to allow for consumption. Roughly speaking, we set the utility function constant in its decreasing part.

These new extensions to portfolio optimization problems with processes of unbounded jump size potentially leading to “negative pricing” measures are part of Bender and Niethammer [14] and presented in Chapter 4.

The conditions in this chapter are rather weak and easily verified character- izations are provided. The chapter therefore can be considered to contain

a full treatment of theq-optimal martingale measure (the dual solution of the p-th problem) and its behavior when q tends to 1 in a market driven by exponential L´evy processes. To the best of our knowledge, this is the first treatment when martingale measures can become zero or even nega- tive. Furthermore, our idea of finding optimal (signed) martingale measure by simultaneously working out an optimal solution of the portfolio optimiza- tion problem in general semimartingale models seems to be rather promising to tackle even more general problems in future research.

Finally, we are interested in the parameterization of the optimal portfo- lio. Examples are given in Section 5.2. A method to actually estimate the covariance structure is described in Section 5.1.

A Guided Tour

In detail, the thesis is organized as follows. The thesis treats portfolio op- timization in its classical form by consideringN risky assets and a risk-free bond. In Section 1.1, we start presenting an introduction to L´evy processes which is supposed to provide the reader with the relevant results on such processes. In Section 1.2, we then provide a detailed introduction to the general market model (risky assets and a bond) and some - in part known - results on portfolio optimization problems. The corresponding literature is discussed in Section 1.3. In the sequel, at first we specialize the model to risky assets with continuous paths (Chapter 2) and continue with exponen- tial L´evy processes with upward bounded jumps in a second step (Chapter 3). Summaries of the results are given in Section 2.4 and 3.4. In Chapter 4, we discuss the case when jumps become unbounded. We develop a theory on q-optimal signed/absolutely continuous martingale measures. This chapter covers, in one-dimensional case, the most general case treated in this thesis.

After studying Chapter 1.2, Chapter 4 can be read almost independent of Chapter 2 and 3. A summary of Chapter 4 is given in Section 4.7. A com- parison of continuous and jump processes is postponed to Chapter 5. Some techniques to estimate the covariance structure of the processes and a test on the existence of jumps are described in the same chapter.

Summary of the Results We finally list our results:

Chapter 1: Toolbox and some Basic Results 1. Introduction to L´evy processes

2. Presentation of the general problems of portfolio optimization, in par- ticular the connection between dynamic, static, and dual solution

• Derivation of the optimal terminal value of the so-called “p-th problem” (for “isoelastic-type” utility functions −|1−^αx_p |^p)

• Convergence of the optimal terminal wealth of the p-th problem to the exponential problem (−e^−αx) under general assumptions

• Overview about the literature

Chapter 2: Portfolio Optimization in a Continuous Semimartin- gale Model

• Convergence of the optimal terminal wealth of thep-th problem to the exponential problem (−e^−αx) under the Reverse H¨older inequality

• Existence and explicit form of the optimal portfolio ϑ^(p) for the p- th problem (w.r.t.−|1−^αx_p |^p) under the condition of a deterministic mean-variance-tradeoff process

• Convergence of the portfolioϑ^(p)to the optimal portfolio of the expo- nential utility function−e^−αx

Chapter 3: Portfolio Optimization for Exponential L´evy Processes

• Results on q-optimal martingale measures and the minimal entropy martingale measure basing on assumption C_q introduced in Jean- blanc et al. [69] (to gain an explicit representation of theq-optimal equivalent martingale measure)

• Convergence of the optimal terminal wealth of thep-th problem to the exponential problem under conditionCq

• Derivation of the optimal portfolio ϑ^(p) for the p-th problem under condition C_q

• Convergence of the portfolio ϑ^(p) to the optimal portfolio of exponen- tial utility function

Chapter 4: On q-Optimal Martingale Measures or Portfolio Opti- mization for Unbounded Jump Processes

• Discussion of the restrictiveness of condition C_q - root and positivity condition, in particular for models with unbounded jumps

• Presentation of necessary and sufficient, but weaker than C_q, condi- tions for the existence of the root leading to different versions of the q-optimal martingale measure

• A verification procedure based on a hedging problem for theq-optimal signed martingale measure

• A verification procedure based on a superhedging problem for the q- optimal absolutely continuous martingale measure

• Based on the verification procedure, explicit forms of the q-optimal signed and absolutely continuous martingale measure are established, nevertheless restrictive conditions (C_q) for the equivalent case can be dropped

