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Article: Applebaum, D. (2004) Levy processes - from probability theory to finance and quantum groups. Notices of the American Mathematical Society, 51 (11). pp. 1336-1347. ISSN 0002-9920
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[email protected] https://eprints.whiterose.ac.uk/ Lévy Processes—From Probability to Finance and Quantum Groups David Applebaum
he theory of stochastic processes was be embedded into a suitable noncommutative struc- one of the most important mathematical ture. developments of the twentieth century. Stochastic processes are not only mathematically Intuitively, it aims to model the interac- rich objects. They also have an extensive range of Ttion of “chance” with “time”. The tools applications in, e.g., physics, engineering, ecology, with which this is made precise were provided by and economics—indeed, it is difficult to conceive the great Russian mathematician A. N. Kolmogorov of a quantitative discipline in which they do not fea- in the 1930s. He realized that probability can be ture. There is a limited amount that can be said rigorously founded on measure theory, and then about the general concept, and much of both the- a stochastic process is a family of random variables ory and applications focusses on the properties of (X(t),t 0) defined on a probability space (Ω, ,P) specific classes of process that possess additional and taking≥ values in a measurable space (E,F ). structure. Many of these, such as random walks and Here Ω is a set (the sample space of possible out-E Markov chains, will be well known to readers. Oth- comes), is a σ-algebra of subsets of Ω (the events), ers, such as semimartingales and measure-valued and P is Fa positive measure of total mass 1 on (Ω, ) diffusions, are more esoteric. In this article, I will (the probability). E is sometimes called the stateF space. Each X(t) is a ( , ) measurable mapping give an introduction to a class of stochastic from Ω to E and shouldF beE thought of as a random processes called Lévy processes, in honor of the observation made on E at time t. For many devel- great French probabilist Paul Lévy, who first stud- opments, both theoretical and applied, E is Eu- ied them in the 1930s. Their basic structure was clidean space Rd (often with d 1); however, there understood during the “heroic age” of probability is also considerable interest in= the case where E is in the 1930s and 1940s and much of this was due an infinite dimensional Hilbert or Banach space, or to Paul Lévy himself, the Russian mathematician a finite-dimensional Lie group or manifold. In all A. N. Khintchine, and to K. Itô in Japan. During the of these cases can be taken to be the Borel σ- past ten years, there has been a great revival of in- E algebra generated by the open sets. To model prob- terest in these processes, due to new theoretical de- abilities arising within quantum theory, the scheme velopments and also a wealth of novel applica- described above is insufficiently general and must tions—particularly to option pricing in mathematical finance. As well as a vast number of David Applebaum is professor of probability and statistics at the University of Sheffield. His email address is research papers, a number of books on the subject [email protected]. He is the author of have been published ([3], [11], [1], [2], [12]) and Lévy Processes and Stochastic Calculus, Cambridge Uni- there have been annual international conferences versity Press, 2004, on which part of this article is based. devoted to these processes since 1998. Before we Work carried out at The Nottingham Trent University. begin the main part of the article, it is worth
1336 NOTICES OF THE AMS VOLUME 51, NUMBER 11 listing some of the reasons why Lévy processes are characteristic function of X(t) is the mapping d so important: φt : R C defined by → • There are many important examples, such as iu X(t) iu y φt (u) E(e · ) e · pt (dy), Brownian motion, the Poisson process, stable = = Rd processes, and subordinators. where p is the law (or distribution) of X(t), i.e., • They are generalizations of random walks to t p P X(t) 1 , and E denotes expectation. φ is continuous time. t − t continuous= ◦ and positive definite; indeed, a famous • They are the simplest class of processes whose theorem of Bochner asserts that all continuous paths consist of continuous motion interspersed positive definite mappings from Rd to C are Fourier with jump discontinuities of random size ap- transforms of finite measures on Rd. pearing at random times. It follows from the axiom (L1) that each X(t) is • Their structure contains many features, within infinitely divisible, i.e., for each n N, there exists a relatively simple context, that generalize nat- a probability measure p on Rd with∈ characteris- urally to much wider classes of processes, such t,n tic function φ such that φ (u) (φ (u))n, for as semimartingales, Feller-Markov processes, t,n t t,n each u Rd. The characteristic functions= of infi- processes associated to Dirichlet forms, and nitely divisible∈ probability measures were com- (generalizing the strictly stable Lévy processes) pletely characterized by Lévy and Khintchine in the self-similar processes. 1930s. Their result, which we now state, is funda- • They are a natural model of noise that can be mental for all that follows: used to build stochastic integrals and to drive stochastic differential equations. Theorem 0.1 [The Lévy-Khintchine Formula]. If • Their structure is mathematically robust and X (X(t),t 0) is a Lévy process, then = tη(u≥) d generalizes from Euclidean space to Banach and φt (u) e , for each t 0,u R , where = ≥ ∈ Hilbert spaces, Lie groups, and symmetric 1 spaces, and algebraically to quantum groups. (0.1) η(u) ib u u au = · − 2 · + iu y The Structure of Lévy Processes [e · 1 iu y1 y <1(y)]ν(dy), d Rd 0 − − · || || We will take E R throughout the first part of this −{ } article. = for some b Rd, a non-negative definite symmet- Definition. A Lévy process X (X(t),t 0) is a ∈ = ≥ ric d d matrix a and a Borel measure ν on stochastic process satisfying the following: d × 2 R 0 for which d ( y 1)ν(dy) < . (L1) X has independent and stationary incre- R 0 Conversely,−{ } given a mapping−{ } || of|| the∧ form (0.1) ∞we ments, 1 can always construct a Lévy process for which (L2) Each X(0) 0 (with probability one ), tη(u) = φt (u) e . (L3) X is stochastically continuous, i.e., for all = a>0 and for all s 0, limt s P( X(t) X(s) >a) One of our goals is to give a probabilistic inter- ≥ → | − | =0. pretation to the Lévy-Khintchine formula. The map- Of these three axioms, (L1) is the most important, ping η : Rd C is called the characteristic exponent and we begin by explaining what it means. of X. It is → conditionally positive definite in that n N It focusses on the increments X(t) X(s); i,j 1 ci cj η(ui uj ) 0 , for all n ,c1,..., { − n 0 s t< . Stationarity of these means that c =C with − c ≥0.A theorem due to Schoen- ∈ ≤ ≤ ∞} n i 1 i P(X(t) X(s) A) P(X(t s) X(0) A) for all berg∈ asserts that= all= continuous, hermitian (i.e., − ∈ = − − ∈ Borel sets A, i.e., the distribution of X(t) X(s) is in- η(u) η( u), for all u Rd), conditionally positive − variant under shifts (s,t) (s h, t h). Indepen- maps= from− Rd to C that∈ satisfy η(0) 0 must take → + + dence means that given any finite ordered sequence the form (0.1). The triple (b, a, ν) is called= the char- of times 0 t1 t2 tn < , the random vari- acteristics of X. It determines the law p . The mea- ≤ ≤ ≤···≤ ∞ t ables X(t1) X(0),X(t2) X(t1),... ,X(tn) X(tn 1) sures ν that can appear in (0.1) are called Lévy − − − − are (statistically) independent. We emphasize again measures. that (L1) is the key defining axiom for Lévy processes; We begin the task of interpreting (0.1) by ex- indeed, for many years they were known as “processes amining some examples. The first two that we con- with stationary and independent increments”. Of the sider are very well known in probability theory— other axioms, (L2) is a convenient normalization and indeed, each has an extensive theoretical (L3) is a technical (but important) assumption that development in its own right with many applica- enables us to do serious analysis. tions. The Lévy-Khintchine Formula Examples of Lévy Processes To understand the structure of a generic Lévy 1. Brownian Motion and Gaussian Processes process, we employ Fourier analysis. The We define a Brownian motion Ba (Ba(t),t 0) = ≥ 1To be denoted w.p.1, henceforth. to be a Lévy process with characteristics (0,a,0).
