Affine Markov processes on a general state space

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Veerman, E.

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2011

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Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.

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Affine Markov Processes

on a General State Space

Enno Veerman

Affine Mark

ov P

rocesses on a General State Space

Enno V

eerman

Uitnodiging

tot het bijwonen van de openbare verdediging van mijn proefschrift

Affine Markov Processes

on a General State Space

op woensdag 6 juli 2011 om 14:00 uur

in de Agnietenkapel van de Universiteit van Amsterdam Oudezijds Voorburgwal 231

te Amsterdam

Na afloop is er een receptie

Enno Veerman e.veerman@uva.nl

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Enno Veerman

Korteweg-De Vries Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Proefschrift Universiteit van Amsterdam

Printed & Lay Out by: Proefschriftmaken.nl || Printyourthesis.com Published by: Uitgeverij BOXPress, Oisterwijk

ISBN 978-90-8891-304-4

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Affine Markov Processes on

a General State Space

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus prof. dr. D.C. van den Boom

ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel

op woensdag 6 juli 2011, te 14:00 uur

door

Enno Veerman

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Promotor: prof. dr. C.A.J. Klaassen Copromotor: dr. P.J.C. Spreij

Overige leden: prof. dr. A. Bagchi prof. dr. M.R.H. Mandjes prof. dr. J.M. Schumacher prof. dr. J. Teichmann prof. dr. M.H. Vellekoop prof. dr. J.H. van Zanten dr. A.J. van Es

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And now for something completely different

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Voorwoord

Na ruim vier jaar noeste arbeid ligt thans voor u het resultaat van mijn promotie-onderzoek in de wiskunde. Het onderwerp van dit proefschrift betreft de theorie van affiene Markov processen, een vakgebied binnen de stochastiek en financi¨ele wiskunde. Hoewel affiene Markov processen zeer fundamenteel zijn voor financi¨ele toepassingen en derhalve ook van algemeen belang zijn, heeft dit proefschrift toch een ietwat esoterisch karakter. Het zal dan ook niet helemaal weglezen als een roman. Om u enig inzicht te verschaffen in het heikele onderwerp van affiene pro-cessen verwijs ik u graag naar de samenvatting aan het einde van dit proefschrift, alsmede naar de eerste paragraaf van de introductie even verderop. Het overige en grootste gedeelte zal voor velen waarschijnlijk een magisch schouwspel van sym-bolen zijn, vrij van enige semantiek, wat uiteraard spijtig is. Maar ik ben zeer trots op het bereikte resultaat, dat ik niet zou hebben behaald zonder de hulp van anderen. Een aantal belangrijke personen die er op directe dan wel indirecte wijze aan hebben bijgedragen, wil ik hier nu bedanken.

Allereerst bedank ik mijn copromotor Peter Spreij. Peter, feitelijk alles wat ik van kansrekening en stochastiek weet, heb ik van jou geleerd, of uit de boeken uit jouw priv´ebibliotheek. Tijdens de afgelopen vier jaar (maar ook daarvoor) heb je altijd vol belangstelling en met een opgeruimde blik mijn talloze manuscripten met berekeningen grondig doorgelezen, waarbij je nooit een spier vertrok als ik vervolgens voorstelde om het allemaal eens heel anders te gaan doen, wat het daarvoor gelezen werk dan weer obsoleet maakte. Dankzij jouw expertise en kennis, maar ook je positieve instelling en goedgehumeurdheid, heb ik dit proefschrift en mijn AiO-schap met veel plezier kunnen voltooien.

Als tweede wil ik mijn promotor Chris Klaassen bedanken. Chris, jij hebt me altijd alle ruimte gegeven en vrij gelaten in de invulling van mijn onderzoek.

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L´evy processen betrof, bleek ik meer affiniteit te hebben met een ander vakgebied in de stochastiek. Voor je vertrouwen dat alles goed zou komen ben ik je zeer dankbaar. Ook de zorgvuldige nalezing van en correcties bij eerdere versies van dit proefschrift waren zeer waardevol voor mij.

Ik dank de overige leden van de promotiecommissie voor hun bereidheid deel te nemen aan de zitting en het beoordelen van mijn proefschrift. In particular I like to thank Josef Teichmann for his hospitality during my visit at the ETH Z¨urich and his careful annotations, which account for a significant improvement of this thesis.

Het onderzoek waaruit dit proefschrift is voortgekomen, is verricht aan het Korteweg-de Vries Instituut (KdVI) voor Wiskunde aan de Universiteit van Am-sterdam. Ik heb hier altijd met veel genoegen gewerkt en wil mijn collega’s van het KdVI hartelijk danken voor de plezierige tijd. For being such good friends and organizing dinners for all fellow PhD students (as well as for sharing their incredi-ble LaTeX-skills), I would like to thank Zhenya (aka Jevgenijs) and Ricardo. Ook dank ik de andere mede-promovendi en mijn reeds gepromoveerde vrienden Abdel, Andries, Benjamin, Bert, Frank, Kamil, Loek, Michel, Nabi, Naser, Pascal, Paul, Piotr, Ramon en Roland, alsmede Evelien en Hanneke voor hun gezelligheid. Laat ik ook de portier Ren´e niet vergeten, die mijn leidinggevende capaciteiten al in een vroeg stadium op de juiste waarde wist te schatten.

Mijn familie en vrienden wil ik danken voor hun liefde, steun, toeverlaat, vriendschap en alle andere zaken des levens die een wiskundige zo nu en dan nodig heeft om op het juiste pad te geraken. Mijn vader en moeder zijn hierbij van onschatbare waarde. Ik ben dankbaar dat ik altijd bij jullie terecht kan voor advies en dat ik al die jaren van jullie wijsheid heb mogen genieten. Zeer verheugd ben ik dat mijn broer Christiaan en mijn neef Jacob de rol van paranimf op zich willen nemen; ik ben jullie hiervoor zeer erkentelijk. Als allerlaatste dank ik mijn lieve vriendin Nadette. Zonder jouw liefde en steun was ik nooit zover gekomen.

Enno Veerman Amsterdam, mei 2011

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Notation

Let E ⊂ Rp _{be a closed set. Throughout we use the following notation.}

EC _{the class of functions on E with values in C}

B(E) the class of bounded measurable functions in EC

C(E) the class of continuous functions in EC

Cb(E) the class of bounded continuous functions in EC

Cc(E) the class of continuous functions in ECwith compact support

C0(E) the class of continuous functions in ECvanishing at infinity

Ck_(E) _{the class of k-times continuously differentiable functions in E}C

M (E) the class of measurable functions in EC

Rn×m the set of (n × m)-matrices with real-valued coefficients B(Rp₎

the Borel σ-algebra on Rp R+ [0, ∞), likewise R−= (−∞, 0]

R++ (0, ∞), likewise R−−= (−∞, 0)

C+ R++ iR, likewise C−= R−+ iR

Sp _{the set of symmetric matrices in R}p×p

S₊p _{the cone of positive semi-definite matrices in R}p×p

∂x, ∂x+ short hand notation for ∂

∂x and the right-hand side derivative

DE[0, ∞) the class of c`adl`ag functions f : [0, ∞) → E.

I the identity matrix ei the i-th unit vector

|v| Euclidean norm of a vector v

|K|(dz) the variation of a signed measure K(dz) = K+_{(dz) − K}−_(dz),

defined as |K|(dz) = K+_{(dz) + K}−_(dz)

fu the function on E given by x 7→ exp(u>x), for some u ∈ Cp

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Introduction

Since the rediscovery of Bachelier’s work on modeling stock prices with Brown-ian motion, many attempts in finance and mathematical finance have been made to capture the movements of the financial market in a mathematical framework. Traditionally, a main instrument for modeling asset prices has been the class of semimartingales, in particular continuous diffusions that arise as solutions to cer-tain stochastic differential equations (SDEs). We mention the models introduced in the last decades of the previous century, such as the Black-Scholes model [3], where the stock price is modeled by a geometric Brownian motion, the Vasicek model [53], where the short interest rate is described as an Ornstein-Uhlenbeck process, the Cox-Ingersoll-Ross model [9], which uses a square root SDE for the interest rate, the affine term structure model by Duffie and Kan [18], which extends the CIR-model to multiple dimensions, and the Heston model [29], which incor-porates stochastic volatility into the Black-Scholes model by adding a square root SDE for the variance. All these models share the desired property that they are mathematically tractable. They do not only capture the dynamics of the market (reasonably) well, but also allow for mathematical analysis and the performance of calculations. In particular they make calibration possible.

