Feller Processes: The Next Generation in Modeling. , Le´vy Processes and Beyond

Bjo¨ rn Bo¨ ttcher* TU Dresden, Institut fu¨r Mathematische Stochastik, Dresden, Germany

Abstract We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent of Le´vy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Le´vy processes, of which Brownian Motion is a special case, have become increasingly popular. Le´vy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Le´vy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.

Citation: Bo¨ttcher B (2010) Feller Processes: The Next Generation in Modeling. Brownian Motion, Le´vy Processes and Beyond. PLoS ONE 5(12): e15102. doi:10.1371/journal.pone.0015102 Editor: Enrico Scalas, Universita` del Piemonte Orientale, Italy Received September 24, 2010; Accepted October 20, 2010; Published December 3, 2010 Copyright: ß 2010 Bjo¨rn Bo¨ttcher. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: The author has no support or funding to report. Competing Interests: The author has declared that no competing interests exist. * E-mail: [email protected]

Introduction N In geology also stable processes are used in models for the temperature change. Based on ice-core data the temperatures in The paper is written especially for practitioners and applied the last-glacial and Holocene periods are recorded. Statistical scientists. It is based on two recent papers in stochastic analysis analysis showed that the temperature change in the last-glacial [1,2]. We will start with a survey of applications of Feller processes. periods is stable with index 1.75 and in the Holocene periods it is Thereafter we recall some existence and approximation results. In Gaussian, i.e. stable with index 2 (see Fig. 4 in [10]). the last part of the introduction we give the necessary definitions. N For a technical example from physics note that the fluctuations The main part of the paper contains a simple existence result for of the ion saturation current measured by Langmuir probes in Feller processes and a description of the general simulation the edge plasma of the Uragan-3M stellarator-torsatron are scheme. These results will be followed by several examples. alpha-stable and the alpha depends on the distance from the The source code for the simulations can be found as supporting plasma boundary [11]. information (Appendix S1). N Anomalous diffusive behavior has been observed in various Motivation physical systems and a standard model for this behavior are continuous time random walks (CTRWs) [12,13]. To study Brownian motion and more general Le´vy processes are used as these systems the limiting particle distribution is a major tool, models in many areas: For example in medicine to model the which is in fact a Feller process [14]. spreading of diseases [3], in genetics in connection with the maximal segmental score [4], in biology for the movement N In the idea of extending Le´vy processes patterns of various animals (cf. [5] and the references therein), for to Le´vy-like Feller processes was first introduced in [15]. The various phenomena in physics [6] and in financial mathematics proposed procedure is simple: A given Le´vy model usually uses [7]. In these models the spatial homogeneity is often assumed for a parameter dependent class of Le´vy processes. Now one simplicity, but empirical data or theoretical considerations suggest makes the parameters of the Le´vy process (in its characteristic that the underlying process is actually state space dependent. Thus exponent) price-dependent, i.e. the increment of the process Feller processes would serve as more realistic models. We give shall depend on the current price. This procedure is applicable some explicit examples: to every class of Le´vy processes, but the existence has to be shown for each class separately [15–17]. N In hydrology stable processes are used as models for the movement of particles in contaminated ground water. It has Thus there is plenty of evidence that Feller processes can be been shown that state space dependent models provide a better used as suitable models for real-world phenomena. fit to empirical data [8,9]. Also on an intuitive level it seems natural that different kinds of soils have different properties. Existence and Approximation Thus the movement of a particle should depend on its current Up to now general Feller processes were not very popular in position, i.e. the soil it is currently in. applications. This might be due to the fact that the existence and