DECEMBER 2004 NOTICES OF THE AMS 1337 1.2
0.8 Brownian motion with drift is the Lévy process Ca,b (Ca,b(t),t 0), with characteristics (b, a, 0). = ≥ 0.4 Each Ca,b(t) is a Gaussian random variable having mean vector tb and covariance matrix ta. In fact
0 each Ca,b(t) bt Ba(t). A Lévy process has con- = + tinuous sample paths (w.p.1), or is Gaussian if and −0.4 only if it is a Brownian motion with drift. 2. The Poisson Process −0.8 A Poisson process Nλ (Nλ(t),t 0) with in- = ≥ −1.2 tensity λ>0 is a Lévy process with characteristics (0, 0,λδ1), where δ1 is a Dirac mass concentrated −1.6 at 1. Nλ takes non-negative integer values, and we have the Poisson distribution: − 2.0 e λt (λt)n 0 1 2 3 4 5 − P(Nλ(t) n) . = = n!
Figure 1. Simulation of standard Brownian motion. The path is The paths of Nλ are piecewise constant on each continuous but nowhere differentiable. If you were to zoom finite interval, with jumps of size 1 at the random in, the fractal nature of the path would become apparent and times τn inf t 0,Nλ(t) n . this reflects the self-similarity of the process. = { ≥ = } 3. The Compound Poisson Process
E i Let (Yn,n N) be a sequence of independent It has mean zero and covariance (Ba(s) ∈ j ij i identically distributed random variables with com- Ba(t)) a (s t) (where Ba(s) is the ith compo- = ∧ mon law q and let N be an independent Poisson nent of the vector Ba(s)). If a is positive definite, λ process. The compound Poisson process is the then each Ba(t) has a normal distribution with den- Lévy process Z (t) Nλ(t) Y . It has characteristic sity fa,t where λ j 1 j = iu= y exponent η(u) Rd (e · 1)λq(dy) . The com- 1 1 1 = − fa,t (x) d exp (x a− x) . pound Poisson process (with d 1) can be used to = (2πt) 2 det(a) −2t · = model the takings at a till in a supermarket, where
Nλ(t) is the number of customers in the queue at When d 1, we write B1 B and call it a stan- = = dard Brownian motion. Brownian motion has a fas- time t and Yj is the amount paid by the jth cus- cinating history. It is named after the botanist tomer. Robert Brown, who first observed, in the 1820s, the 4. Interlacing Processes irregular motion of pollen grains immersed in We can define a Lévy process by the prescrip-
water. By the end of the nineteenth century, the phe- tion X(t) Ca,b(t) Zλ(t), provided the two sum- = + nomenon was understood by means of kinetic the- mands are assumed to be independent. We call ory as a result of molecular bombardment. Indeed, this an interlacing process since its paths have the in 1905, Einstein, although ignorant of the dis- form of continuous motion interlaced with ran- covery of the phenomenon and of previous work dom jumps of size Yn occurring at the random on it, predicted its existence from purely theoret- || || times τn (where the Yns are as in Example 3 above). ical considerations. Five years earlier L. Bachelier X has characteristic exponent had employed it to model the stock market, where the analogue of molecular bombardment is the in- 1 iu y (0.2) η(u) ib u u au (e · 1)λq(dy), terplay of the myriad of individual market decisions = · − 2 · + Rd − that determine the market price. Standard Brownian motion was rigorously con- which is quite close to the general form (0.1). In- structed by N. Wiener in the 1920s as a family of deed (0.2) was proposed as the form of the most general η by the Italian mathematician B. de Finetti functionals on the space 0([0, ), R) of real- valued continuous functionsC=C on [0, ∞) that vanish in the 1920s. His error was in failing to appreciate at zero. In so doing, he equipped∞ the infinite- that the finite measure λq can be replaced by a σ- iu y dimensional space with a Gaussian measure that finite Lévy measure ν. But if we do this, (e · 1) − is now called WienerC measure in his honour. It fol- may not be ν-integrable and hence we must adjust lows that the paths t Ba(t)(ω), where ω , are the integrand. Probabilistically, this corresponds to continuous. In the 1930s→ Wiener, together∈C with a lack of convergence of a countable number of R. Paley and A. Zygmund, showed that the paths “small jumps”, as we will see in the next section. are nowhere differentiable (w.p.1). Although (0.2) is incorrect, the most general η can Figure 1 presents a simulation of the paths of be obtained as a pointwise limit of terms of simi- standard Brownian motion. lar type, i.e., η(u) limn ηn(u), where each = →∞
1338 NOTICES OF THE AMS VOLUME 51, NUMBER 11 80
40 ηn(u) i b yν(dy) u = − 1 < y <1 · n || || 0 1 iu y u au (e · 1)λq(dy), −2 · + y 1 − −40 || ||≥ n and the integrals must be combined together be- − fore the passage to the limit. In the next section we 80 will see the intuition behind this. −120 From the above examples, the reader may be for- given for thinking that a Lévy process is nothing −160 but the interplay of Gaussian and Poisson measures. In a sense this is correct; however, note that the −200 Gaussian and Poisson measures give rise to ex- treme points of the convex cone of all character- −240 istic exponents. As the following shows, there are 0 1 2 3 4 5 some interesting inhabitants of the interior. 