AN INTRODUCTION TO LEVY´ PROCESSES ERIK BAURDOUX AND ANTONIS PAPAPANTOLEON Abstract. Lecture notes from courses at TU Berlin, LSE and CIMAT Contents 1. Introduction 2 2. Definition of L´evy processes 2 3. Toy example: a L´evyjump-diffusion 3 4. Infinitely Divisible distributions 6 5. The L´evy{Khintchine representation 11 6. The L´evy{It^odecomposition 19 7. The L´evymeasure and path properties 32 8. Elementary operations 38 9. Moments and Martingales 41 10. Popular examples 46 11. Simulation of L´evyprocesses 52 12. Stochastic integration 53 References 60 Date: February 6, 2015. 1 2 ERIK BAURDOUX AND ANTONIS PAPAPANTOLEON 1. Introduction These lecture notes aim at providing a (mostly) self-contained introduc- tion to L´evyprocesses. We start by defining L´evyprocesses in Section 2 and study the simple but very important example of a L´evyjump-diffusion in Section 3. In Section 4 we discuss infinitely divisible distributions, as the distribution of a L´evyprocess at any fixed time has this property. In the sub- sequent section we prove the L´evy{Khintchine formula, which characterises all infinitely divisible distributions. We then prove the L´evy{It^odecomposi- tion of a L´evyprocess, which is an explicit existence result explaining how to construct a L´evyprocess based on a given infinitely divisible distribution. In Section 7, we make use of the L´evy{It^odecomposition to derive necesarry and sufficient conditions for a L´evyprocess to be of (in)finite variation and a subordinator, respectively. In Section 8 we study elementary operations such as linear transformations, projections and subordination. We then move on to moments and martingales followed by a brief section on simulation fol- lowed by a number of popular models used mostly in mathematical finance. Then there is a section on simulation and we finish with a brief treatment of stochastic integration. 2. Definition of Levy´ processes 2.1. Notation and auxiliary definitions. Let (Ω; F; F; IP) denote a sto- chastic basis, or filtered probability space, i.e. a probability space (Ω; F; IP) endowed with a filtration F = (Ft)t≥0. A filtration is an increasing family of sub-σ-algebras of F, i.e. Fs ⊂ Ft W for s ≤ t. By convention F1 = F and F∞− = s≥0 Fs. A stochastic basis satisfies the usual conditions if it is right-continuous, T i.e. Ft = Ft+, where Ft+ = s>t Fs, and is complete, i.e. the σ-algebra F is IP-complete and every Ft contains all IP-null sets of F. Definition 2.1. A stochastic process X = (Xt)t≥0 has independent incre- ments if, for any n ≥ 1 and 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn, the random variables Xt0 ;Xt1 − Xt0 ;:::;Xtn − Xtn−1 are independent. Alternatively, we say that X has independent increments if, for any 0 ≤ s < t, Xt − Xs is independent of Fs. Definition 2.2. A stochastic process X = (Xt)t≥0 has stationary incre- ments if, for any s; t ≥ 0, the distribution of Xt+s − Xs does not depend on s. Alternatively, we say that X has stationary increments if, for any 0 ≤ s ≤ t, Xt − Xs is equal in distribution to Xt−s. Definition 2.3. A stochastic process X = (Xt)t≥0 is stochastically contin- uous if, for every t ≥ 0 and > 0 lim IP(jXs − Xtj > ) = 0: s!t 2.2. Definition of L´evyprocesses. We will now define L´evyprocesses and then present some well-known examples, like the Brownian motion and the Poisson process. d Definition 2.4 (L´evyprocess). An adapted, R -valued stochastic process X = (Xt)t≥0 with X0 = 0 a.s. is called a L´evyprocess if: AN INTRODUCTION TO LEVY´ PROCESSES 3 (L1) X has independent increments, (L2) X has stationary increments, (L3) X is stochastically continuous. In the sequel, we will always assume that X has c`adl`agpaths. The next two results provide the justification. Lemma 2.5. If X is a L´evyprocess and Y is a modification of X (i.e. IP(Xt 6= Yt) = 0 a.s. for each t ≥ 0), then Y is a L´evyprocess and has the same characteristics as X. Proof. [App09, Lemma 1.4.8]. Theorem 2.6. Every L´evyprocess has a unique c`adl`agmodification that is itself a L´evyprocess. Proof. [App09, Theorem 2.1.8] or [Pro04, Theorem I.30]. 