At the end of the previous century, the above mentioned “classical” models were all subsumed into one framework in the pioneering paper [19] by Duffie, Pan and Singleton. They introduced a class of processes, adopted with the name affine jump-diffusions, that are characterized as solutions of multi-dimensional SDEs, in-cluding jumps, where the drift vector, the instantaneous diffusion matrix and the arrival rate of the jumps all depend in an affine way on the current state of the pro-cess. Analytic expressions were provided for various transforms of these processes, including Fourier and Laplace transforms. Elaborating on this, in the seminal

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paper [17] by Duffie, Filipovi´c and Schachermayer, a fundamental and rigorous mathematical treatment has been given for an even more general class of affine jump-diffusions, called affine processes, which also allows for killing and explosion. These processes are characterized as Markov processes where the logarithm of the characteristic function of the transition function has affine dependence on the ini-tial state of the process. Having a closed form expression for the Fourier transform at hand, Duffie, Filipovi´c and Schachermayer provide examples of financial appli-cations of affine processes, including interest rate term structure modeling, option pricing and risk modeling.

The class of affine processes can be regarded as a complete generalization of and an improvement upon the previously mentioned classical models. Affine pro-cesses not only preserve the desired property of mathematical tractability from the old models, but also realize a better fit of the market. The latter is due to their flexibility, since multiple (macro-economical) factors can be included in an affine process. Moreover, as observed in [20], modeling the driving force of stock movements solely by Brownian motion appears to be unsatisfactory, for the tails of the Gaussian distribution are too thin. Incorporating jumps in the model is a possible remedy for this deficiency.

This is not the end of the story though (indeed, it’s the start). A standing as-sumption in [17] is that the state space in which the affine process takes its values, is of the so-called canonical form Rm+× R

p−m_{, a notion that was introduced in [14].}

A consequence of this is that the instantaneous covariance matrix is essentially a diagonal matrix, in other words, infinitesimal small movements of the diffusion part are mutually uncorrelated. For applications this might be problematic, see for instance [50], where an affine model for interest rate and inflation is consid-ered. In that paper it is shown that the mathematical restrictions implied by the shape of the state space, are in contrast with economic principles for the interplay between interest rate and inflation, wherefore they are simply dropped. Though the resulting model is not of the affine type anymore, still the bond prices can be accurately approximated by the closed form expressions for affine processes, as shown by Monte Carlo simulations.

An alternative way to overcome these modeling issues, is to extend the flexi-bility of affine processes by seeking for other state spaces than the canonical one. It has already been observed in [17] that affine processes are not limited to take values in the canonical state space, but they might also assume their values in a parabolic state space. For the 2-dimensional diffusion case this was further in-vestigated in [27]. Recently the attention has been drawn to matrix-valued affine

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Introduction 3

processes, which assume their values in the cone of positive semi-definite matrices, see for instance [10, 11, 25, 28, 44]. These processes are typically used for model-ing instantaneous stochastic covariance in multi-variate extensions of the Heston model. A mathematical foundation for matrix-valued affine processes has been provided in [10].

This thesis contributes further to the theory of affine processes with a non-canonical state space and complements [17] and [10]. We develop a mathematical foundation for affine processes with a general state space, including the canonical state space and the cone of positive semi-definite matrices, as well as the parabolic state space and the Lorentz cone. The main contributions in these thesis are as follows.

• First, a full characterization is given for affine processes on a rather general (convex) state space. We show the equivalence between an affine process and an affine jump-diffusion by means of the Feller property. The necessary and sufficient conditions on the affine characteristics, called admissibility conditions, are presented in a general form, based on the positive maximum principle.

• Second, we determine all possible polyhedral and quadratic state spaces on which an affine process exists, under the presence of a diffusion part, and we work out the admissibility conditions for these.

• Third, we extend the validity of the exponential affine expression for the Fourier-Laplace transform of an affine process beyond its natural domain. In particular we obtain conditions under which an affine process has a finite exponential moment.

In order to understand the topic of this thesis better and as a warming up for the next chapters, we discuss below the main ideas in the theory of affine processes and give a mathematical overview.

Mathematical overview

We consider a very simple affine process (namely the square root process, as used in the Cox-Ingersoll-Ross model) and calculate its conditional exponential moments, using heuristic arguments. Let (Ω, F , (Ft), P) be a filtered probability space and

X a stochastic process that solves the 1-dimensional square root SDE

dXt= (aXt+ b)dt +

p

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with a ∈ R, b ≥ 0, x ≥ 0 and W a 1-dimensional Brownian motion. Observe that X is an affine process with state space R+. Indeed, the drift aXt+ b and the

instantaneous variance matrix Xt(the square of the term in front of the Brownian

motion) clearly depend in an affine way on Xt, while Xt≥ 0 for all t, due to the

condition b ≥ 0. We aim to calculate the conditional exponential moment

E(exp(uXt)|Fs), for s ≤ t, some u ∈ C.

Fix t ≥ 0. As an “ansatz”, we try f (s, Xs), with f : [0, t] × [0, ∞) → C a C1,2

-function given by

f (s, x) = exp(φ(t − s) + ψ(t − s)x),

for some C1_{-functions φ : [0, t] → C, ψ : [0, t] → C with φ(0) = 0, ψ(0) = u. Note} that f (t, Xt) = exp(uXt). Therefore, it suffices to choose φ and ψ in such a way

that f (s, Xs) is a martingale, since in that case we have

E(exp(uXt)|Fs) = E(f (t, Xt)|Fs) = f (s, Xs).

To determine φ and ψ, we apply Itˆo’s formula, which yields (suppressing the arguments of φ and ψ)

∂sf (s, Xs) = ∂sf (s, Xs)ds + ∂xf (s, Xs)dXs+1₂∂xxf (s, Xs)dhXis

= f (s, Xs)(− ˙φ − ˙ψXs+ ψ(aXs+ b) +1₂ψ2Xs)ds + f (s, Xs)ψ

p

XsdWs.

Suppose that the stochastic integral is a proper martingale. Then f (s, Xs) is a

martingale under the additional condition that the drift term vanishes, which holds if (φ, ψ) satisfies the Riccati equations

˙ φ = bψ

˙

ψ = aψ +1₂ψ2.

Thus we have found an “exponential affine” expression for the conditional exponen-tial moments of the square root process. The above derivation can easily be gen-eralized to multi-dimensional diffusions with affine drift and affine instantaneous covariance matrix, or even to jump-diffusions where in addition the jump-rates are affine, see [19].

For turning the previous derivation into a rigorous mathematical proof, some gaps need to be filled, to wit:

• The solutions (φ, ψ) to the Riccati equations might explode in finite time, due to the quadratic term 1₂ψ2_{. It is obvious that E exp(uX}t) is finite for

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Introduction 5

u ∈ C−, but a careful analysis is needed to show that explosion of (φ, ψ) for

u ∈ C−is impossible. In addition, E exp(uXt) = 0 for some complex u ∈ C−

is not excluded a priori, which would correspond with an explosion of ψ.

• It is assumed that the stochastic integral R·

0f (s, Xs)ψ(t − s)

√

XsdWs is a

proper martingale. This is not immediately clear and needs further verifica-tion.

These issues are taken care of by Duffie, Filipovi´c and Schachermayer in [17] (and generalized to Rm+ × R

p−m_{-valued jump-diffusions). In addition, they show the}

reverse statement, in the sense that a Markov process X necessarily solves a square root SDE, whenever its exponential moments are exponential affine.

Let us explain the importance of having a closed form expression for the (con-ditional) exponential moments for financial applications, following [17, Section 13]. A very popular application of affine processes is modeling the term structure of interest rates. In general, the pricing formula based on no-arbitrage reasoning (see for instance [31]) yields that the price Dt,T of a zero-coupon bond at time t paying

one unit of money at maturity time T, is given by

Dt,T = E(exp(−

Z T

t

rsds)|Ft),

where r denotes the short interest rate process and the expectation is taken under the risk-neutral measure. In a short-rate model, one chooses a stochastic process for modeling the short interest rate, from which the dynamics of the bond price can be deduced by the above relation. A desirable feature of a short rate model is that the obtained formula for the bond price can be calculated analytically, rather than by Monte Carlo simulations, so that calibration is possible. One way to achieve this, is by modeling r as an affine transformation of an affine process, due to the exponential affine expression for the conditional exponential moments. This can be shown as follows.

Let rt= δ0+ δ>Xt, for some δ0 ∈ R, δ ∈ Rp, with Xt an affine diffusion on

some state space E ⊂ Rp_{, say with affine drift b(X}

t) and affine diffusion matrix

c(Xt). Define Yt= −R t

0rsds. Then it is easy to see that Zt= (Xt, Yt) is an affine

diffusion with state space E × R and drift and diffusion matrix given by

b(Xt) −δ0− δ>Xt ! and c(Xt) 0 0 0 ! .