5. Stable Lévy Processes Figure 2. Simulation of the Cauchy process. The Cauchy Stable probability distributions arise as the pos- process is stable with α =1. Jump discontinuities are sible weak limits of normalized sums of i.i.d. (i.e., represented by vertical lines. This process is also self-similar independent, identically distributed) random vari- so the path has a fractal nature. α ables in the central limit theorem. The normal dis- tails”, i.e. P(X>y) behaves asymptotically like y − tribution is stable and corresponds to the case in as y , as opposed to the exponential decay →∞ which each of the i.i.d. random variables has finite found in the Gaussian case. Such behavior has mean and variance. Stable random variables are been found in models of telecommunications traf- those whose laws are stable. They are characterized fic on the Internet. by the property that if X and X are independent 1 2 6. Relativistic Processes copies of a stable random variable X, then for d 1905 was a busy year for Albert Einstein. As well each c1,c2 > 0, there exists c>0 and d R such ∈ as his work on Brownian motion, mentioned above, that cX d has the same law as c1X1 c2X2. A Lévy process+ is stable if each X(t) is +stable in he also gave a quantum mechanical explanation of this sense. The characteristics of a stable Lévy the photoelectric effect (for which he won his Nobel process are either of the form (b, a, 0) (so it is a Prize) and developed the special theory of relativ- Brownian motion with drift) or (b, 0,ν), where ity. According to the latter, a particle of rest mass C m moving with momentum p has kinetic energy ν(dx) dx, with 0 <α<2 and C>0. α is 2 x α d E(p) m2c4 c2 p 2 mc , where c is the ve- = + = + | | − | | locity of light. If we define η(p) E(p) , then η is called the index of stability. With the sole exception =− of the Brownian motions with drift, the random the characteristic exponent of a Lévy process. We variables of a stable Lévy process all have infinite will explore some consequences of this below. variance, and if α 1, they also have infinite mean. 7. Subordinators ≤ One example of interest (in the case d 1) for A subordinator is a one-dimensional Lévy = which α 1 is the Cauchy process, which has process (T (t),t 0) that is nondecreasing (w.p.1). = ≥ t In this case, the Fourier transform that defines the the density ft (x) . Figure 2 presents a = π(x2 t2) characteristic function can be analytically contin- + simulation of its paths in which jump discontinu- ued to yield the Laplace transform uT(t) tψ(u) ities are represented by vertical lines. E(e− ) e− , for each u>0, where With a little calculus, the characteristic exponent = uy can be transformed to a more useful form. This ψ(u) η(iu) bu (1 e− )λ(dy). =− = + (0, ) − is particularly simple when X is rotationally ∞ invariant, i.e., P(X(t) OA) P(X(t) A) , for all Here b 0 and λ is a Lévy measure that satisfies ≥ O O(d),t 0, and Borel∈ sets= A. We∈ then obtain the additional constraints λ( , 0) 0 and ∈ α ≥ α −∞ = η(u) σ u , where σ>0. Rotationally invari- (0, )(y 1)λ(dy) < . =− | | ∞ ∧ ∞ ant stable processes are an important class of ψ is called the Laplace exponent of the subor- self-similar processes, i.e., (X(ct),t 0) and dinator. The set of all of these is in one-to-one cor- 1 ≥ (c α X(t),t 0) have the same finite dimensional respondence with the set of Bernstein functions for ≥ distributions (for each c>0), and this is one which limx 0 f (x) 0, where we recall that an → = reason why such processes are important in infinitely differentiable function f on (0, ) is a ∞ applications. Another reason, applying to general Bernstein function if and only if f 0 and ≥ stable random variables X, is that they have “heavy ( 1)nf (n) 0, for all n N. − ≤ ∈
DECEMBER 2004 NOTICES OF THE AMS 1339 5
the structure of the sample paths of Lévy processes. 4 Given a characteristic exponent, we can always as- sociate to it a Lévy process whose paths are right continuous with left limits (w.p.1). It follows that this process X can only have jump discontinuities, 3 and there are, at most, a countable number of these on each closed interval. We formally write ∆ X(t) Xc (t) 0 s t X(s) , where Xc has continu- = + ≤ ≤ 2 ous paths (w.p.1) and ∆X(s) X(s) X(s ) is the = − − “jump” at time s where X(s ) limu s X(u) is the left limit. − = ↑ 1 We can describe Xc quite easily. It is a Brownian motion with drift, Xc (t) bt Ba(t) (although this is by no means easy to prove).= + The second term is more problematic—in particular, the sum may not 0 converge. It turns out to be helpful to count the 0 1 2 3 4 5 jumps up to time t that are in a given Borel set A and to introduce Figure 3. Simulation of the gamma subordinator. In contrast N(t,A) # 0 s t; ∆X(s) A . to the cases shown by the previous two figures, the sample = { ≤ ≤ ∈ } paths of subordinators are considerably more regular. The N is a very interesting object—it is in fact a func- path is a non-decreasing step function with jump tion of three variables—time t, the set A, and the discontinuities again shown as vertical lines. sample point ω. If we fix t and ω, we get a d Examples of subordinators include the α-stable σ-finite measure on the Borel sets of R . On the ones (0 <α<1) that have Laplace exponent other hand, if we fix the set A and ensure that it ψ(u) uα. For the case α 1, each T (t) is the first is bounded away from zero, we get a Poisson = = 2 process with intensity λ ν(A). For these reasons hitting time of a standard Brownian motion to a = level, i.e., T (t) inf s>0; B(s) t . Furthermore, N is called a Poisson random measure. = { = √2 } each T (t) has a Lévy distribution with density In any finite time, X can have only a finite num- 2 t 3 t ber of jumps of size greater than 1 (or indeed ft (s) s− 2 e− 4s . Another well-known example = 2√π greater than any ǫ>0). We can write this finite sum of a subordinator, where each T (t) has a gamma of jumps as xN(t,dx). Similarly, the sum of distribution, is depicted in Figure 3. x >1 all the jumps|| of|| size greater than 1 but less than An important application of subordinators is n 1 is 1 xN(t,dx); however, the limit may not to the time change of Lévy processes. If X is a Lévy n < x <1 converge|| ||as n . Paul Lévy argued that the ac- process with characteristic exponent ηX and T →∞ is an independent subordinator with Laplace cumulation of a large number of very small jumps exponent λ, then Y (t) X(T (t)) is a new Lévy may be difficult to distinguish from bursts of de- = terministic motion, so one should consider process with characteristic exponent ηY = Mn(t) 1 