• Convergence of the q-optimal measures to the minimal entropy mar- tingale measure

• Existence and explicit form of the optimal portfolio ϑ^(p) for the p- th problem (w.r.t. −|1− ^αx_p |^p) in the presence of unbounded jumps (without consumption)

• Existence and explicit form of the optimal portfolioϑ^(p,C)and the opti- mal consumption processC^(p,C)for thep-th problem (w.r.t.−|1−^αx_p |^p) in the presence of unbounded jumps when we allow for consumption

• Convergence of the portfolio ϑ^(p) resp. ϑ^(p,C) to the optimal portfo- lio of the exponential utility function, −e^−αx. Consumption C^(p,C) converges to zero.

• Verification of our new conditions for many practically relevant models like the Merton, Kou, or Variance-Gamma model, which do not satisfy assumption C_q

Chapter 5: Implementation and Application

• Short introduction to an estimation method for covariances and jump components

• Illustration of the results and comparison of the optimal portfolios

• An application to credit risk management

Acknowledgements

I would like to thank Professor Dr. Michael Kohlmann (University of Kon- stanz) for an excellent guidance bringing me to a high research level, pushing me to my first publication by initializing this fruitful topic, but most im- portant his constant support; Professor Dr. Ludger Overbeck (University of Gießen) for several suggestions, for being interested in my previous research, and for motivating and supporting me to continue on this topic next to our daily work on credit risk; Christian for many fruitful discussions leading to our first joint work, and above all his never ending patience and support in many difficult situations; Professor Dr. Claudia Kl¨uppelberg (TU Munich), Professor Dr. Walter Schachermayer, Professor Dr. Uwe Schmock (TU Vi- enna) for giving me the opportunity to present parts of this thesis in their research seminars/conferences and jointly with Professor Dr. Jan Kallsen (University of Kiel) and Professor Dr. Yoshio Miyahara (Nagoya City Uni- versity) for the subsequent, very fruitful discussions; my colleagues, fellow students, and several members of Uni Konstanz, JLU Gießen, LMU and TU Munich, Yale University, and Uni Ulm for providing an amazing and familiar working and research environment, in particular my colleagues Dr. Sascha Meyer-Dautrich, Dr. Thomas Seifert (HVB Munich), Dr. Christoph G¨ossl (HVB London), Dr. Christoph Wagner (Allianz Risk Transfer), Mikhail Krayzler (TU Munich), and my fellow students Dr. Steffi Kammer, Swantje Becker, Rolf Klaas (University of Gießen), and Gustavo Soares (Yale Univer- sity) for helpful communication and critical questions concerning my work;

Dr. Clemens Prestele (University of Ulm) for several non-mathematical discussions between two mathematicians and not to forget Andreas Grau, Mr. Thomas “Lieblingsossi” Fischer, my two “nurses” Barbara and Kasia for taking care of my physical and mental shape in my daily professional life; Dr. St´ephane Capet, Stephen Dell, Dr. Martin Krekel, Julia Hein, for proofreading the manuscript; last but not least my parents and my friends for their patience and their warm-hearted support.

Disclaimer

Financial support by UniCredit, Markets and Investment Banking is grate- fully acknowledged. In particular, I like to thank Dr. Thomas Bretzger and Professor Dr. Ludger Overbeck for arranging this unique doctoral position.

However, this thesis does not reflect the opinion of UniCredit, Markets and Investment Banking or any other subsidiary of the UniCredit Group, it is the personal view of the author.

Contents

Abstract iii

Zusammenfassung v

Introduction xi

Acknowledgements xix

1 Toolbox and some Basic Results 1

1.1 On L´evy Processes . . . 1 1.1.1 Motivation . . . 1 1.1.2 Definition and some Relevant Facts on L´evy Processes 4 1.1.3 Some Typical Models based on L´evy Processes . . . . 15 1.1.4 Some Illustrations . . . 20 1.2 Portfolio Optimization and Optimal Martingale Measures . . 22 1.2.1 The Market Model . . . 22 1.2.2 Trading Strategies and Optimization Problems . . . . 24 1.2.3 Solving Static Utility Optimization Problems . . . 34 1.2.4 Martingale Measures . . . 36 1.2.5 Exponential Problem and its Approximation . . . 38 1.2.6 Convergence of the Terminal Values and the Value