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Now the assertion follows from the observation that E(exp(− Z T t rsds)|Ft) = exp(−Yt)E(exp(YT)|Ft) = exp(−Yt)E(exp(u>ZT)|Ft), with u ∈ Rp+1 _{given by u} i= 0 for i ≤ p and up+1 = 1.

A second application can be found in option pricing. Consider the price of a European put-option that gives the buyer the right (but not the obligation) to sell a certain stock at time T for a fixed price K > 0. Suppose the price of the stock at time T is given by f (XT) for some positive function f and X a stochastic process.

Then the pricing formula based on no-arbitrage yields that the fair price of the option at time 0 equals (with rtthe short rate process)

KE(exp(− Z T 0 rtdt)1{f (XT)≤K}) − E(exp(− Z T 0 rtdt)f (XT)1{f (XT)≤K}).

Let the price of the stock be modeled as k0exp(k>Xt) and the short rate as

r(Xt) = δ0+ δ>Xt, for some p-dimensional affine process X, with k0, δ0∈ R and

k, δ ∈ Rp_{. Then both expectations on the right-hand side of the above display are}

of the form E(exp(− Z T 0 r(Xt)dt)ea >_X T₁ {b>_X_T_≤c}), for some a, b ∈ Rp

, c ∈ R. This expectation can be regarded as the distribution function of b>XT at c under the measure

exp(− Z T 0 r(Xt)dt)ea >_X T_dP.

Hence to determine the expectation it is enough to compute the Fourier transform of the above measure, in view of Plancherel’s Theorem, see e.g. [5]. The Fourier transform can be seen to be of the form

E(exp(− Z T 0 r(Xt)dt)eu >_X T_),

for some u ∈ Cp. As in the case of the zero-coupon bond price (corresponding with u = 0), this expectation is of the exponential affine form, which can explicitly be determined by solving a system of Riccati equations.

We now return to affine processes with a general state space E. For ease of exposition, we consider the continuous diffusion case. Suppose X is an affine diffusion with state space E, given as the solution to the multi-dimensional SDE

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Introduction 7

with affine drift b(Xt) and affine diffusion matrix c(Xt). The following observations

are crucial.

• In order to have existence of the square root, it is necessary that c(x) is positive semi-definite for all x ∈ E. This imposes restrictions on both the diffusion matrix and the state space E.

• The process X is not allowed to leave the state space E (a notion called stochastic invariance of E). Therefore, conditions on the behavior of the drift and diffusion matrix on the boundary of the state space have to be imposed. In particular, the drift b(x) should point inwards and the diffusion matrix c(x) should vanish parallel to the boundary, for all x ∈ ∂E. The necessary and sufficient conditions on b and c (and also on the jumps for general jump-diffusions) are called admissibility conditions.

• Uniqueness of the solution to the SDE is not immediate. Since x 7→ c(x)1/2_is

in general not Lipschitz-continuous for x on the boundary, standard unique-ness results for SDEs fail. In addition, we note that [55, Theorem 1] is only applicable for affine diffusions on the canonical state space Rm+ × Rp−m, but

not for general state spaces.

• The behavior of the solutions (φ, ψ) to the Riccati equations is crucial for deriving uniqueness. When the exponential affine expression is established for the characteristic function, uniqueness of the affine diffusion follows from uniqueness of (φ, ψ).

The main challenge in developing a theory for affine processes on an arbitrary state space, compared to the canonical state space Rm

+ × Rp−m, is that the

ad-missibility conditions are much more delicate. This is due to singularities on and curvedness of the boundary, resulting in an additional “Stratonovich term” when deriving the boundary conditions required for stochastic invariance. As a conse-quence, it is much harder for general state spaces to analyze the solutions (φ, ψ) to the Riccati equations directly by means of these admissibility conditions. To circumvent this difficulty, we use an indirect approach and rely on probabilistic methods instead. For our analysis we extensively make use of stochastic calculus and the theory on semimartingales.

The contents of the thesis are as follows. In the first three chapters we recall and extend theory for general stochastic processes, which we use in the remaining chapters for the analysis of affine processes. In Chapter 1 we treat general semi-martingales, while in Chapter 2 we consider jump-diffusions and analyze them by

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means of the martingale problem. Using the positive maximum principle, we are able to derive necessary and sufficient conditions for the existence of general jump-diffusions, see Proposition 2.15. This proposition will be used for proving existence of affine processes in Chapter 4. In addition, we infer a result on the martingale property of a stochastic exponential, see Proposition 2.23, which will be used in Chapter 6 for establishing the validity of the affine transform formula. Chapter 3 is devoted to general Markov and Feller processes that live on a state space E of the form E = X × Rp−m

, where X ⊂ Rm_{is a closed convex set satisfying certain}

properties. We characterize regular Feller processes as the solution of a martingale problem in Theorem 3.20.

Next, we establish existence and uniqueness of affine processes living on an arbitrary state space E of the aforementioned form X × Rp−m _{in Chapter 4. The}

main result of this chapter is Theorem 4.4. Here, the admissibility conditions are given in a general form, which are explicitly worked out for polyhedral and quadratic state spaces in Chapter 5, see Theorems 5.12, 5.17 and 5.22. In this chapter we also specify the form of a general polyhedral state space and charac-terize all quadratic state spaces. It turns out that the parabolic state space and the Lorentz cone are the only possibilities for the latter. Finally, in Chapter 6 we aim to extend the validity of the exponential affine expression for exponential moments. The main results here are Theorems 6.4 and 6.7. With the aid of the first theorem, we provide tractable conditions under which the Fourier-Laplace function does not vanish in Theorem 6.5.

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Chapter

1

Semimartingales

Since a large part of this thesis heavily relies on methods from stochastic calculus for semimartingales, we recall some notation, definitions and results. For a com-plete overview the reader is referred to [33]; see also [35] for a didactic note. This chapter is not only for the sake of making this thesis self-contained. We also adapt and restate known results into a form tailored to our needs in subsequent chap-ters. In addition, some results for semimartingales we want to apply, are scattered through the literature, or are well-known, but not easy to find. These are also included here. The chapter starts with a summary of notation and concepts in the theory of semimartingales, including stochastic integration with respect to a semimartingale and with respect to a random measure, as well as square and angle bracket processes. In Section 1.2 we recall the definition of the characteristics of a semimartingale and state Itˆo’s formula in terms of these, together with some corollaries. Next we consider stochastic exponentials and measure changes in Sec-tion 1.3 and we prove a version of Girsanov’s Theorem in order to derive the form of the “inverse” of a stochastic exponential. The latter is stated in Proposition 1.12, a result which is surprisingly hard to find in the literature. It is applied in Chap-ter 2 to obtain a sufficient condition for the martingale property of a stochastic exponential.

1.1 Definitions

Let (Ω, F , (Ft)t≥0, P) be a filtered probability space with a right-continuous

filtra-tion (Ft)t≥0. An (Ft)-adapted, c`adl`ag stochastic process X with values in R is

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called a semimartingale if it admits a decomposition X = X0+ M + A, where M

is a local martingale and A is a c`adl`ag adapted process with locally finite varia-tion, with M0= A0= 0. By convention, every local martingale is assumed to be

c`adl`ag. We call the semimartingale special if A can be chosen to be predictable (i.e. measurable with respect to the σ-algebra that is generated by all left-continuous adapted processes, viewed as mappings on Ω × R+). Every local martingale M can

be decomposed uniquely as M = M0+ Mc+ Md, where Mc is a continuous local

martingale and Md _{is a purely discontinuous local martingale (i.e. M}d_{N is a local}

martingale for all continuous local martingales N ), with Mc

0 = M0d = 0. There

exists a unique continuous local martingale denoted by Xc_{, such that X}c _{= M}c

for every decomposition X = X0+ M + A as above. We call Xc the continuous

martingale part of X.

Every semimartingale has a decomposition such that M is a local L2_-martingale.

Under this decomposition, the stochastic integral of a predictable locally bounded process H with respect to the semimartingale X, denoted by H · X, is defined by H · X = H · M + H · A, where H · M is the stochastic integral in the sense of [33, I.4.40] and H · A is defined pathwise as the Lebesgue-Stieltjes integral R·

0Ht(ω)dAt(ω).

The quadratic covariation or square bracket process of two semimartingales X and Y is defined by

[X, Y ] = XY − X0Y0− X−· Y − Y−· X.