PLoS ONE | www.plosone.org 1 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling construction of Feller processes is a major problem. There are The most popular Le´vy process is Brownian motion 1 many approaches: Using the Hille-Yosida theorem and Kolmo- (y(j)~ jjj2), which has the special property that almost every gorov’s construction [18,19], solving the associated evolution 2 equation (Kolmogorov’s backwards equation) [20–23], proving the sample path is continuous. In general, Le´vy processes have well-posedness of the martingale problem [16,19,24], solving a discontinuous sample paths, some examples with their corre- ij stochastic differential equation [25–27]. The conditions for these sponding exponents are the Poisson process (y(j)~e {1), the a constructions are usually quite technical. Nevertheless, let us stress symmetric a- (y(j)~jjj with a [ (0,2), the Gamma that the proof of the very existence is crucial for the use of Feller process (y(j)~ln(1zij)) andqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the normalq inverseffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gaussian processes. Some explicit examples to illustrate this will be given at process (y(j)~{imjzd a2{(bzij)2{ a2{b2 with the end of the next section. 0ƒ b va, dw0, m [ R Our construction will not yield processes as general as the j j ). previous ones, but it will still provide a rich class of examples. In fact Classes of Le´vy exponents depend, especially in modeling, on the presented method is just a simple consequence of a recent result some parameters. Thus one can easily construct a family of Le´vy on the solutions to certain stochastic differential equations [2]. processes by replacing these parameters by state space dependent functions. Another approach to construct families of Le´vy Furthermore each of the above mentioned methods also processes is to introduce a state space dependent mixing of some provides an approximation to the constructed Feller process. given Le´vy processes. We will elaborate this in the next section. Most of them are not usable for simulations or work only under  ðÞx technical conditions. Also further general approximation schemes Given a family of Le´vy processes Lt , i.e. given exist, for example the approximation in [28]. But t [ ½0,?) x [ R a family of characteristic exponents (yx)x [ R, we can construct for also the latter is not useful for simulations, since the explicit R N distribution of the increments of the chain is unknown. fixed x0 [ , T [ ½0,?), n [ a Markov chain as follows: In contrast to these we derived in [1] a very general 1. The chain starts at time 0 in x0. approximation scheme for Feller processes which is also usable 1 ðÞx0 2. The first step is at time and it is distributed as L1 . The for simulations. We will present here this method for practitioners. n n chain reaches some point x1.

Le´vy processes and Feller processes 2 ðÞx1 3. The second step is at time and it is distributed as L1 . The Within different fields the terms Le´vy process and Feller process are n n sometimes used for different objects. Thus we will clarify our chain reaches some point x2. 3 ðÞx2 notion by giving precise definitions and mentioning some of the 4. The third step is at time and it is distributed as L1 . The n n common uses of these terms. chain reaches some point x3. A is a family of random variables indexed by 5. etc. until time T. a time parameter t [ ½0,?) on a space (V,F,P). For simplicity we concentrate on one-dimensional processes. The This Markov chain is spatially inhomogeneous since the expectation with respect to the measure P will be denoted by E. distribution of the next step always depends on the current Although this will not appear explicitly in the sequel, a process position. If the chain converges (in distribution for n?? and will always be equipped with its so-called natural filtration, which every fixed T [ ½0,?)) then the limit is - under very mild is a formal way of taking into account all the information related to conditions (see [30] and also Theorem 2.5 by [31]) - a Feller the history of the process. Technically the filtration, which is an process. Formally, a Feller process is a stochastic process increasing family of sigma fields indexed by time, is important (Xt)t [ ½0,?) such that the operators since a change from the natural filtration to another filtration might alter the properties of the process dramatically. Ttf (x) : ~E(f (Xt)jX0~x), t [ ½0,?), x [ R A Le´vy process starting in L0 : ~0 is a stochastic process (Lt)t [ ½0,?) with L ,L {L , - independent increments: The random variables t1 t2 t1 satisfy { Lt3 Lt2 , ... are independent for every increasing sequence (tn)n [ N, T0~id, Ts0Tt~Tszt (s,t§0) (semigroup property) - stationary increments: Lt{Ls has the same distribution as Lt{s for all svt, -ca`dla`g paths: Almost every sample path is a right continuous and function with left limits. For equivalent definitions and a comprehensive mathematical treatment of Le´vy processes and their properties see [29]. lim sup jTtf (x){f (x)j~0 (strong continuity) t?0 R Note that the term Le´vy flight often refers to a process which is a x [ continuous time (CTRW) with spatial increments from a one-sided or two-sided (the former is also for all f which are continuous and vanish at infinity. called Le´vy distribution). In our notion the processes associated A Feller processes is sometimes also called: Le´vy-type process, with these increments are Le´vy processes which are called stable jump-diffusion, process generated by a pseudo-differential opera- subordinator and stable process, respectively. tor, process with a Le´vy generator or process with a Le´vy-type ALe´vy process (Lt)t [ ½0,?) on its probability space is completely operator as generator. Note that in mathematical finance often the characterized by its Le´vy exponent j.y(j) calculated via the Cox-Ingersoll-Ross process [32] is called the Feller process, but in our characteristic function notion this is a Feller diffusion in the sense of [33]. For a comprehensive mathematical treatment of Feller processes and E(eijLt )~e{ty(j): their properties see [18].