x(N(t,dx) tν(dx)). (Mn,n N) λ ηX. This procedure was first investigated by n < x <1 − ◦− = || || − ∈ S. Bochner in the 1950s and is sometimes called is a sequence of square-integrable, mean zero mar- “subordination in the sense of Bochner” in his tingales and hence is a very pleasant object from honor. In particular, if X is a Brownian motion both a probabilistic and an analytic viewpoint. In (with a a multiple of the identity) and T is an particular the sequence converges in mean square to a martingale M(t) xN˜(t,dx) , where independent α-stable subordinator, then Y is a 0< x <1 N˜(t,dx) N(t,dx) tν=(dx) ||is ||called a compensated 2α-stable rotationally invariant Lévy process. Poisson random= measure− . Lévy’s intuition was made 8. The Riemann-Zeta Process precise by K. Itô, and we can now give the celebrated Readers who are interested in number theory Lévy-Itô decomposition for the sample paths of a may find this example of interest. If ζ is the usual Lévy process: Riemann zeta function, we obtain a Lévy process
for each u>1 by the following prescription for the (0.3) X(t) bt Ba(t) xN˜(t,dx) = + + x <1 + characteristic exponent, | | ζ(u iv) ηu(v) log + . xN(t,dx). x 1 = ζ(u) | |≥ This was established by Khintchine in the 1930s. Readers should beware of generalizing from the The Lévy-Itô Decomposition Gaussian to this more general case. For example, With the insight we obtained from Example 4, we bt is not in general the mean of X(t)—indeed, as can now return to the task of trying to understand we saw in Example 5, this may not exist. The
1340 NOTICES OF THE AMS VOLUME 51, NUMBER 11 “martingale part” of X(t) , i.e., the process the point of view of the Lévy-Itô decomposition ˜ M(t) Ba(t), has moments to all orders, so if X(t) (0.3), where the small jumps term x <1 xN(t,dx) de- + | | itself fails to have an nth moment this is entirely scribes the day-to-day jitter that causes minor fluc- due to the influence of “large jumps”. tuations in stock prices, while the big jumps term
x 1 xN(t,dx) describes large stock price move- Applications to Finance | |≥ ments caused by major market upsets arising from, A sociologist investigating the behavior of the prob- e.g., earthquakes or terrorist atrocities. ability community during the early 1990s would If we set aside Brownian motion, there are a surely report an interesting phenomenon. Many of plethora of Lévy processes to choose from, and our the best minds of this (or any other) generation choice must enable us to derive a pricing formula began concentrating their research in the area of that market analysts can compute with. One in- mathematical finance. The main reason for this teresting group of candidates is the (symmetric) hy- can be summed up in two words—option pricing. perbolic Lévy processes, whose financial applications Essentially, an option is a contract that confers have been extensively developed by E. Eberlein and upon the holder the right, but not the obligation, his group in Freiburg, Germany. These are processes to purchase (or sell) a unit of a certain stock for a with no Brownian motion part in (0.3), and the fixed price k on (or perhaps before) a fixed expiry characteristic function is given by date T, after which the option becomes worthless. t 2 2 2 For the option to make sense, k should be consid- ζ K1( ζ δ u ) φ (u) + , erably less than the current price of the stock. If t 2 2 2 = K1(ζ) ζ δ u the stock price rises above k, the holder of the op- + tion may make a considerable profit; on the other where K1 is a Bessel function of the third kind, and hand, if the stock price falls dramatically, losses ζ and δ are non-negative parameters. will be considerably less through buying options Hyperbolic Lévy processes were discovered by than by purchasing the stock itself. O. Barndorff-Nielsen the 1970s and used as mod- The key question is—does the market deter- els for the distribution of particle size in wind- mine a unique price for a given option, and if so, blown sand deposits. N. H. Bingham and R. Keisel can this price be explicitly computed? Much of the make an interesting analogy between the dynam- current interest in the subject derives from Nobel- ics of sand production and stock prices in that prize winning work of F. Black, M. Scholes and R. just as large rocks are broken down to smaller and Merton in the 1970s who gave a positive answer to smaller particles “this ‘energy cascade effect’ might this question. Underlying their analysis was a model be paralleled in the ‘information cascade effect’, of stock prices that improved upon that of Bache- whereby price-sensitive information originates in, lier by using geometric Brownian motion; i.e., the say, a global newsflash and trickles down through price S(t) of a given stock at time t is national and local level to smaller and smaller units 1 of the economic and social environment.” S(t) S(0) exp µ σ 2 t σB(t) . A problem with non-Gaussian option pricing is = − 2 + that the market is “incomplete”, i.e., there may be The constant µ R is the (logarithmic) expected more than one possible pricing formula. This is rate of return, while∈ σ>0, called the volatility, is clearly undesirable, and a number of selection prin- a measure of the excitability of the market. We will ciples, such as entropy minimization, have been em- have more to say about volatility below. Black and ployed to overcome this problem. For hyperbolic Scholes obtained an exact formula for the unique processes, a pricing formula has been developed price of a European option (i.e., one that can only that has minimum entropy and that is claimed to be exercised at time T) using the normal distribu- be an improvement on the Black-Scholes formula. tion. The derivation of this formula involves the use Another problem with the Black-Scholes- of tools such as martingales and Girsanov trans- Merton