By [33, I.4.50.a and I.4.55.b], a local martingale M is purely discontinuous if and only if [M, N ] = 0 for all continuous local martingales N . In case [X, Y ] has locally integrable variation (i.e. there exists stopping times Tn↑ ∞ such that the variation

on the stopped interval [0, Tn] has finite expectation), it has a predictable

compen-sator called the predictable quadratic covariation or angle bracket process, which is denoted by hX, Y i and defined as the predictable process of locally bounded variation such that [X, Y ] − hX, Y i is a local martingale. We write hXi = hX, Xi. If M is a local L2_{-martingale, then hM i is the predictable compensator of M}2_{, i.e.}

M2_{− hM i is a local martingale.}

In the sequel we consider p-dimensional semimartingales. We call a p-dimensi-onal process X = (X1, . . . , Xp) a semimartingale in Rpif all its components Xiare

semimartingales in R. For a predictable locally bounded process H in Rp_{we define}

H ·X :=Pp

i=1Hi·Xi. The square (resp. angle) brackets between a semimartingale

X in Rp

and a semimartingale Y in Rq_{is defined as the matrix-valued process with}

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1.2. Itˆo’s formula and characteristics 11

In addition to stochastic integrals with respect to semimartingales, we also need stochastic integration with respect to random measures. Therefore, let us introduce the random measure associated to the jumps of X, which is denoted by µX _{and defined as µ}X_{([0, t] × B) =} P

s≤t1{∆Xs∈B\{0}}, for t ≥ 0, B ∈ B(R

p_).

There exists a predictable random measure νX_{, called the compensator of µ}X_,

with the property that W ∗ µX_{− W ∗ ν}X _{is a local martingale for all predictable}

W : Ω × R+× Rp → R where |W | ∗ µX is locally integrable. Here, the processes

W ∗ µX _{and W ∗ ν}X _{are defined pathwise as integrals}

W ∗ µX_t (ω) = Z Z [0,t]×Rp W (ω, s, z)µX(ω; ds, dz) (1.1) W ∗ ν_tX(ω) = Z Z [0,t]×Rp W (ω, s, z)νX(ω; ds, dz). (1.2)

Let Gloc(µX) denote the set of all predictable W : Ω × R+× Rp → R with the

property that (P

s≤tXe_s2)1/2 is locally integrable, where we write

e Xs= Z W (s, z)(µX− νX_{)({s} × dz) =}Z Z {s}×Rp W (r, z)µX(dr, dz) − Z Z {s}×Rp W (r, z)νX(dr, dz).

Then for W ∈ Gloc(µX) the stochastic integral W ∗ (µX− νX) is defined as the

unique purely discontinuous local martingale starting at 0 with jumps equal to eXt.

1.2 Itˆ

o’s formula and characteristics

The cornerstone of stochastic calculus is Itˆo’s formula, which takes the following form in the present setting.

Theorem 1.1 (Itˆo’s formula). If f ∈ C2

(Rp

) and X is a semimartingale in Rp_,

then f (X) is a semimartingale and

f (X) = f (X0) + ∇f (X−) · X +12tr Z · 0 ∇2_{f (X} t−)dhXcit + (f (Xt−+ z) − f (Xt−) − ∇f (Xt−)>z) ∗ µX,

with all terms well-defined.

Here, the Lebesgue-Stieltjes integral R₀·∇2_{f (X}

t−)dhXcitis defined as the

ma-trix M with components

Mij= p X k=1 Z · 0 ∂i∂kf (Xt−)dhXkc, Xjcit.

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Our aim is to state Itˆo’s formula in terms of the characteristics of X, which are defined as follows. Fix a truncation function χ : Rp _{→ R}p_{, i.e. χ is bounded}

and χ(z) = z in a neighborhood of 0 (for example χ(z) = z1_{|z|≤1}). For every semimartingale X in Rp _{there exists a triplet (B, C, ν}X_{) of characteristics relative}

to the truncation function χ. This triplet consists of a predictable process B in Rp of locally bounded variation, a continuous process C in S+p of locally bounded

finite variation, namely C = hXc_{i (i.e. the quadratic variation of the continuous}

martingale part of X), and a predictable random measure νX

on R+× Rp, namely

the compensator of the random measure µX _{associated to the jumps of X. It}

holds that χi ∈ Gloc(µX) for all i and the semimartingale X can be decomposed

according to its characteristics as

X = X0+ B + Xc+ χ ∗ (µX− νX) + (z − χ(z)) ∗ µX. (1.3)

Here, χ ∗ (µX_{− ν}X_{) is p-dimensional and should be read componentwise. In case}

X is a special semimartingale we have z ∈ Gloc(µX) and (after modifying B) X

admits the decomposition

X = X0+ B + Xc+ z ∗ (µX− νX),

in other words, in (1.3) one can replace χ(z) with z, see [33, II.2.38]. For special semimartingales we therefore often use the improper truncation function χ(z) = z. Throughout and henceforth we restrict ourselves to semimartingales where the characteristics (B, C, νX_{) are absolutely continuous with respect to the Lebesgue}

measure, in the sense that

Bt= Z t 0 bsds, Ct= Z t 0 csds, νX(ω; [0, t], A) = Z t 0 Kω,s(A)ds,

for some adapted processes b in Rp_{, c in S}p

+ and a predictable transition kernel

Kω,t(dz) from Ω × R+ into Rp\{0} satisfying

Z

(|z|2∧ 1)Kω,t(dz) < ∞, for all ω ∈ Ω, t ∈ R+. (1.4)

We call (b, c, K) the differential characteristics of X. Note that in this case (B, C, νX_{) is a “good” version of the characteristics in the sense of [33, II.2.9].}

We call X a time-homogeneous jump-diffusion if the differential characteristics take the form

bt(ω) = b(Xt(ω)),

ct(ω) = c(Xt(ω)),

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for some measurable functions b : Rp

→ Rp

, c : Rp _{→ S}p

+ and a transition kernel

K from Rp

into Rp_{\{0}. In that case, X can be written as the solution to an}

SDE with jumps, see [33, III.2.26]. For continuous diffusions, this comes down to the existence of a Brownian motion W on some probability space together with a process Y that solves the SDE

dYt= b(Yt)dt + c(Yt)1/2dWt,

such that the law of X is equal to the law of Y .

We are now able to state a version of Itˆo’s formula in terms of the differential characteristics of X.

Theorem 1.2. Suppose that X is a semimartingale with differential characteris-tics (b, c, K) and f (X) is a special semimartingale for some f ∈ C2

(Rp_{). Then it}

holds that f (X) = f (X0) + Mc+ Md+ A, with Mc= ∇f (X−) · Xc the continuous

martingale part of f (X), Md_{= W ∗ (µ}X_{− ν}X_{) a purely discontinuous local}

mar-tingale, where W (t, z) = f (Xt−+ z) − f (Xt−), and A a continuous process given

by At= Z t 0 ∇f (Xs)>bs+1₂tr (∇2f (Xs)cs) + Z (f (Xs+ z) − f (Xs) − ∇f (Xs)>χ(z))Ks(dz) ds.

Proof. By Itˆo’s formula and [33, II.1.30] we can write

f (X) − f (X0) =

Z ·

0

∇f (Xt)>bt+₂1tr (∇2f (Xt)c(Xt)) dt + V ∗ µX

+ ∇f (X−) · Xc+ ∇f (X−)>χ ∗ (µX− νX),

where V (t, z) = f (Xt−+ z) − f (Xt−) − ∇f (Xt−)>χ(z). The last two terms on the

right are local martingales, while the first term on the right has locally integrable variation, as it is continuous. Since f (X) is a special semimartingale, it follows from [33, I.4.23] that V ∗ µX _{has locally integrable variation, whence V ∈ G}

loc(µX)

and V ∗ µX = V ∗ (µX− νX_{) + V ∗ ν}X _{by [33, II.1.28]. This yields the assertion,}

as W = V + ∇f (X−)>χ.

A sufficient criterion for f (X) being a special semimartingale is stated in the next proposition.

Proposition 1.3. Let f ∈ C2

(Rp_{) and X be a semimartingale. If |f (z)|1}

{|z|>1}∗

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Proof. By Theorem 1.1, f (X) is a semimartingale. It is a special semimartingale if and only if Yt:= sups≤t|f (Xs) − f (X0)| is locally integrable in view of [33, I.4.23].

Let Sn ↑ ∞ be a sequence of stopping times such that E|f(z)|1{|z|>1}∗ νSXn< ∞.

Define

Tn= inf{t ≥ 0 : |Xt−| ≥ n or |Xt| ≥ n} ∧ Sn.