PLoS ONE | www.plosone.org 2 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling

ðÞ2 ðÞ2 The generator A of a Feller process is defined via a (x)~0 for all x [ K1 and a (x)~1 for all x [ K2:

Now set for j ,j ,x [ R TtfxðÞ{fxðÞ 1 2 lim sup Af (x){ ~0 t?0 R t ! x [  T j aðÞ1 ðÞx y 1 : ~yðÞ1 (j )zyðÞ2 (j ), W(x) : ~ [ R1|2 1 2 ðÞ2 for all f such that the limit exists. Moreover, if the limit exists for j2 a ðÞx arbitrarily often differentiable functions with compact support then the operator A has on these functions the representation and note that for x,j [ R ð ð  T ~ (1) ðÞ1 z ðÞ2 ðÞ2 ~{ ixj {1 {iyj y(W (x)j) y a ðÞx j y a ðÞx j Af (x) e yx(j) (2p) e f (y)dy dj, R R holds. Thus corresponding to the family of Le´vy processes defined R where for each fixed x [ the function j.yx(j) is a Le´vy by the Le´vy exponents exponent. Thus a family of Le´vy processes with Le´vy exponents  (y ) R corresponds to the Feller process (X ) with x x [ t t [ ½0,?) y (j) : ~yðÞ1 aðÞ1 ðÞx j zyðÞ2 aðÞ2 ðÞx j ~yðÞWtðÞx j generator A as above. x If the corresponding family of Le´vy processes is a subset of a there exists a Feller process as a consequence of Corollary 5.2 from named class of Le´vy processes, one calls the Feller process also by ~ (i) ~ the name of the class and adds -like or -type to it. Thus for example [2] and yx(j) y (j) for x [ Ki holds (i 1,2), i.e. (Xt)t§0 ðÞi a Feller process corresponding to a class of symmetric stable behaves like L on Ki for i~1,2. processes is called symmetric stable-like process. Note that the theorem extends to any finite number of Le´vy In general, as mentioned in the previous section, the ðÞi processes Lt (i~1,::,n) with corresponding regions Ki. construction of a Feller process corresponding to a given family t§0 of Le´vy processes is very complicated. It even might be impossible More generally for any finite number of independent Le´vy ðÞx ðÞi as the following examples show: Let Lt be the family of processes Lt (i~1,::,n) with corresponding characteristic t§0 t§0 ~{ Le´vy processes with characteristic exponents yx(j) ia(x)j, i.e. yðÞi ðÞx exponents and non-negative bounded and Lipschitz contin- the Le´vy processes have deterministic paths L ~axðÞt. Now if ðÞi t uous functions x.a ðÞx the family (yx)x [ R with a(x)~x a corresponding Feller process exists, starting in x it has pffiffiffi   the path X ~xet. But for a(x)~x2 and a(x)~sgn(x) x a t ~ ðÞ1 ðÞ1 z ðÞ2 ðÞ2 z z ðÞn ðÞn corresponding Feller process does not exist, a(x)~x2 yields paths yx(j) : y a ðÞx j y a ðÞx j ... y a ðÞx j which do not tend to negative infinity as x?{? and  pffiffiffi  ~ ~ðÞx a(x) sgnðÞx x yields paths which are not continuous with defines a family of Le´vy processes Lt and there § respect to the starting position. t 0 x [ R However, we will present in the next section a very simple exists a corresponding Feller process (Xt)t§0. method to construct Feller processes. To avoid pathological cases one should assume aðÞ1 ðÞx zaðÞ2 (x)z ...zaðÞn (x)w0 for all x. Further note that Results and Discussion the following equality in distribution holds for all x and t Construction of Feller processes by mixing Le´vy ~(x) ~d (1) (1)z (2) (2)z z (n) (n) processes Lt a (x)Lt a (x)Lt ... a (x)Lt : Suppose we know (for example based on an empirical study) that the process we want to model behaves like a Le´vy pro- ðÞi ðÞ1 Thus if one knows how to simulate increments of the Lt one cess Lt in a region K1 and like a different Le´vy process ~(x)  t§0 can also simulate increments of Lt . We will see in the next section ðÞ2 that simulation of increments of the corresponding family of Le´vy Lt in a region K2. Then we know that a Feller process t§0 processes is the key to the simulation of the Feller process. which models this behavior exists by the following result: Theorem. If the sets K1, K2 are uniformly separated, i.e. Simulation of Feller processes w there exists an e 0 such that Given a Feller process (X ) with corresponding family of t t [ ½0,?) Le´vy processes LðÞx we can use the following inf Ex{yEwe t t [ ½0,?) x [ R x [ K1, y [ K2 scheme to approximate the sample path of Xt:

ðÞ1 hen there exists a Feller process (Xt)t§0 which behaves like L on 1. Select a starting point x0, the time interval ½0,T and the time- ðÞ2 K1 and like L on K2.  step size h. ðÞi ðÞi Proof. Let y be the characteristic exponent of Lt for ~ ~ t§0 2. The first point of the sample path is x x0 at time t 0. i~1,2. Under the above condition there exist non-negative (x) 3. Draw a random number z from the distribution of Lh (x is the bounded and Lipschitz continuous functions aðÞ1 and aðÞ2 such that current position of the sample path).

4. The next point of the sample path is x > x + z at time t > t + h. ðÞ1 ðÞ1 a (x)~1 for all x [ K1 and a (x)~0 for all x [ K2, 5. Repeat 3. and 4. until t [ ðT{h,T .

PLoS ONE | www.plosone.org 3 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling

The simulated path is an approximation of the sample path of Examples the Feller process, in the sense that for h?0 it converges toward We will now present some examples of Feller processes together the sample path of the Feller process on ½0,T. To be precise, for with simulations of their sample paths. The first example will show the convergence the Feller process has to be unique for its the generality of the mixture approach, the remaining examples generator restricted to the test functions and the family of Le´vy are special cases for which the existence has been shown by (x) processes Lt has to satisfy some mild condition on the x- different techniques. dependence: The Le´vy exponent yx(j) has to be bounded by All simulations are done with the software package R [36] and 2 some constant times 1zjjj uniformly in x, see [1] for further the source code of the figures can be found as supporting details. This condition is satisfied for many common examples of information (Appendix S1). Feller processes, in particular for the processes constructed in the previous section. Brownian-Poisson-Cauchy-mixture Feller process The reader familiar with the Euler scheme for Brownian or To show the range of possibilities which are covered by the Le´vy-driven stochastic differential equations (SDEs) will note that mixture approach we construct a process which behaves like the approximation looks like an Euler scheme for an SDE. In fact it is an Euler scheme, but the corresponding SDE does not have { { such a nice form as for example the Le´vy-driven SDEs discussed in Brownian motion on ( ?, 6), [34]. This is due to the fact that in their case for a particular a Poisson process on ({4,4), increment all jumps of the driving term are transformed in the a on (6,?): same manner, but in the general Feller case the transformation of each jump can depend explicitly on the jump size. More details on the relation of this scheme to an Euler scheme can be found in a For this we just define a family of Le´vy processes by the family forthcoming paper [35]. of characteristic exponents (yx)x [ R with