Functions . . . 42 1.3 Literature . . . 47 1.3.1 General Overview . . . 47 1.3.2 Embedding Continuous Semimartingales and L´evy

Processes in a General Framework . . . 48 1.3.3 Some Literature on the General Jump Case . . . 50 2 Portfolio Optimization in a Continuous Semimartingale

Model 55

2.1 Assumptions and a Superhedging Result . . . 55 2.2 Convergence of the Terminal Values and the Value Functions 57 2.2.1 Some Facts about Martingale Measures . . . 57 2.2.2 Replacing Technical Assumptions . . . 59

xxi

2.3 Convergence to the Optimal Portfolio for an Exponential Util- ity Function . . . 61 2.4 Concluding Remarks . . . 66 3 Portfolio Optimization for Exponential L´evy Processes 69 3.1 Exponential L´evy Processes and the Structure Condition . . . 70 3.2 Static Solution and Duality . . . 72 3.2.1 The Minimal Entropy Martingale Measure . . . 73 3.2.2 The q-Optimal (Equivalent) Martingale Measure . . . 74 3.2.3 Convergence to the Minimal Entropy Martingale Mea-

sure . . . 76 3.3 Convergence to the Exponential Utility Problem . . . 83

3.3.1 Convergence of the Terminal Values and the Value Functions . . . 83 3.3.2 Convergence to the Optimal Portfolio . . . 84 3.4 Concluding Remarks . . . 92

4 On q-Optimal Martingale Measures 95

4.1 Discussion of ConditionCq . . . 96 4.1.1 Review . . . 96 4.1.2 Reformulation and Discussion of Condition C_q . . . . 98 4.2 Theq-Optimal Signed Martingale Measure . . . 102 4.2.1 Relaxing Assumption C_q. . . 102 4.2.2 Verification of theq-Optimal Signed Solution . . . 109 4.3 Theq-Optimal Absolutely Continuous Martingale Measure . 114 4.3.1 Relaxing ConditionC_q . . . 114 4.3.2 Verification of the q-Optimal Absolutely Continuous

Solution . . . 121 4.4 Convergence to the Minimal Entropy Martingale Measure . . 127 4.5 Some Consequences for Portfolio Optimization . . . 137 4.6 Existence of theq-Optimal Equivalent Martingale Measure in

the Presence of Unbounded Jumps . . . 142 4.7 Concluding Remarks . . . 147 5 The Implementation of Models with Jumps and Applica-

tions 151

5.1 Covariance Estimation and Testing on the Existence of Jumps 151 5.2 Comparison: Brownian vs. L´evy Model . . . 154 5.3 Concluding Remarks . . . 157

References 161

Index 171

List of Figures

1.1 Jump diffusions . . . 20 1.2 Exponential normal inverse Gaussian vs. geometric Brownian

motion . . . 21 5.1 Influence of jump size and intensity on the optimal portfolio . 155 5.2 Decomposition of the variance of a stock . . . 157 5.3 Convergence of the portfolio in a model with two stocks . . . 158 5.4 Convergence of the portfolio in a model with three stocks . . 159

xxiii

List of Tables

4.1 Conditions leading to theq-optimal martingale measure . . . 149 4.2 Versions of the q-optimal martingale measure (existence and

conditions) . . . 149 4.3 Convergence to the minimal entropy martingale measure - dif-

ferent conditions for different versions of the q-optimal mar- tingale measure. . . 149

xxv

Toolbox and some Basic Results

This chapter is devoted to a brief introduction to L´evy processes and some often applied models within this framework. Moreover, we give the necessary background on utility maximization and optimal martingale measures and prove some basic results on those. A comprehensive overview about the surrounding literature is given at the end of the chapter.

1.1 On L´ evy Processes

This section is devoted to an introduction to L´evy processes. We summarize several results that we will apply in the sequel. This is for the reader’s convenience to provide her/him with the material on L´evy processes to be applied below which is widely spread over many books and articles in the recent literature; the most important results are quoted below.

We start with a short motivation why L´evy processes present an intuitive extension of deterministic relationships. The idea is taken from a manuscript by Jan Kallsen, “L´evy-Prozesse anschaulich” [73]. He provides an intuitive and illustrative way to introduce L´evy processes. We close the motivation at a certain point where mere intuition becomes difficult and continue with a rigorous treatment. Finally, we present some models often applied for pricing issues. In the following chapters, we will then discuss if and when our assumptions are satisfied in these models.