Then it holds that

EYTn≤ 2n + E|f(∆XTn)| ≤ 2n + sup |z|≤1 |f (z)| + E(|f(z)|1{|z|>1}∗ µXSn) = 2n + sup |z|≤1|f (z)| + E(|f(z)|1{|z|>1} ∗ νX Sn) < ∞,

which gives the result.

The following theorem gives an important characterization for X being a semi-martingale with differential characteristics (b, c, K). One direction is a consequence of Theorem 1.2. For a full proof we refer to [33, II.2.42].

Theorem 1.4. Let us be given predictable processes b in Rp_{, c in S}p

+ and a

pre-dictable transition kernel Kω,t(dz) from Ω × R+ into Rp satisfying (1.4). Then a

c`adl`ag adapted process X is a semimartingale and it admits the differential char-acteristics (b, c, K) if and only if

M_tf = f (Xt) − f (X0)− Z t 0 ∇f (Xs)>bs+1₂tr (∇2f (Xs)cs) + Z (f (Xs+ z) − f (Xs) − ∇f (Xs)>χ(z))Ks(dz) ds (1.5)

is a local martingale for all f ∈ C2 b(R

p_{), which in turn holds if and only if M}f _is

a local martingale for all f of the form f (x) = exp(u>_{x) with u ∈ iR}p_.

As a corollary we determine the differential characteristics of a semimartingale under stopping.

Proposition 1.5. Suppose X is a semimartingale with differential characteristics (b, c, K) and T is a stopping time. Then Y := XT is a semimartingale with differential characteristics (b1[0,T ], c1[0,T ], K(dz)1[0,T ]). Moreover, Yc = (Xc)T

and for W ∈ Gloc(µX) it holds that W 1[0,T ]∈ Gloc(µX), W ∈ Gloc(µY) and

(W ∗ (µX− νX₎₎T _{= W 1}

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Proof. The first assertion is an immediate consequence of Theorem 1.4. Suppose W ∈ Gloc(µX). Then obviously also W ∈ Gloc(µY), whence the stochastic integral

W ∗(µY_−νY_{) exists. It is the unique purely discontinuous local martingale starting}

at 0 with jumps equal to

W (t, ∆Yt)1{∆Yt6=0}= W (t, ∆Xt)1{∆Xt6=0}1[0,T ](t).

By [33, II.1.30.a], we have (W ∗ (µX− νX₎₎T _{= W 1}

[0,T ]∗ (µX− νX), so it is a

purely discontinuous local martingale with the same jumps as W ∗ (µY _{− ν}Y_{). It}

follows that (W ∗ (µX− νX₎₎T _{= W ∗ (µ}Y _{− ν}Y_{). This yields}

Y = Y0+ BT + (Xc)T + χ(z) ∗ (µY − νY) + (z − χ(z)) ∗ µY,

which implies that (Xc₎T _{= Y}c _{by the uniqueness of the continuous martingale}

part.

In the next proposition, inspired by [37, Problem 5.3.15], we establish finite second moments for a semimartingale under a growth condition. We write kZkt=

sup_s≤t|Zs|, where Z is a stochastic process.

Proposition 1.6. Let X be a special semimartingale with differential character-istics (b, c, K) relative to the (improper) truncation function χ(z) = z. Assume E|X0|2< ∞ and

|bt|2+ tr ct+

Z |z|2_K

t(dz) ≤ C(1 + kXk2t), for some C > 0, all t ≥ 0, P-a.s.

(1.6) Then for all t ≥ 0

EkXk2t ≤ 4(E|X0|2+ Ct(8 + t)) exp(4Ct(8 + t))

holds. In addition, Xc _{and z ∗ (µ}X_{− ν}X_{) are proper martingales.}

Proof. Define stopping times Tn= inf{t ≥ 0 : |Xt| ≥ n or |Xt−| ≥ n}. By

Propo-sition 1.5, Y := XTn _{is a special semimartingale with differential characteristics}

(b1[0,Tn], c1[0,Tn], K1[0,Tn]) relative to χ. It holds that

1 4kY k 2 t ≤ |X0|2+ k Z · 0 bs1[0,Tn](s)dsk 2 t+ kY c k2t+ kz ∗ (µ Y − νY)k2_t.

The Cauchy-Schwarz inequality gives

sup u≤t Z u 0 bs1[0,Tn](s)ds 2 ≤ t Z t 0 |bs|21[0,Tn](s)ds ≤ Ct Z t 0 (1 + kY k2s)ds,

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while Doob’s inequality [33, I.1.43] yields EkYck2t ≤ 4E|Y c t| 2 = 4Etr hYci∞= 4E Z t 0 tr cs1[0,Tn]ds ≤ 4C Z t 0 (1 + EkY k 2 s)ds and Ekz ∗ (µY − νY)k2t ≤ 4E|z ∗ (µ Y _{− ν}Y₎ t|2= 4Etr hz ∗ (µY − νY)it= 4E|z|2∗ νYt = E Z t 0 Z |z|21[0,Tn](s)Ks(dz)ds ≤ 4C Z t 0 (1 + EkY k2s)ds It follows that EkY k2t ≤ 4|X0|2+ 4Ct(8 + t) + 4C(8 + t) Z t 0 EkY k2sds. Since

EkXTnk2t ≤ E|X0|2+ n2+ E|∆Xt∧Tn|

2 ≤ E|X0|2+ n2+ E(|z|2∗ µXt∧Tn) = E|X0|2+ n2+ E(|z|2∗ νt∧TX n) ≤ E|X0|2+ n2+ CE Z t∧Tn 0 (1 + kXk2_s)ds < ∞,

the integral form of the Gronwall-Bellman inequality yields

EkXTnk2t ≤ 4(E|X0|2+ Ct(8 + t)) exp(4Ct(8 + t)).

Let n → ∞, then the left-hand side converges by the Monotone Convergence Theorem to EkXk2

t, which is bounded by the right-hand side. This yields the first

assertion of the lemma. The second assertion is an immediate consequence in view of [33, I.4.50 and II.1.33.a], since both

E[Xic]t= EhXicit= E Z t 0 cii,sds and Ehzi∗ (µX− νX)it= E(z2i ∗ ν X t ) = E Z t 0 Z z_i2Ks(dz)ds

are finite due to the growth-condition (1.6) and the derived moment inequality for kXk2

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1.3. Girsanov’s Theorem 17

1.3 Girsanov’s Theorem

We finish this chapter by discussing locally absolutely continuous measure changes. Recall that a probability measure Q is said to be locally absolutely continuous with respect to P with density process Z if Q|Ft is absolutely continuous with respect

to P|Ft for all t ≥ 0 and Z is the density process defined as

Zt= dQ|Ft

dP|Ft

,

which is a P-martingale. We consider the case that the density process is given by a stochastic exponential . The stochastic exponential Z = E (X) of a semimartingale X is defined as the unique solution to the stochastic differential equation Z = 1 + Z−· X, which exists and is equal to

E(X)t= exp(Xt−1₂hXcit)

Y

s≤t

(1 + ∆Xs) exp(−∆Xs).

In case ∆Xt> −1, the above expression can be simplified to

E(X)t= exp(Xt−1₂hXcit+ (log(1 + z) − z) ∗ µXt ). (1.7)

We need the following version of the “classical” Girsanov’s Theorem.

Theorem 1.7 (Girsanov). Suppose that Q is locally absolutely continuous with respect to P with density process Z. If M is a P-local martingale with bounded jumps, then

M − 1 Z−

· hM, Zi

is well-defined Q-a.s. and is a Q-local martingale. Consequently, if Z is of the form Z = E (Y ) for some P-local martingale Y , then M − hM, Y i is a Q-local martingale.

Proof. The first assertion follows from [33, III.3.11 and III.3.14]. For the second assertion, [33, III.3.14] yields that both [M, Z] and [M, Y ] have locally integrable variation. Therefore, their compensators hM, Zi respectively hM, Y i exist. It holds that [M, Z] = [M, Z−· Y ] = Z−· [M, Y ] by [33, I.4.54]. The compensator of the

latter process equals Z−· hM, Y i by [33, I.3.18]. Plugging in hM, Zi = Z−· hM, Y i

in the above display gives the result.

In the following, L2

loc(Xc) denotes the class of predictable processes H with

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stochastic integral H·Xc_{is defined as in [33, III.4.5]. In order to clarify under which}

measure the stochastic integral is taken, we also write H ·P_Xc _{and L}2

loc(Xc, P), as

well as W ∗P_(µX_{− ν}X_{) and G}

loc(µX, P). Likewise we write hXiP, [X]P, etc. We

are interested in how the differential characteristics of a semimartingale translate under a change of measure where the density process Z takes the typical form

Z = E (H · Xc+ W ∗ (µX− νX_)), _(1.8)

for some H ∈ L2 loc(X

c_{), W ∈ G}

loc(µX). For this, we have to calculate the angle

brackets, for which we use the following lemma.