formula is the constancy of the volatility. forms, and it is this link with stochastic analysis Empirical studies suggest that this should vary to that so excited the probabilistic community. give a curve called the “volatility smile”. This has Although very elegant, the Black-Scholes-Merton prompted some authors to propose “stochastic model has limitations and possible defects that volatility models” wherein σ is replaced in the have led many probabilists to query it. Indeed, em- standard Black-Scholes model by a random process pirical studies of stock prices have found evidence that solves a stochastic differential equation. There of heavy tails, which is incompatible with a Gauss- are a number of different approaches to this; e.g., ian model, and this suggests that it might be fruit- O. Barndorff-Nielsen and N. Shephard have recently ful to replace Brownian motion with a more gen- proposed that (σ (t)2,t 0) should be an Ornstein- ≥ eral Lévy process. Indeed, H. Geman, D. Madan and Uhlenbeck process driven by a subordinator M. Yor have argued that this is quite natural from (T (t),t 0), i.e., ≥
DECEMBER 2004 NOTICES OF THE AMS 1341 t so that T is a pseudodifferential operator with 2 λt 2 λ(t s) t σ (t) e− σ (0) e− − dT(λs), symbol etη. Formal differentiation can be justified, = + 0 and we find that where λ>0. As T has finite variation (w.p.1), the d i(u,x) integral is well defined in the random Lebesgue- (Af )(x) (2π)− 2 e η(u)fˆ(u)du, Stieltjes sense. = Rd Readers who want to learn more about “Lévy fi- so A is also a pseudodifferential operator, with sym- nance” should consult [12], [4] , chapter 5 of [1], bol η. Using the Lévy-Khintchine formula (0.1) and and references therein. elementary properties of the Fourier transform, we obtain the following explicit form for the action Markov Processes, Semigroups, and of the generator on S(Rd ) Pseudodifferential Operators Lévy processes are, in particular, Markov processes, (0.4) i.e., their past and future are independent, given d 1 d (Af )(x) b ∂ f (x) a ∂ ∂ f (x) the present. This is formulated precisely using i i 2 ij i j = i 1 + i,j 1 = = the conditional expectation: E(f (X(t u)) t ) = + |F d E(f (X(t u)) X(t)) ,for all t,u 0 and all f (x y) f (x) yi ∂i f (x)1 y <1(y) ν(dy). Rd || || +d | ≥ + 0 + − − i 1 f Bb(R )—the Banach space, under the supre- −{ } = ∈ mum norm, of all bounded Borel measurable Using more sophisticated methods the domain in d functions on R . Here “the past” t is the smallest F (0.4) can be extended to a larger space of twice dif- sub-σ-algebra of with respect to which all Rd F ferentiable functions in C0( ). Here are some spe- X(s)(0 s t) are measurable. We define a two- ≤ ≤ cific examples of interesting generators: parameter family of linear contractions 1. Brownian motion (with a I) is generated by (T ;0 s t< ) on B (Rd ) by the prescription s,t b (one-half times) the = Laplacian, i.e., ≤ ≤E ∞ (Ts,tf )(x) (f (X(t)) X(s) x) Rd f (x y)pt (dy) . 1 d 2 1 ∆ = | = = + A 2 i 1 ∂i 2 . Then the Markov property implies that these form = = = an evolution, i.e., Tr,sTs,t Tr,t, for all r s t. 2. Rotationally invariant α-stable processes (with = ≤ ≤ Note that these operators all commute with the nat- σ 1) are generated by fractional powers of = α Rd Rd the Laplacian: A ( ∆) 2 . ural action of the translation group of on Bb( ). =−− Lévy processes form a nice subclass of Markov 3. For the relativistic process, we have processes. First, they are time-homogeneous, A (√m2c4 c2∆ mc2). =− − − i.e., Ts,t T0,t s for all s t. If we now write Tt T0,t, = − ≤ = In the last example, A is called a relativistic the evolution property becomes the semigroup Schrödinger operator in quantum− theory. Note that law Ts Tt Ts t. Second, Lévy processes are Feller = + A is obtained from its symbol through the corre- processes, i.e., each Tt preserves the Banach space spondence p i , which is precisely the usual Rd Rd ↔−∇ C0( ) of continuous functions on that vanish rule for quantization, although this is more natu- at infinity and lim t 0 Tt f f 0 , for all rally carried out in a Hilbert space setting (see d ↓ || − || = f C0(R ). Hence (Tt ,t 0) is a strongly continu- below). ∈ ≥ ous, one-parameter contraction semigroup on If AZ is the generator of the Lévy d C0(R ), and by the general theory of such process Z(t) X(T (t)) obtained from a Lévy = semigroups, we can assert the existence of the process X with characteristic exponent ηX, associ- X T (t)f f ated semigroup (Tt ,t 0), and generator AX using generator Af limt 0 − , for all f DA. The ≥ = ↓ t ∈ an independent subordinator T with Laplace ex- d domain DA is a linear space that is dense in C0(R ) ponent ψ, then the identity ηZ ψ ηX , quan- =− ◦− and A is a closed linear operator. We can explicitly tizes nicely to yield AZ ψ( AX ) . In particular, =− − compute the semigroup and its generator as pseu- we can use the α-stable subordinators to define dodifferential operators. For convenience, we work fractional powers of AX using the following beau- − in Schwartz space S(Rd )—the space of all smooth tiful formula functions on Rd that are such that they and all their α ds ( A )αf (T X f f ) . derivatives decay to zero at infinity faster than any X Γ s 1 α − − = (1 α) (0, ) − s + d d − ∞ negative power of x . S(R ) is dense in C0(R ) and | | A deep generalization due to R. S. Phillips allows is a natural domain for the Fourier transform X ˆ d i(u,x) the replacement of AX and Tt with the generator f (u) (2π)− 2 Rd e− f (x)dx . Fourier inversion = d ˆ i(u,x) of a general contraction semigroup on a Banach then yields f (x) (2π)− 2 Rd f (u)e du. Applying = space. theorem 0.1, we compute The semigroup associated with each Lévy d p d 2 i(u,x) tη(u) ˆ process also operates in each L (R )(1 p< ) (Tt f )(x) (2π)− e e f (u)du, ≤ ∞ = Rd and is again strongly continuous and contractive.