1.1.1 Motivation

The aim of this thesis is to optimize the expected utility of a portfolio of stock returns with respect to a certain utility function. So one of the first steps is to model these stocks. As we go beyond a single time step model, we need to model the whole dynamics of the stocks, i.e. a function depending

1

on time and randomness. InKohlmann and Niethammer[76] dynamics are described by continuous semimartingales. In the simplest case, the pro- cess evolves like a diffusion - a Brownian motion - which has been observed for the first time by a biologist. In finance one considers the exponential of a Brownian motion (geometric Brownian motion) to guarantee positivity of the process. The problem of such models in finance is that sudden and not predictable events cannot be modeled adequately. The variance of the process has to be increased tremendously such that this extreme behavior admits a realistic probability of appearance. However, such a high variance would not meet the normal evolution of a stock. It is therefore natural to introduce jumps. We will remain with L´evy processes, as they are quite flexible while possessing analytical tractability. One the one hand its char- acteristic function can easily be described (L´evy-Khinichin, see Theorem 1.1.14) and on the other hand, one obtains an explicit representation of the process. Every L´evy process can be represented as a sum of a drift term, a Brownian motion times a volatility factor, and an integral with respect to a (compensated) Poisson random measure (L´evy-Itˆo-decomposition, see Theorem 1.1.11).

We start supposing that we are able to describe the world in a determin- istic setting and dynamics A depend on dynamics B. If we change dynamics B, dynamics A is supposed to change as well. In Economics, we would say, we like to derive the sensitivity of A with respect to B. If the func- tion describing the relation between A and B is differentiable, we ask for a derivative. Hence, it makes sense to assume an approximately linear de- pendency for small changes. However many dynamics in reality cannot be determined by deterministic dynamics. We thus like to have a stochastic analogue to a linear behavior. So what are the characteristics of a linear function: ˇX_t = ˇX₀ +bt. It is determined by its starting value and the constant increase of the function in an arbitrary time interval with certain length. The increase from 0 to 1, coincides with the increase from 1 to 2 and is equal to b. So how shall we translate this constant increase to the stochastic world? One way is to assume that the increments (the increase in the observed process, ˇX_s+∆t−Xˇ_s) have the same (“stationary”) distribution and that they are independent of each other for disjoint time intervals. In a discrete time setting, this describes a random walk. For a generalization to continuous time, at first we choose an infinitely small time interval dt, as- sume thatX₀= 0 and add stationary and independent increments on these infinitely small intervals. Then we have roughly defined a L´evy process (of course except for some technical conditions also on the distribution of the process).

We next ask why can this processes be interpreted as linear and how can we describe the distribution of such a process? We can answer these two questions together. Notice that ˇX_t can be described by the sum of its increments in the infinitely small time intervals (0, dt) for n = 1 and

[(n−1)dt, ndt) forn >1:

t/dt

X

n=1

( ˇX_ndt−Xˇ_(n−1)dt) (1.1)

We start to determine its distribution. As the increments are independent, the characteristic function of ˇX_tis the characteristic function of the _dt^t-power of the characteristic function of ˇX_dt. From the characteristic function, we can then derive a distribution. If ˇX_dt possesses a characteristic function of log linear form e^ψ(·)dt, we obtain e^ψ(·)t for ˇX_t. Thanks to (1.1), we get E( ˇX₁) = ^{E( ˇX}_dt^dt) and so E( ˇX_dt) = bdt for a b ∈ R. If ˇX_dt is already deter- ministic, we have a linear function and its characteristic function is equal to exp(iubdt).If not, then the variance of ˇX_dtis of orderdtasV( ˇX₁) = ^{V( ˇX}_dt^dt). To induce such a property in the two simplest cases, firstly one can for example assume that the process increases with a probability of γdt by a random variable with distribution F_J and stays constant with probability 1−γdt. Secondly, we might assume that ˇX_dt/dt changes with probability distributionF_B with constant variance. In both cases the variance of ˇX_dt is of orderdt.The latter describes a diffusion leading to a characteristic func- tion that can be approximated by exp(−¹₂σ²u²dt).The first example leads to a compound Poisson process. Its characteristic function is approximately exp(R

(e^iux−1)γFJ(dx)) ase^x ≈x+ 1 for smallx. Assuming both processes are independent and by adding a drift (the linear deterministic part), one gets the following first form of a characteristic function:

exp((iub−1

2σ²u²+ Z

(e^iux−1)γF_J(dx))dt)

ν =γF_J will describe the well-known L´evy measure. This measure together with the drift and the volatility of the Brownian motion σ - the charac- teristic triplet - uniquely determines the distribution of the L´evy process.