Lemma 1.8. Let X be a semimartingale with differential characteristics (b, c, K). Suppose Y = H · Xc_{+ W ∗ (µ}X_{− ν}X_{), e}_{Y = e}_{H · X}c_{+ f}_{W ∗ (µ}X_{− ν}X_{) for some}

H, eH ∈ L2 loc(X

c_{) and some bounded W, f}_{W ∈ G}

loc(µX). Then it holds that

hY, eY it=

Z t

0

H_s>csHesds + W fW ∗ νtX.

Proof. By [33, I.4.52 and III.4.5.c] it holds that

[Y, eY ]t= hYc, eYcit+ X s≤t ∆Ys∆ eYs = hH · Xc, eH · Xcit+ X s≤t W (s, ∆Xs)fW (s, ∆Xs)1{∆Xs6=0} = Z t 0 H_s>csHesds + W fW ∗ µXt .

The predictable compensator hY, eY it exists by [33, III.3.14] and is equal to the

processRt

0H >

s csHesds + W fW ∗ νtX, since νX is the compensator of µX.

The next proposition describes the behavior of the differential characteristics under an absolutely continuous change of measure. The same proposition is also stated in [35, Proposition 4], but without proof. However, a proof can be found inside the proof of [6, Theorem 2.4]. For completeness we include it here.

Proposition 1.9. Let X be a semimartingale with differential characteristics (b, c, K). Suppose that Q is locally absolutely continuous with respect to P with density process

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for some H ∈ L2 loc(X

c

, P), W ∈ Gloc(µX, P) with W ≥ −1. Then X is a

semi-martingale under Q with differential characteristics (eb,ec, eK) given by

ebt= bt+ ctHt+ Z W (t, z)χ(z)Kt(dz) e ct= ct e Kt(dz) = (1 + W (t, z))Kt(dz). (1.9)

Proof. First we observe that P-a.s. (whence Q-a.s.) it holds that ebtis well-defined,

Rt

0ebsds exists and eK satisfies (1.4). Indeed, W ∈ Gloc(µ X

, P) implies that (W21_{{|W |≤1}}+ |W |1_{{|W |>1}}) ∗ ν_tX

is locally P-integrable by [33, II.1.33.c]. In particular it is finite for all t ≥ 0, P-a.s. Since also χ ∈ Gloc(µX, P), one can derive the desired properties with the aid of

Cauchy-Schwarz, which is left to the reader.

Now let Mf _{be given by (1.5) and similarly let f}_Mf _{be given by the same}

expression with b and K replaced by eb and eK. Note that fMf is related to Mf by

f M_tf = M_tf− Z t 0 ∇f (Xs)>csHs+ Z (f (Xs+ z) − f (Xs))W (s, z)Ks(dz) ds.

In view of Theorem 1.4, we have to show that fMf

is a Q-local martingale for all f ∈ C2

b(Rp). Since f (X) is bounded, it is a special semimartingale, whence by

Theorem 1.2 we can write

Mf = ∇f (X−) · Xc+ (f (Xt−+ z) − f (Xt−)) ∗ (µX− νX).

It holds that Mf

is a P-local martingale by Theorem 1.4. Moreover, |∆Mtf| =

|∆f (Xt)| ≤ kf k∞, so Theorem 1.7 yields that

Mf− hMf_{, H · X}c_{+ W ∗ (µ}X_{− ν}X_)i

is a Q-local martingale. From Lemma 1.8 it follows that the angle brackets in the above display are equal to

Z t 0 ∇f (Xs)>csHs+ Z (f (Xs+ z) − f (Xs))W (s, z)Ks(dz) ds,

which yields the result.

Proposition 1.10. Let X be a semimartingale with differential characteristics (b, c, K). Suppose H ∈ L2_loc(Xc_{, P) and W ∈ G}loc(µX, P) with W > −1 are such

that

E exp(12

Z ∞

0

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Then Z = E (H · Xc_{+ W ∗ (µ}X_{− ν}X_{)) is a uniformly integrable martingale.}

Proof. This follows from [39, Theorem IV.3].

We now consider locally equivalent measure changes. In Proposition 1.12 below we show that if the density process Z is a strictly positive stochastic exponential of the form (1.8), then Z−1 is also a stochastic exponential of the form (1.8), and we determine its explicit expression. This result extends a result in [37], which is limited to continuous semimartingales. To handle the jumps, we need the following lemma.

Lemma 1.11. Consider the situation of Proposition 1.9 and suppose Z > 0. If V ∈ Gloc(µX, P) and V is bounded, then V ∈ Gloc(µX, Q) and

V ∗Q_(µX₋

e

νX) = V ∗P_(µX_{− ν}X_{) − V W ∗ ν}X_, _(1.10)

where the stochastic integration on the left- resp. right-hand side is taken under Q resp. P, andνe

X_{(dt, dz) = e}_K

t(dz)dt denotes the compensator of µX under Q.

Proof. Since V ∈ Gloc(µX, P) and V is bounded, it holds that V2∗ νX is locally

integrable, by [33, II.1.33.c]. This yields that V2_∗

e

νX _{= V}2_{(W + 1) ∗ ν}X _{is locally}

integrable. Indeed, we have

V2|W | ∗ νX _{≤ V}2_{∗ ν}X_{+ kV k}2

∞|W |1{|W |>1}∗ νX.

The second term on the right is locally integrable by [33, II.1.33.c], since W ∈ Gloc(µX, P). To show (1.10), it suffices to show that the right-hand side is a

purely discontinuous Q-local martingale, as it has the same jumps as the left-hand side. It is a Q-local martingale by Theorem 1.7 and Lemma 1.8. To prove that Y := V ∗P_(µX_{− ν}X_{) − V W ∗ ν}X _{is purely discontinuous, we have to show that}

[Y, N ]_Q _{= 0 for all continuous Q-local martingales N . By [33, III.3.13], N is a} continuous semimartingale under P and the square brackets are the same under P and Q. Under P, the continuous martingale part Yc of Y is 0, as V ∗P_(µX_{− ν}X₎

is purely discontinuous and V W ∗ νX _{is of locally bounded variation. Since in}

addition ∆Nt= 0, it follows from [33, I.4.52] that

[Y, N ]_P = hYc, N i_P = 0,

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Proposition 1.12. Consider the situation of Proposition 1.9. Suppose W > −1. Then it holds that −H ∈ L2

loc( eXc, Q) and −W/(W + 1) ∈ Gloc(µX, Q). Moreover,

Z > 0 and Z−1= E (−H ·Q e Xc− W/(W + 1) ∗Q_(µX₋ e νX)), (1.11) where eXc

denotes the continuous martingale part of X under Q andeν

X_{(dt, dz) =}

e

Kt(dz)dt the compensator of µX under Q.

Proof. Write V = −W/(W + 1). Then V > −1 and one verifies that

(1 −√1 + V )2∗ν_eX= (1 −√1 + W )2∗ νX_.

Note that the latter process is continuous (as νX(dt, dz) = Kt(dz)dt). Therefore,

it is locally integrable under Q if and only if it is locally integrable under P, so W ∈ Gloc(µX, P) if and only if V ∈ Gloc(µX, Q) by [33, II.1.33.d]. Likewise we

have −H ∈ L2

loc( eXc, P). That Z > 0 holds, follows from [33, I.4.61.c], since we

have that ∆(W ∗ (µX− νX₎₎

t= W (t, ∆Xt)1{∆Xt6=0} > −1.

It remains to show (1.11). By using (1.7) we infer that the right-hand side equals exp(−H ·Q e Xc− W/(W + 1) ∗Q_(µX₋ e νX) −1 2 Z · 0 H_t>ctHtdt + (W − log(1 + W )) ∗ µX− W2/(W + 1) ∗ µX),

and the left-hand side equals

exp(−H ·P_Xc_{− W ∗}P_(µX_{− ν}X_{) +}1 2

Z ·

0

Ht>ctHtdt + (W − log(1 + W )) ∗ µX).