1342 NOTICES OF THE AMS VOLUME 51, NUMBER 11 Since S(Rd ) is dense in each Lp(Rd ), the pseudo- dodifferential operators should be aware that the differential operator representations discussed map x η(x, u) does not, in general, have nice → above still hold here. From now on, we take p 2. smoothness properties. = The generator corresponding to the symbol η Recurrence, Transience, and Bound States d has maximal domain η(R )—the nonisotropic From an intuitive point of view a stochastic process H Sobolev space of all f L2(Rd ) for which is recurrent at a point x if it visits any arbitrarily 2 ˆ 2 ∈ Rd η(u) f (u) du < . small neighborhood of that point an infinite num- | | | | ∞ Standard semigroup theory tells us that a nec- ber of times (w.p.1), and it is transient if each such essary and sufficient condition for each Tt to be self- neighborhood is only visited finitely many times adjoint is that A is positive, self-adjoint. A nec- (w.p.1). More precisely, a Lévy process is recurrent − essary and sufficient condition for this is that the (at the origin) if lim inft X(t) 0 (w.p.1) and →∞ | |= associated Lévy process is symmetric, i.e., transient (at the origin) if limt X(t) (w.p.1). P(X(t) A) P(X(t) A), and this holds if and The recurrence/transience →∞ dichotomy| |=∞ holds in only if∈ = ∈− that every Lévy process is either recurrent or 1 transient. In the 1960s, S. C. Port and C. J. Stone η(u) u au (cos(u y) 1)ν(dy). proved that a Lévy process is recurrent if and only =−2 · + Rd 0 · − 1 −{ } if u 0. It follows || || ℜ − =∞ This yields a probabilistic proof of self-adjoint- that Brownian motion is recurrent for d 1, 2 and d = ness (on η(R )) of each of the three operators dis- that for d 1 every α-stable process is recurrent H if 1 α<=2 and transient if 0 <α<1. For d 3, cussed above. ≤ ≥ Let A be the self-adjoint generator of a sym- every Lévy process is transient. Rd In the 1990s, R. Carmona, W. C. Masters, and metric Lévy process and for each f,g Cc∞( ), define (f,g)
DECEMBER 2004 NOTICES OF THE AMS 1343 1 n n probability measures on G, so that in particular where b (b ,...,b ) R ,a (aij) , is a non- 1 = ∈ = ps t (A) G pt (τ− A)ps (dτ) . negative definite, symmetric n n real-valued ma- + = × There are three cases of interest—locally com- trix and ν is a Lévy measure on G e . −{ } pact abelian groups (LCA groups), Lie groups, and general locally compact groups. The LCA case was Conversely, any linear operator with a representa- extensively studied during the 1960s. The fact that tion as in (0.5) is the restriction to C2(G) of the in- the dual group Gˆ of characters is itself an LCA finitesimal generator of some Lévy process. group allows a natural generalization of the Fourier The characteristics (b, a, ν) of a Lévy process de- d transform from R to G, and a Lévy-Khintchine for- termine its law, just as in the Euclidean case. mula that characterizes Lévy processes can hence In the 1990s, H. Kunita and the author were be developed similarly to the Euclidean case. We able to generalize the Lévy-Itô decomposition to the will not dwell further on this topic here; interested extent that for each f C2(G), the real-valued readers are directed to section 5.6 in [6]. process f (X) (f (X(t),t∈ 0) can be described = ≥ The case in which G is a Lie group has been ex- (using stochastic integrals in the sense of K. Itô) in tensively studied. For non-abelian G, there is no di- terms of a Brownian motion on Rd and a Poisson rect analogue of the Fourier transform available, random measure on R+ (G e ). We now give × −{ } and one of the joys of the subject is the challenge some examples of Lévy processes on a Lie group of surmounting this obstacle using tools from G: semigroup theory, stochastic analysis, group rep- 1. Brownian motion in G. resentations, and noncommutative harmonic analy- sis. The first important step in this direction was This is a Lévy process that has characteristics (0,I,0). It has continuous sample paths (w.p.1), taken by G. A. Hunt in 1956. He effectively char- and its generator is (up to the usual factor of one- acterized Lévy processes in Lie groups by gener- ∆ d 2 d half) a left-invariant Laplacian on G, G i 1 Yj . alizing the formula (0.4) for the generator in R . To = The basis dependence is a nuisance here.= It can be be precise, let X (X(t),t 0) be a Lévy process on = ≥ dispensed with by equipping G with a left-invari- a d-dimensional Lie group G and let pt be the law ant Riemannian metric m, with respect to which of each X(t). We obtain a one-parameter, strongly Y1,...,Yd is orthonormal. ∆G is then the Laplace- continuous, contraction semigroup (Tt ,t 0) with { } ≥ Beltrami operator associated to (G, m) and the cor- generator A on C0(G) by the prescription responding Brownian motion is a geometrically in- trinsic object—indeed, it has played a central role E (Tt f )(τ) (f (τX(t))) f (τσ)pt (dσ). in recent years within the development of analy- = = G sis in path and loop spaces. Note that Tt commutes with left translations. Now 2. The Compound Poisson Process let Y1,...,Yd be a fixed basis for the Lie algebra { } g of left-invariant vector fields on G. Define a Let (γn,n N) be a sequence of i.i.d. random ∈ linear manifold C2(G) that is dense in C0(G) by the variables taking values in G with common law µ prescription C2(G) f C0(G); Yi f C0(G) and and let (N(t),t 0) be an independent Poisson ={ ∈ ∈ ≥ Yi Yj f C0(G) for all 1 i,j n . Hunt showed process with intensity λ>0. We define the com- ∈ ≤ ≤ } that there exist functions xi C2(G), 1 i n pound Poisson process in G by Y (t) γ1γ2 ...γN(t). ∈ ≤ ≤ = so that (x1,...,xn) is a system of canonical In this case the generator is bounded and is given by ( f )(τ) (f (τσ) f (τ))ν(dσ), for each coordinates for G at e. A Lévy measure ν is A = G − f C0(G) where the Lévy measure ν( ) λµ( ) is a Borel measure on G e for which ∈ · = · d 2 −{ } finite. G e i 1 xi (σ ) 1 ν(dσ) < . Hunt was −{ } = ∧ ∞ then able to obtain the following key result: 3. Stable Processes Theorem 0.2 [Hunt’s Theorem]. If X is a Lévy The theory of stable processes in Lie groups process in G with infinitesimal generator , then was developed by H. Kunita in the 1990s. His ap- A proach was to generalize the self-similarity prop- 1. C2(G) Dom( ). erty, and for this he needed a notion of scaling. This ⊆ A 2. For each τ G, f C2(G), is provided by a dilation, i.e., a family of automor- ∈ ∈ phisms δ (δ(r),r >0) for which δ(r)δ(s) δ(rs) (0.5) = = d d for all r,s > 0, which also possess suitable conti- 1 ( f )(τ) bi Yi f (τ) aijYi Yj f (τ) nuity properties. A Lévy process X in G is stable A = + 2 i 1 i,j 1 with respect to the dilation δ if δ(r)X(s) has the = = d same law as X(rs) for each r,s > 0. Dilations (and f (τσ) f (τ) xi (σ )Yi f (τ) ν(dσ), hence stable Lévy processes) can exist only on sim- + G e − − −{ } i1 ply connected nilpotent groups. Stable processes =