We proceed and ask what happens if jumps are very large or appear too often, perhaps infinitely often? Large jumps produce a problem only if ˇX_t has no first finite moment, see Theorem 1.1.17 below. Later we will exclude this case in our considerations, as mathematical finance with infinite returns is rather unrealistic. If there is an infinite number of jumps around zero, R(e^iux−1)γF_J(dx) might not exist, asF_J((−ǫ, ǫ)) is infinite for an arbitrary ǫ > 0. Fortunately, it is known that R

|x|≤1x²γF_J(dx) < ∞ which implies thatR

|x|≤1(e^iux−1−x)γF_J(dx) is finite. As also F_J(R\[−1,1])<∞ (large jumps can be included) and the characteristic function of ˇX_t is the charac- teristic function of the _dt^t-power of the characteristic function of ˇX_dt,we get an intuitively derived version of the so-called L´evy-Khinchin representation for the characteristic function of a limiting L´evy process ˇX_t:

exp((iu˜b− 1

2σ²u²+ Z

(e^iux−1−iux1_|x|≤1)γF_J(dx))t),

where ˜b = b+R

x1_|x|≤1γF_J(dx). We could now examine all the different variants testing on their flexibility. But sooner or later these heuristics become inconvenient. We therefore continue with a rigorous treatment next.

We closely follow Cont and Tankov [32].

1.1.2 Definition and some Relevant Facts on L´evy Processes We next continue with a rigorous treatment of L´evy processes. L´evy pro- cesses perfectly make sense as an extension of their discrete time equivalent - random walks: a random walk is determined by its initial starting point and the evolution of the following independent steps - increments. These new steps are always drawn from the same “stationary” distribution. A nat- ural generalization in continuous time is thus given by L´evy processes. We consider those processes on a finite time interval [0, T], T <∞ throughout the thesis:

Definition 1.1.1. A RCLL (right-continuous left limits/cadlag) stochastic process( ˇX_t)_t∈[0,T]on (Ω,F, P)with values in R^N such thatXˇ₀ = 0is called a L´evy process if it possesses the following properties:

1. Independent Increments: for every increasing sequence of times t0, ..., tn the random variables Xˇt0,Xˇt1−Xˇt0, ...,Xˇtn−Xˇtn−1 are inde- pendent.

2. Stationary increments: the law ofXˇ_t+h−Xˇ_t does not depend on t.

3. Stochastic continuity: ∀ǫ >0, lim_h→0P(|Xˇ_t+h−Xˇ_t|> ǫ) = 0.

Note, the third condition does not imply that the sample paths are con- tinuous. We start with the simplest counterexample - a Poisson process:

Example 1.1.2 (Poisson process). Let (v_i)_i be a sequence of independent exponential random variables with parameterγ >0 and ˇv_n=P_n

i=1v_i.The process

Ut=X 1t≥ˇvn

is called a Poisson process with intensityγ.

Clearly the process is almost surely finite, piecewise constant, right con- tinuous with left limits (cadlag), and Markovian. It has stationary and independent increments and satisfies stochastic continuity. For a full list of properties see [32, Proposition 2.12]. We however wish to recall two impor- tant facts. Firstly, for every t > 0, U_t follows a Poisson distribution with parameterγt,i.e. for alln∈N, P(Ut=n) =e^−γt(γt)_n!ⁿ.This further implies the special form of the characteristic function ofU_t:

E(e^iuUt) = exp{γt(e^iu−1)} (1.2)

Further it can be easily seen that the sum of two independent Poisson pro- cesses is again a Poisson process with respect to the sum of the intensities (see [32, Proposition 2.13]). Moreover, a Poisson process counts the number of jump times in [0, t]:

U_t= #{n≥1,vˇ_n∈[0, t]}, U_t−U_s= #{n≥1,vˇ_n∈(s, t]}

So for any measurable set D ⊂ [0, T], a random measure G is defined by setting

G(ω, D) = #{n≥1,vˇ_n(ω)∈D}.