Therefore, it suffices to show

H ·Q e Xc= H ·P_Xc₋ Z · 0 H_t>ctHtdt, (1.12) W/(W + 1) ∗Q_(µX₋ e νX) = W ∗P_(µX_{− ν}X_{) − W}2_{/(W + 1) ∗ µ}X_. _(1.13)

We start with the latter. It holds that |W |1{|W |>1}∗ νX is locally integrable by

[33, II.1.33.c], whence [33, II.1.28] enables us to split the integrals to obtain

W 1{|W |>1}∗P(µX− νX) − W2/(W + 1)1{|W |>1}∗ µX

= W/(W + 1)1{|W |>1}∗ µX− W/(W + 1)1{|W |>1}∗eν

X

= W/(W + 1)1{|W |>1}∗Q(µX−eν

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We now show that W 1_{{|W |≤1}}∗P_(µX_{− ν}X_{) −} W 2 W + 11{|W |≤1}∗ µ X₌ W W + 11{|W |≤1}∗ Q_(µX₋ e νX). Applying Lemma 1.11 with V = W 1{|W |≤1} yields

W 1{|W |≤1}∗Q(µX−eν X_{) = W 1} {|W |≤1}∗P(µX− νX) − W2 W + 11{|W |≤1}∗eν X_. (1.14) Since W21{|W |≤1}∗ νX is locally integrable by [33, II.1.33.c], it follows from [33,

II.1.28] that W2_{/(W + 1)1} {|W |≤1}∈ Gloc(µX, Q) and W2 W + 11{|W |≤1}∗ Q_(µX₋ e νX) = W 2 W + 11{|W |≤1}∗ µ X₋ W 2 W + 11{|W |≤1}∗eν X_.

Subtracting this from (1.14) gives the result. Thus we have proved (1.12). To show (1.13) we note that Theorem 1.7 and Lemma 1.8 yield that

Xc− hXc_{, M i} P = X c₋ Z · 0 ctHtdt

is a continuous Q-local martingale. Moreover, from Lemma 1.11 it follows that χ ∗Q_(µX₋

e

νX) = χ ∗P_(µX_{− ν}X_{) − W χ ∗ ν}X_,

whence X can be decomposed as

X = X0+ Z · 0 e btdt + Xc− Z · 0 ctHtdt + χ ∗Q(µX−eν X_{) + (z − χ(z)) ∗ µ}X_.

By the uniqueness of the decomposition of a local martingale in its continuous and purely discontinuous part, it follows that eXc _{= X}c₋R·

0ctHtdt. Now by [33,

III.4.5.b], H ·Q

e

Xc _{is characterized as the unique continuous Q-local martingale,} null at 0, such that

hH ·Q

e

Xc, Y i_Q= H · h eXc, Y i_Q,

for all local L2_{(P)-martingales Y . From [33, I.4.52 and III.3.13] one infers that} h eXc, Y i_Q= [ eXc, Y ]_Q= [ eXc, Y ]_P= [Xc, Y ]_P. Likewise we have hH ·P_Xc − Z · 0 H_s>csHsds, Y iQ= [H ·PX c − Z · 0 H_s>csHsds, Y ]Q = [H ·P_Xc₋ Z · 0 H_s>csHsds, Y ]P = [H ·P_Xc_{, Y ]} P = H · [Xc, Y ]_P.

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Theorem 1.7 and Lemma 1.8 yield that

H ·P_Xc t−

Z t

0

H_s>csHsds

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Chapter

2

The martingale problem

This chapter concerns the theory of the martingale problem. Originally developed by Stroock and Varadhan as presented in their classical work [51], this theory char-acterizes jump-diffusions as solutions of a martingale problem, which has become a fundamental tool for the analysis of stochastic processes. For a full overview we refer to [21].

In this chapter we consider jump-diffusions that assume their values in a certain state space E ⊂ Rpand characterize them as solutions of the aforementioned mar-tingale problem. We include the possibility of explosion and killing by a potential, wherefore we follow the framework in [6] (with a few modifications). We revisit the latter paper and deduce an important criterium in Section 2.2 for a solution of the martingale problem, see Proposition 2.5. In Section 2.3 we apply this proposition to derive the Markov property for general jump-diffusions, under well-posedness. Although these results are not new, to the best of our knowledge they are not stated in the literature in the particular form as presented here. We emphasize that Proposition 2.7 is crucial in proving existence and uniqueness for affine pro-cesses in Chapter 4. In addition, we give in this section a well-known stochastic representation of a solution to Kolmogorov’s backward equation, extending known results to the general jump-diffusion case.

In Section 2.4 we discuss the positive maximum principle. A consequence of [21, Theorem 4.5.4] is that existence of a solution of the martingale problem is equivalent with the positive maximum principle. Using a technique involving a change of measure, we derive necessary and sufficient conditions for the positive maximum principle in Proposition 2.13, which we use extensively for our analysis of

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affine processes in later chapters when we introduce the admissibility conditions. In Section 2.5 we build further on the main result of [6] and deduce sufficient conditions for the martingale property of a stochastic exponential. This will be needed in Chapter 6 to establish the validity of the affine transform formula.

2.1 Set-up and notation

Let E ⊂ Rp _{be the closure of an unbounded open set and E}

∆ = E ∪ {∆} the

one-point compactification of E, where ∆ 6∈ Rp _{corresponds with “the point at}

infinity”. Every measurable function f on E is extended to E∆by setting f (∆) =

0, except the norm-function, for which we take |∆| = ∞. Note that the derivatives of f ∈ C2_{(E) are well-defined on E, as they are determined by the values of f on}

E◦_{, by the assumption that E = E}◦_{. Throughout this chapter, Ω denotes a subset}

of DE∆[0, ∞), the space of c`adl`ag functions ω : [0, ∞) → E∆. Unless mentioned

otherwise, Ω is equipped with the σ-algebra FX_{= σ(X}

s: s ≥ 0) and the filtration

(FX

t ) with FtX= σ(Xs: 0 ≤ s ≤ t), generated by the coordinate process X given

by Xt(ω) = ω(t).

Let us be given measurable functions b : E → Rp_{, c : E → S}p

+, γ : E → R+

and K a transition kernel from E to Rp\{0} with supp K(x, dz) ⊂ E − x for all x ∈ E. Assume that

b(·), c(·), γ(·) and Z

(|z|2∧ 1)dK(·, dz) are bounded on compacta of E, (2.1)

and let χ : Rp_{→ R}p _{denote a truncation function. We are given a linear operator}

A : C2_{(E) → M (E) by} Af (x) = ∇f (x)>b(x) +1₂tr (∇2f (x)c(x)) − γ(x)f (x) + Z (f (x + z) − f (x) − ∇f (x)>χ(z))K(x, dz), (2.2)

and we assume A(C2

c(E)) ⊂ B(E), so that

Rt

0Af (Xs)ds is well-defined pathwise

for f ∈ C2

c(E). The example below demonstrates that the assumption A(Cc2(E)) ⊂

B(E) is not redundant.

Example 2.1. Let E = R, take b = c = γ = 0 and define the transition kernel K by K(x, dz) = x4

µ(dz), with µ a measure with support N and given by µ({k}) = 1/k2 _{for k ∈ N. Then} Z (|z|2∧ 1)K(x, dz) = x4 ∞ X k=1 1/k2,

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2.1. Set-up and notation 27

whence x 7→R (|z|2_{∧ 1)K(x, dz) is bounded on compacta. Take f ∈ C}2

c(R) such

that 1{|x|≤1}≤ f (x) ≤ 1{|x|≤M }for some M > 1. Then for |x| > M with 1 − x ∈ N

it holds that Af (x) = Z f (x + z)K(x, dz) ≥ x4 Z 1{z=1−x}µ(dz) = x4/(1 − x)2,

which can be made arbitrarily large. Hence A(C2

c(E)) 6⊂ B(E).

Definition 2.2. A probability measure P on (Ω, FX_{) is called a solution of the}

martingale problem for A in Ω if

M_tf= f (Xt) − f (X0) −

Z t 0

Af (Xs)ds (2.3)

is a P-martingale with respect to (FX

t ) for all f ∈ Cc2(E). If in addition λ is a

probability measure on E∆ such that P ◦ X0−1 = λ, then we say P is a solution

of the martingale problem for (A, λ) and we often write P = Pλ. If λ = δx, the

Dirac-measure at x for some x ∈ E∆, then we write Px instead. Likewise, Eλ

denotes the expectation with respect to Pλ and Ex the expectation with respect

to Px. We call the martingale problem for A well-posed in Ω if for all probability

measures λ on E∆there exists a unique solution Pλon (Ω, FX) of the martingale

problem for (A, λ).