1344 NOTICES OF THE AMS VOLUME 51, NUMBER 11 in such groups have some surprising properties, gated by A. Bendikov and L. Saloff-Coste at Cornell. e.g., Kunita has shown that there is no dilation It will be interesting to see if the new techniques with respect to which Brownian motion in the they’ve developed can be applied to more general Heisenberg group is stable. It is however possible classes of Lévy processes. to construct a stable process on this group whose first two components are Brownian motion whereas Lévy Processes in Quantum Groups the third is a Cauchy process. Through the work of physicists such as N. Bohr, M. Born, and W. Heisenberg and its mathematical 4. Subordinated Processes formulation by J. von Neumann, we came to a dual Let Y (Y (t),t 0) be a Lévy process on G and understanding of quantum mechanics. On the one T (T (t)=,t 0) be≥ a subordinator that is inde- hand, physical observables such as position, mo- pendent= of ≥Y. Just as in the Euclidean case, we can mentum, energy, and spin should be described as construct a new Lévy process Z (Z(t),t 0) by (not necessarily bounded) self-adjoint linear oper- the prescription Z(t) Y (T (t)), for= each t ≥ 0. ators acting in a complex Hilbert space. On the Lévy processes in Lie= groups is a subject≥ that is other hand, these observables are also random currently undergoing intense development—see quantities whose statistical properties are deter- the author’s survey article in [2] and the recent book mined by a unit vector in Hilbert space (for pure by M. Liao [8]. The latter contains a lot of interest- states) or a more general density matrix (for mixed ing material on the asymptotics of Lévy processes states). However, the celebrated Heisenberg un- on noncompact semisimple Lie groups, as t . certainty principle tells us that certain pairs of Liao has also found some classes of →∞Lévy these operators, such as those representing posi- processes on compact Lie groups that have L2- tion and momentum, fail to commute. Conse- densities. The density then has a “noncommutative quently they cannot both be described together as Fourier series” expansion via the Peter-Weyl theo- measurable functions on the same probability space rem. In the special case of Brownian motion on using Kolmogorov’s prescription, and hence they SU(2), Liao obtains the following beautiful formula cannot have a joint probability distribution. for its density ρt at time t: To describe the probabilistic features of quan- tum theoretic phenomena systematically, we need ∞ (n2 1)t sin(2πnθ) to take an algebraic viewpoint. We define a quan- ρ (θ) n exp − , t 2 (B,ω) B = n 1 − 64π sin(2πθ) tum probability space to be a pair where = is a complex -algebra (with identity I) and ω is a where θ (0, 1] parameterizes the maximal torus state on B, i.e.,∗ a positive, linear map for which 2∈πiθ 2πiθ diag e ,e− ,θ [0, 1) . ω(I) 1. If B is a C∗-algebra, we can recover a { ∈ } = Another important theme, originally due to R. Hilbert space viewpoint by taking the Gelfand- Gangolli in the 1960s, is to study spherically sym- Naimark-Segal representation. metric Lévy processes on semisimple Lie groups G Quantum stochastic processes were introduced (i.e., those whose laws are bi-invariant under the by L. Accardi, A. Frigerio, and J. T. Lewis in the action of a fixed compact subgroup K). Using Har- 1980s. Every “classical” stochastic process ish-Chandra’s theory of spherical functions, one can (X(t),t 0) with state space E gives rise to a fam- ≥ carry out “Fourier analysis” and obtain a Lévy- ily of -homomorphisms (jt ,t 0) from the ∗ ≥ Khintchine-type formula. One of the reasons why -algebra Bb(E) of bounded measurable functions ∗ this is interesting is that G/K is a Riemannian on E into the -algebra L∞(Ω, ,P) by the pre- ∗ F (globally) symmetric space and all such spaces can scription jt (f ) f X(t). Given a quantum proba- be obtained in this way. The Lévy process in G pro- bility space (B,ω= )◦and a -algebra A, a quantum ∗ jects to a Lévy process in G/K, and this is the pro- stochastic process is a family (jt ,t 0) of totype for constructions of Lévy processes in more -homomorphisms from A into B. Many concrete≥ general Riemannian manifolds. ∗examples of these have been constructed using Before leaving the subject of Lévy processes in the quantum stochastic calculus of R. L. Hudson groups, we briefly mention the general locally com- and K. R. Parthasarathy as solutions of operator- pact case. Work on Hilbert’s fifth problem during valued stochastic differential equations driven by the 1950s established that every such group has “quantum noise”, i.e., the creation, conservation, an open subgroup of the identity that is a projec- and annihilation processes acting in a suitable Fock tive limit of Lie groups. This enables the use of Lie space. group methods within the more general case, and In order to clarify the last remark, we make a there has been intensive work on this subject since brief diversion. Fock space Γ (h) over a complex Γ (n) the 1970s by the German school of H. Heyer, W. Hilbert space h is (h): n∞ 0 h , where (0) C (1) (n) = = Hazod, E. Siebert, and their students ([6]). Quite re- h ,h h, and h is the tensor product of cently, the path properties of Brownian motion in n copies= of =h. It is often desirable to restrict to general locally compact groups have been investi- boson (symmetric) or fermion (antisymmetric) Fock
DECEMBER 2004 NOTICES OF THE AMS 1345 space, which are the closed subspaces obtained by ∆ : A A A and a co-unit ε : A C which satisfy → ⊗ → restricting to symmetric or antisymmetric tensors, the co-associativity and co-unit axioms: respectively. For each f h the creation operator ∆ ∆ ∆ ∆ (n) (∈n 1) (id ) ( id) , a†(f ) maps each h to h + while the annihilation ⊗ ◦ = ⊗ ◦ (n) (n 1) (id ε) ∆ (ε id) ∆, operator a(f ) maps each h to h − . For each self- ⊗ ◦ = ⊗ ◦ adjoint T acting in h, the conservation operator where id is the identity mapping. dΓ (T ) maps h(n) to itself. All three types of opera- If A is a -bialgebra, we obtain a quantum Lévy ∗ tor are densely defined linear operators in Γ (h) process on A when we augment the generalizations (see, e.g., [9] for precise definitions). As a by- of (L1) to (L3) with an additional axiom product of work on factorizable representations of (L0) kr,s ks,t kr,t, for all 0 r s t< , current groups in the 1960s and 1970s it was found ∗ = ≤ ≤ ≤ ∞ d that any Lévy process X (X(t),t 0) on R can where the convolution is given by = ≥ be realized as a family of self-adjoint operators act- kr,s ks,t mB (kr,s ks,t) ∆; ing in a symmetric Fock space, where the Lévy-Itô ∗ = ◦ ⊗ ◦ decomposition (0.3) appears as a certain combi- here mB denotes the multiplication in B. nation of creation, conservation, and annihilation To understand the meaning of (L0) in the sim- operators. In the 1980s, Hudson and Parthasarathy plest possible context, let X be a Lévy process in a realized that they could build interesting classes finite group G, and take A to be the -bialgebra of ∗ of quantum stochastic processes by developing a all complex valued functions on G with the usual stochastic calculus in which each of the creation, pointwise algebra operations and comultiplication