2.3. Examples. The following are some well-known examples of L´evypro- cesses: • The linear drift is the simplest L´evyprocess, a deterministic process; see Figure 5.3 for a sample path. • The Brownian motion is the only non-deterministic L´evyprocess with continuous sample paths; see Figure 5.3 for a sample path. • The Poisson, the compound Poisson and the compensated (com- pound) Poisson processes are also examples of L´evyprocesses; see Figure 5.3 for a sample path of a compound Poisson process. The sum of a linear drift, a Brownian motion and a (compound or com- pensated) Poisson process is again a L´evyprocess. It is often called a \jump- diffusion” process. We shall call it a L´evyjump-diffusion process, since there exist jump-diffusion processes which are not L´evyprocesses. See Figure 5.3 for a sample path of a L´evyjump-diffusion process. 3. Toy example: a Levy´ jump-diffusion Let us study the L´evyjump-diffusion process more closely; it is the sim- plest L´evyprocess we have encountered so far that contains both a diffusive part and a jump part. We will calculate the characteristic function of the L´evyjump-diffusion, since it offers significant insight into the structure of the characteristic function of general L´evyprocesses. Assume that the process X = (Xt)t≥0 is a L´evyjump-diffusion, i.e. a linear deterministic process, plus a Brownian motion, plus a compensated compound Poisson process. The paths of this process are described by N Xt Xt = bt + σWt + Jk − tλβ ; (3.1) k=1 where b 2 R, σ 2 R>0, W = (Wt)t≥0 is a standard Brownian motion, N = (Nt)t≥0 is a Poisson process with intensity λ 2 R>0 (i.e. IE[Nt] = λt), and J = (Jk)k≥1 is an i.i.d. sequence of random variables with probability distribution F and IE[Jk] = β < 1. Here F describes the distribution of the jumps, which arrive according to the Poisson process N. All sources of randomness are assumed mutually independent. 4 ERIK BAURDOUX AND ANTONIS PAPAPANTOLEON 0.2 0.05 0.04 0.1 0.03 0.0 0.02 0.01 −0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.8 0.3 0.6 0.4 0.2 0.2 0.1 0.0 −0.2 0.0 −0.4 −0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2.1. Sample paths of a linear drift processs (top- left), a Brownian motion (top-right), a compound Poisson process (bottom-left) and a L´evyjump-diffusion. The characteristic function of Xt, taking into account that all sources of randomness are independent, is h Nt i iuXt X IE e = IE exp iu bt + σWt + Jk − tλβ k=1 Nt h i h X i = exp iubt IE exp iuσWt IE exp iu Jk − iutλβ ; k=1 recalling that the characteristic functions of the normal and the compound Poisson distributions are 1 2 2 iuσWt − σ u t IE[e ] = e 2 ;Wt ∼ N (0; t) PNt iuJ iu Jk λt(IE[e k −1]) IE[e k=1 ] = e ;Nt ∼ Poi(λt) (cf. Example 4.14 and Exercise 1), we get h 1 i h i = exp iubt exp − u2σ2t exp λtIE[eiuJk − 1] − iuIE[J ] 2 k h 1 i h i = exp iubt exp − u2σ2t exp λtIE[eiuJk − 1 − iuJ ] ; 2 k AN INTRODUCTION TO LEVY´ PROCESSES 5 and since the distribution of Jk is F we have h 1 i h Z i = exp iubt exp − u2σ2t exp λt eiux − 1 − iuxF (dx) : 2 R Finally, since t is a common factor, we can rewrite the above equation as u2σ2 Z IEeiuXt = exp t iub − + (eiux − 1 − iux)λF (dx) : (3.2) 2 R We can make the following observations based on the structure of the char- acteristic function of the random variable Xt from the L´evyjump-diffusion: (O1) time and space factorize; (O2) the drift, the diffusion and the jump parts are separated; (O3) the jump part decomposes to λ × F , where λ is the expected number of jumps and F is the distribution of jump size. One would naturally ask if these observations are true for any L´evyprocess. The answer for (O1) and (O2) is yes, because L´evy processes have stationary and independent increments. The answer for (O3) is no, because there exist L´evyprocesses with infinitely many jumps (on any compact time interval), thus their expected number of jumps is also infinite. Since the characteristic function of a random variable determines its dis- tribution, (3.2) provides a characterization of the distribution of the random variables Xt from the L´evyjump-diffusion X. We will soon see that this dis- tribution belongs to the class of infinitely divisible distributions and that equation (3.2) is a special case of the celebrated L´evy-Khintchineformula. 3.1. The basic connections. The next sections will be devoted to estab- lishing the connection between the following mathematical objects: • L´evyprocesses X = (Xt)t≥0 • infinitely divisible distributions ρ = L(X1) • L´evytriplets (b; c; ν).