There is a close relation between the martingale problem and jump-diffusions. For Ω = DE[0, ∞) and γ = 0 this is easy to see. Indeed, if X is a semimartingale on

(DE[0, ∞), (Ft+X), P) with differential characteristics (b(X), c(X), K(X, dz)), then

P is a solution of the martingale problem for A in DE[0, ∞) (with γ = 0), in view

of (1.4). For general Ω ⊂ DE∆[0, ∞) and γ, we need to extend the definition of

jump-diffusions by allowing the possibility of explosion and killing by a potential. Let X be a process in DE∆[0, ∞) with absorbing cemetery state ∆, i.e. if Xt−= ∆

or Xt= ∆, then Xs= ∆ for all s ≥ t. Define the stopping time

Sn = inf{t ≥ 0 : |Xt−| ≥ n or |Xt| ≥ n}.

Then X is an exploding jump-diffusion if for all n ∈ N it holds that XSn _{is a}

semimartingale with differential characteristics

(b(Xt)1[0,Sn](t), c(Xt)1[0,Sn](t), K(Xt, dz)1[0,Sn](t)),

and if in addition we have Texpl:= limn→∞Sn < ∞ with positive probability as

well as limt↑Texpl|Xt| = ∞ for Texpl< ∞. Let e be an independent exponentially

distributed random variable with rate 1 and define the stopping time

Tkill= inf{t ≥ 0 :

Z t

0

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If X is a (possibly exploding) jump-diffusion, then

Xt1[0,Tkill)(t) + ∆1[Tkill,∞)(t)

denotes a killed jump-diffusion, with killing rate γ. Note that if Tkill < Texpl, then

a transition from E to ∆ occurs by a jump, while if Tkill≥ Texpl, then a transition

to ∆ occurs by an explosion. In the latter case the killing is redundant.

With this extended definition in mind, one can show that if P is the distribution of a jump-diffusion, then P solves the martingale problem for A in DE∆[0, ∞). The

next section is devoted to the converse of this assertion.

2.2 Turning X into a semimartingale

As explained in the preceding section, we relate in this section the solution of a martingale problem with the law of a semimartingale. We follow [6], but slightly adapt their proofs in order to obtain a useful characterization of a solution of the martingale problem. As in [6] we define a couple of (FX

t )-stopping times. First,

we define the (FX

t )-stopping time

T∆= inf{t ≥ 0 : Xt−= ∆ or Xt= ∆}, (2.4)

which can be regarded as the lifetime of X in case ∆ is absorbing. Being a first contact time, T∆is indeed an (FtX)-stopping time, in view of [21, Proposition 2.1.5].

To handle an explosion of X we introduce the stopping times

Tn0 = inf{t ≥ 0 : |Xt−| ≥ n or |Xt| ≥ n}.

A transition to ∆ can occur by either a jump (when the process is killed by a potential) or an explosion. Accordingly, we define

Tjump=    T∆, if Tn0 = T∆for some n, ∞ if T_n0 < T∆for all n, (2.5) Texpl=    T∆, if Tn< T∆for all n, ∞ if Tn0 = T∆for some n, (2.6) Tn=    Tn0, if Tn0 < T∆, ∞ if T_n0 = T∆. (2.7)

Note that T∆ = Texpl and Tjump = ∞ in the case that a transition from E to ∆

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2.2. Turning X into a semimartingale 29

occurs by a jump (i.e. by killing). Note also that Tn↑ Texpland that Tjumpdiffers

from Tkill (as defined in the preceding paragraph), since Tjump assumes the value

∞ when an explosion occurs first.

Now we assume that ∆ is an absorbing cemetery state. Therefore, we take Ω to be equal to

Ω = {ω ∈ DE∆[0, ∞) : if ω(t−) = ∆ or ω(t) = ∆, then ω(s) = ∆ for s ≥ t}.

(2.8)

We modify X such that it becomes a c`adl`_{ag process in R}p_{. Following [6], we}

identify a jump to ∆ with a jump to some point ∂ ∈ Rp\E. The existence of such a point can be guaranteed after enlarging the dimension when necessary, see [6, Section 3]. Without loss of generality we may also assume that dist(∂, E) > 0 (where dist denotes the distance) and that χ(∂ − x) = 0 for all x ∈ E. We put

b

Xt= Xt1[0,Tjump)(t) + ∂1[Tjump,∞)(t). (2.9)

Then bX cannot jump to ∆, but an explosion to ∆ is still possible. Stopping bX by Tn we obtain a c`adl`ag process in Rp.

Next we show that bXTn _{is a semimartingale on (Ω, (F}X

t+), P) and determine its

characteristics, in case P solves the martingale problem for A in Ω. In order to do so, we define a second linear operator bA : C2

(Rp) → M (E) by b Af (x) = ∇f (x)>b(x) +1₂tr (∇2f (x)c(x)) + Z (f (x + z) − f (x) − ∇f (x)>χ(z))K∂(x, dz), where K∂_{(x, dz) = K(x, dz) + γ(x)δ}

∂−x(dz) (we restrict the domain of bAf to E,

as b(x), c(x), γ(x) and K(x, dz) are not defined for x 6∈ E∆). Note that

b

Af (x) = Af (x) + f (∂)γ(x). (2.10)

The assumptions in (2.1) yield that the processes

Mf,Tn t = f (X Tn t ) − f (X Tn 0 ) − Z t∧Tn 0 Af (Xt)ds and Mc f,Tn t = f ( bX Tn t ) − f ( bX Tn 0 ) − Z t∧Tn 0 b Af (Xt)ds

are well-defined for f ∈ C2

(Rp_{) and bounded for f ∈ C}2 b(R

p_{). In addition, we have}

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for t ≥ 0, wherefore it follows from (2.10) that c Mf,Tn t = M f,Tn t + f (∂)N Tn t , (2.11) with NTn t := 1{0<Tjump≤t∧Tn}− Z t∧Tn 0 γ(Xs)ds.

Recalling the convention b(∆) = c(∆) = γ(∆) = K(∆, ·) = 0, we can also write

b Af (Xt) = ∇f ( bXt)>b(Xt) +1₂tr (∇2f ( bXt)c(Xt)) + Z (f ( bXt+ z) − f ( bXt) − ∇f ( bXt)>χ(z))K∂(Xt, dz), (2.12)

for t ≥ 0. Now Theorem 1.4 translates as follows.

Proposition 2.3. Let Ω be given by (2.8) and let P be a measure on (Ω, F). It holds that bXTn_{is a semimartingale on (Ω, (F}X

t+), P) with differential characteristics

(b(Xt)1[0,Tn](t), c(Xt)1[0,Tn](t), K

∂_(X

t, dz)1[0,Tn](t)), (2.13)

if and only if cMf,Tn _{is an (F}X

t+, P)-martingale for all f ∈ C 2 b(R

p_{), which in turn}

holds if and only if cMfu,Tn _{is an (F}X

t+, P)-martingale for all u ∈ iRp.

A direct consequence is the following proposition.

Proposition 2.4. Let P be a probability measure on (Ω, FX_{) with Ω given by}

(2.8). Suppose Mfu,Tn _{is an (F}X

t , P)-martingale for all u ∈ iRp, some n ∈ N.

Then it holds that NTn _{is an (F}X

t+, P)-martingale and bXTnis a semimartingale on

(Ω, (FX

t+)t≥0, P) with differential characteristics (2.13).

Proof. Note that if Mfu,Tn _{is an (F}X

t , P)-martingale, then it is also an (Ft+X,

P)-martingale, as it is right-continuous (see [15, Theor`eme VI.1.3]). For u = 0 we have fu(x) = 1 for x ∈ E and fu(∆) = 0, whence

Mf0,Tn _{= 1} {t∧Tn<Tjump}− 1{Tjump>0}+ Z t∧Tn 0 γ(Xs)ds = −NtTn. Thus NTn

t is an (Ft+X, P)-martingale. It follows from (2.11) that cM fu,Tn

t is an

(FX

t+, P)-martingale for all u ∈ iRp. Thus the result follows from Proposition 2.3.

In the following we need a countable collection C ⊂ C2

c(E), defined as follows.

Affine Markov processes on a general state space - Thesis

UvA-DARE (Digital Academic Repository)

Affine Markov processes on a general state space

Veerman, E.

Publication date

2011

Document Version

Final published version

Link to publication

Citation for published version (APA):

Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.

Affine Markov Processes

on a General State Space

Enno Veerman

Affine Mark

ov P

rocesses on a General State Space

Enno V

eerman

Uitnodiging

Affine Markov Processes

on a General State Space

Affine Markov Processes on

a General State Space

ACADEMISCH PROEFSCHRIFT

Voorwoord

Contents

Notation

Introduction

Mathematical overview

Chapter

1

Semimartingales

1.1

Definitions

1.2

Itˆ

o’s formula and characteristics

1.3

Girsanov’s Theorem

Chapter

2

The martingale problem

2.1

Set-up and notation

2.2

Turning X into a semimartingale