conservation, and annihilation parts is treated as (∆f )(σ1,σ2) f (σ1σ2) and co-unit ε(f ) f (e). Take = = 1 a separate operator-valued process rather than in B L∞(Ω, ,P) and each ks,tf f X(s)− X(t) . = F = ◦ a special “classical” self-adjoint combination. Then (L0) precisely expresses the “increment prop- 1 1 1 We can now make an attempt at defining a erty”, X(r)− X(s)X(s)− X(t) X(r)− X(t) . = “quantum Lévy process”. At the very least this Quantum Lévy processes first arose in work by should be a quantum stochastic process (jt ,t 0) W. von Waldenfels on a model of the emission and ≥ where each jt is embedded as k0,t into an associ- absorption of light by atoms interacting with ated two-parameter family of -homomorphisms ∗ “noise”. The quantum stochastic process obtained (ks,t, 0 s t< ) which are the “increments” of ≤ ≤ ∞ appeared to be a noncommutative analogue of a the process. We generalize the key axiom (L1). The Lévy process on the unitary group U(d), and this stationary increments requirement becomes was made precise in terms of quantum Lévy ω(ks,t(a)) ω(k0,t s (a)), for each a A. For inde- = − ∈ processes when U(d) was replaced by a noncom- pendent increments, we have a choice from a num- mutative -bialgebra that generalizes the coeffi- ∗ ber of competing algebraic notions of indepen- cient algebra of U(d). The theory of quantum Lévy dence, each of which will yield a distinct notion of processes has been extensively developed by Lévy process. The simplest, called tensor (or bosonic) M. Schürmann and U. Franz in Greifswald, Ger- independence, requires that n many (see [13] or Chapter 7 of [9]). In particular, all quantum Lévy processes are equivalent to so- ω(ks1,t1 (a1)ks2,t2 (a2) ksn,tn (an)) ω(ksi ,ti (ai )), ··· = i 1 lutions of quantum stochastic differential equations = driven by creation, conservation, and annihilation for all n N,a1,...,an A, 0 s1 t1 s2 ∈ ∈ ≤ ≤ ≤ ≤ processes acting in a suitable Fock space. t2 sn tn < , whenever each pair ks ,t (ai ) ··· ≤ ≤ ∞ i i We briefly describe one interesting application and ksj ,tj (aj ) commute. Other notions of indepen- dence that could be used include the fermionic (or of quantum Lévy processes to classical probability. Z Let B (B(t),t 0) be a one-dimensional Brownian 2 graded version) or the free independence of = ≥ D. Voiculescu. Axioms (L2) and (L3) translate rather motion and g(t) sup 0 s t; B(s) 0 . Azéma’s = { π≤ ≤ = } martingale M(t) sign(B(t)) t g(t) is a easily into this framework; however, the concept = 2 − martingale with respect to the filtration t we have thus obtained is too general, as it is not F = k σ M(s); 0 s t . This process has many intrigu- clear how s,t has captured the notion of “incre- { ≤ ≤ } ment”. ing features, e.g., M. Emery proved that it shares with To overcome this problem, we need to general- Brownian motion and the compensated Poisson ize the group concept algebraically, and this is pre- process the rare property of being “chaotically com- cisely the purpose of quantum groups. More pre- plete” (i.e., the linear span of all multiple Wiener in- cisely, we need A to be a -bialgebra, i.e., a tegrals is dense in the natural L2 space), but it is not ∗ -algebra in which the multiplication and identity a Lévy process on R in the usual sense. However, ∗ have been dualized to give a compatible co- Schürmann has shown that it is a quantum Lévy algebra structure. We thus require that there are process on a certain -bialgebra generated by two in- ∗ two -homomorphisms, a comultiplication determinates. ∗
1346 NOTICES OF THE AMS VOLUME 51, NUMBER 11 Conclusion One way of assessing the health of an area of math- About the Cover ematics is to explore the extent to which it per- Kleinian Pearls meates other aspects of the subject. Another way This month’s cover was created by David is to examine its use in applications. Regarding Wright, who explains it in a brief article in this both of these criteria, Lévy processes appears to issue (pages 1332–1333). He entered it in the be flourishing. Indeed, limitations of space in this 2003 NSF Visualization Challenge, in which it article have prevented me from discussing a host was a semifinalist. Limit sets of Kleinian of other topics, including new theoretical advances groups and how to draw them are major in the fluctuation theory of real-valued Lévy themes of the well-illustrated book Indra’s processes due to J. Bertoin and R. A. Doney and ap- Pearls (Cambridge University Press, 2002), plications to turbulence, time series, and the cod- written by Wright with coauthors David Mum- ification of branching processes. Readers are invited ford and Caroline Series. to join the author in speculating that the interplay of Gaussian continuous motion with Poisson jumps, —Bill Casselman or alternatively its quantum theoretic manifesta- Graphics Editor tion within the dance of creation, conservation, ([email protected]) and annihilation operators, is a universal feature of a class of random motions (both classical and quantum) that is sufficiently wide to keep mathe- maticians busy for many years to come.
Acknowledgement: Thanks are due to Chris Rogers for invaluable advice about simulation. References [1] D. APPLEBAUM, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004. [2] O. E. BARNDORFF-NIELSEN, T. MIKOSCH, and S. RESNICK (Eds.), Lévy Processes: Theory and Applications, Birkhäuser, Basel (2001). [3] J. BERTOIN, Lévy Processes, Cambridge University Press, Cambridge, 1996. [4] R. CONT and P. TANKOV, Financial Modelling with Jump Processes, Chapman and Hall/CRC (2004). [5] P. DIACONIS, Patterns in eigenvalues, the 70th Josiah Willard Gibbs lecture, Bull. Amer. Math. Soc. 40, 155- 79 (2003). [6] H. HEYER, Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin-Heidelberg, 1977. [7] N. JACOB, Pseudo-differential Operators and Markov Processes: 1. Fourier Analysis and Semigroups, World Scientific (2001). 2. Generators and their Potential The- ory, World Scientific (2002). [8] M. LIAO, Lévy Processes in Lie Groups, Cambridge Uni- versity Press, Cambridge, 2004. [9] P.-A. MEYER, Quantum Probability for Probabilists, (sec- ond edition), Lecture Notes in Mathematics Vol. 1538, Springer-Verlag, Berlin, Heidelberg, 1995. [10] L. SALOFF-COSTE, Probability on groups: random walks and invariant diffusions, Notices Amer. Math. Soc., 48, 968-77 (2001). [11] K.-I. SATO, Lévy Processes and Infinite Divisibility, Cambridge University Press, Cambridge, 1999. [12] W. SCHOUTENS, Lévy Processes in Finance: Pricing Fi- nancial Derivatives, Wiley, 2003. [13] M. SCHÜRMANN, White Noise on Bialgebras, Lecture Notes in Mathematics 1544, Springer-Verlag, Berlin, Heidelberg, New York, 1991.
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