For a fixedω the measureG(ω,·) is positive, integer valued, andG(ω, D) is finite for any bounded setD.The process is therefore of the following form:

U_t(ω) =G(ω,[0, t]) =:

Z

[0,t]

G(ω, ds)

Moreover the average value of the random measure atDis thenE(G(·, D)) = γ|D|, where |D| describes the Lebesgue measure of D. γ|D| will be equal to the L´evy measure of D. Finally notice, a Poisson process U is not a martingale. However, we obtain a martingale by compensating with the L´evy measure on [0, t] :

U˜_t=U_t− hUit=U_t−γt,

wherehUiis the predicable quadratic variation ofU, see e.g. [67]. The mar- tingale ˜U is called the compensated Poisson process. A new random measure can be defined by compensating withγ|D|,i.e. ˜G(ω, D) =G(ω, D)−γ|D|. G˜ is signed.

Note, the defining conditions of a L´evy process also do not exclude that a L´evy process can have continuous sample paths. Every Brownian motion has a continuous modification, independent and stationary increments and is therefore a L´evy process. In fact, we will later see that we can split a L´evy process in two parts, a Brownian part and a jump part - the so-called L´evy-Itˆo-decomposition. This implies that every Gaussian L´evy process is continuous and can be represented as a Brownian motion adding a drift term. Before we explain this decomposition in detail, we like to summarize some facts on the characteristic function of a L´evy process, Poisson random measures, the so-called L´evy measure, and illustrate these findings by con- sidering a compound Poisson process. This further leads to a first version of the L´evy-Itˆo-decomposition. We start with a proposition on the characteris- tic function of a L´evy process, see e.g.Cont and Tankov [32, Proposition 3.2]:

Proposition 1.1.3. Let( ˇX_t)_t≥0 be a L´evy process onR^N.Then there exists a continuous function φ :R^N → R called the characteristic exponent of Xˇ such that:

E(e^iu′Xˇt) =e^tφ(u), u∈R^N.

The special form of φ has already been mentioned in the beginning, it is given by the L´evy-Khinchin decomposition. It will be introduced be- low after explaining the L´evy-Itˆo-decomposition. The latter decomposition leads to a natural characterization of the distribution of a L´evy process, the characteristic triplet. By the knowledge of this triplet the L´evy-Khinchin decomposition is then uniquely determined.

We proceed defining a compound Poisson process. Roughly speaking, a Poisson process schedules the jumps of a compound Poisson process and an additional experiment in case of a jump determines the jump size. This additional experiment is distributed with respect to a jump size distribution:

Definition 1.1.4. A compound Poisson process with intensity γ > 0 and jump size distributionf is a stochastic process J_t defined as

J_t=

Ut

X

i=1

X_i,

where jump sizesX_iare i.i.d. with distributionf andU_tis a Poisson Process with intensity γ,independent from (Xi)i.

In principle, by integrating out the jump size distribution, in analogy to a Poisson process, the characteristic function of a compound Poisson process can be obtained, see alsoCont and Tankov [32, Proposition 3.4]:

Proposition 1.1.5. Let J be a compound Poisson process on R^N with in- tensity γ and jump size distribution f. Its characteristic function then has the following representation:

E(e^iuJt) = exp{γt Z

R^N

(e^iux−1)f(dx)}. (1.3) ν(dx) =γf(dx) is the so-called L´evy measure of the compound Poisson process. To explain the L´evy measure in general we need to define a Poisson random measure and present an additional lemma. Before we proceed, note that random measure are defined inJacod and Shiryaev[68, Section II.1]

in a rather general setting. We focus on Poisson random measures as those are sufficient to describe L´evy processes. Apart from being integer-valued, those random measures follow a Poisson distribution for every fixed measur- able set.

We have already become acquainted with a Poisson random measure in Example 1.1.2. Poisson random measures generalize this notion. The

Lebesgue measure is replaced by a Radon measure, i.e. a measure µ on (I,B(R^N)), I ⊂ R^N such that on every bounded, closed, and measurable set D ∈ B(R^N) we have µ(D) < ∞. On the basis of Radon measures, one defines:

Definition 1.1.6. Let (Ω,F, P) be a probability space, I ⊂ R^N, and µ a given (positive) Radon measure on (I,B(R^N)). A Poisson random measure on I with intensity measure µ is an integer valued random measure:

G: Ω× B(R^N)→N, (ω, D)7→G(ω, D) such that the following three assertions hold:

1. For almost allω ∈Ω,G(ω,·)is an integer-valued Radon measure onI, i.e. for any bounded measurable setD⊂I, G(·, D)<∞ is an integer valued random variable.

