This is a repository copy of Levy processes - from probability theory to finance and quantum groups. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/9794/ 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 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [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.