2. For each measurable setD⊂I, G(·, D) =:G(D)is a Poisson random variable with parameter µ(D), i.e. for all k∈N:

P(G(D) =k) =e^µ(D)(µ(D))^k k! .

3. For disjoint measurable sets D₁, ..., D_n ∈ B(R^N), the variables G(D₁),...,G(D_n) are independent.

The compensated Poisson random measure is then defined by G(D) =˜ G(D)−µ(D).

We have E( ˜G(D)) = 0 and V( ˜G(D)) = µ(D) and get the following connection to the L´evy measure ν of a compound Poisson process, see [32, Proposition 3.5]:

Proposition 1.1.7. Let J be a compound Poisson process with intensityγ and jump size distribution f. Then its jump measure N_J on R^N ×[0, T] defined by

Nj(D) = #{(Jt−Jt−, t)∈D}, D ⊂R^N ×[0, T]

is a Poisson random measure with intensity measure µ(dx × dt) = γf(dx)dt:=ν(dx)dt.

Finally, we state the general definition of a L´evy measure:

Definition 1.1.8. Let Xˇ be a L´evy process onR^N. The measure ν on R^N defined by:

ν(B) =E(#{t∈[0,1] : ∆ ˇXt6= 0,∆ ˇXt∈B}), B∈ B(R^N)

is the so-called L´evy measure of Xˇ : ν(B) is the expected number per unit time of jumps whose size belongs toB.

The question arises if for any Radon measure there exists a Poisson random measure with intensity µ. The answer is yes, see e.g. [32, Proposi- tion 2.14]. So for every L´evy measure there exists a corresponding Poisson random measure such that the process can be represented by (an integral w.r.t.) this measure? The answer is almost yes and given by the L´evy-Itˆo- decomposition. The aim is to represent the processes by an integral w.r.t.

its Poisson random measure or its compensated version, respectively. We start defining these integrals and continue with the simple case whenν is a finite measure. In this case the answer to the last question is definitely yes.

At first, set I =R^N\{0} ×[0, T] and notice that every Poisson random measure on I can be associated to a random sequence ( ¯J_n)_n = (( ¯X_n,v¯_n))_n in I. Roughly speaking, a Poisson random measure counts the number of jumps with certain size in the evaluated set, where ¯vn indicates if there is a jump (jump time) and ¯X_n determines the corresponding size. In detail, for every Poisson random measure G on I there exists a sequence of pairs ( ¯X_n(ω),¯v_n(ω))∈I,such that

G(ω, D) =X

n≥1

δ_{( ¯Xn(ω),¯vn(ω))}(D), D⊂I,

where δ denotes the Dirac measure, i.e. δ_(x,y)(D) = 1, if (x, y) ∈ D and 0 otherwise. The assertion is contained in the proof of [32, Proposition 2.14]. In fact, any right continuous process with left limits (RCLL) ˇX in R^N induces an integer-valued random measureNXˇ on R^N×[0, T]:

NXˇ(ω, dx×dt) =X

s

1_{∆ ˇXs6=0}δ_{( ˇXs(ω),s)}(dx, dt). (1.4) SeeJacod and Shiryaev[68, Proposition 1.14/1.16] for further details on general jump measures. We rather proceed with our integration theory in a nutshell:

A Poisson random measure (induced by the sequence ( ¯Xn(ω),¯vn(ω))) is called nonanticipating, if (¯v_n) is Fn-adapted and ¯X_n is Fv¯n-adapted. As G(ω,·) is a measure for every ω, we can define integrals as usual. For a simple functionh(s, x) =P

ic_i1_Di(x, s),where (D_i)_i are disjoint subsets of I and non-negative constants (c_i)_i, we define the new integral as

G(ω, h) =X

i

c_iG(ω, D_i).

Its expectation is µ(h). So for every positive and measurable function h, which can be approximated by an increasing sequence of simple functions (h_n)_n, the integral ofh is defined as the limit of the integrals ofh_n,G(h) = lim_n→∞G(h_n). As usual for an arbitrary measurable function h : [0, T]× R^N →Rsatisfying

µ(|h|) = Z

[0,T]

Z

R^N\{0}|h(s, x)|µ(dx×ds)<∞, (1.5)