Figure 1. Brownian-Poisson-Cauchy-mixture Feller process. Around the origin it behaves like a Poisson process, for smaller values like Brownian motion and a for larger values like a Cauchy process. doi:10.1371/journal.pone.0015102.g001

PLoS ONE | www.plosone.org 4 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling

8 1 > 1,ifxw6 y (j) : ~a (x) jjj2za (x)(1{e{ij)za (x)jjj <> x 1 2 2 3 x{4 a3ðÞx ~ ,ifx [ ðÞ4,6 : > 2 where : 0 , otherwise 8 > 1,ifxv{6 These functions are Lipschitz continuous and thus a corre- <> xz6 sponding Feller process exists. Figure 1 shows some samples of this a1ðÞx ~ 1{ ,ifx [ ðÞ{6,{4 , 1 > 2 process on ½0,20 with time-step size . One can observe that the :> 100 0 , otherwise process behaves like a Poisson process around the origin, like a 8 > { Cauchy process above 6 and like Brownian motion below 26. > 1,ifx [ ½4,4 > > xz6 Symmetric stable-like process <> ,ifx [ ðÞ{6,{4 2 ALe´vy process Lt is a symmetric-a-stable process if there exists a x ~ 2ðÞ > an a [ (0,2 such that its characteristic function is given by > x{4 > 1{ ,ifx [ ðÞ4,6 > 2 { a :> EeijLt ~e tjjj : 0 , otherwise If we now define a function x.a(x) where a(x) takes only and values in (0,2 then there exists a family of of Le´vy processes

 19 x x x x Figure 2. Stable-like processes with a(x) : ~1z { ^ { . Each position is color coded by the corresponding value of the 10 4 4 4 4 exponent. In the yellow regions it behaves almost like Brownian motion, in the red regions it behaves almost like a Cauchy process. doi:10.1371/journal.pone.0015102.g002

PLoS ONE | www.plosone.org 5 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling

 ðÞx (x) Normal inverse Gaussian-like process Lt such that for fixed x the the process Lt has the t [ ½0,?) The characteristic function of a normal inverse Gaussian characteristic function process Lt is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ðÞ { aðÞx ijL 2 2 EeiLt j~e tjjj : Ee t ~ exp tim{td a2{(bzij) { a2{b

where aw0,{avbva,0vd and m.Ifwereplacea,b,d,m by A corresponding Feller process exists and is unique if the arbitrarily often differentiable bounded functions a(x),b(x),d(x),m(x) function x.a(x) is Lipschitz continuous and bounded away from with bounded derivatives and assume that ther exist constants c,Cw0 0 and 2 [16]. such that d(x)wc,a(x){jb(x)jwc,cƒm(x)ƒC, then it was stated Figure 2 shows some samples of a stable-like Feller process on 1 in [15] that a corresponding Feller process exists. Therein was also ½0,20 with time-step size and proposed an example of a mean reverting normal inverse Gaussian-like 10 process, a special case of this model with mean 0 is obtained by setting  19 x x x x a(x) : ~1z { ^ { , 10 4 4 4 4 1 a(x) : ~d(x) : ~1, m(x) : ~0andb(x) : ~{ arctan(x): p i.e. x.a(x) is a function which is Lipschitz continuous (but not smooth) oscillating between 1 and (nearly) 2. To understand the figure note that we color coded the state space: red indicates a&1, Note that the mean reversion is not introduced by using simply yellow indicates a&2 and the values between these extremes are a drift which drags the process back to the origin. It is the choice of b colored with the corresponding shade of orange. Now one can which yields an asymmetric distribution that moves the process observe that the process behaves in the red areas like a Cauchy back to the origin. The mean reversion can be observed in Figure 3 process and the more yellow the state becomes, the more the which shows samples of the normal inverse Gaussian-like process 1 process behaves like Brownian motion. on ½0,1000 with time-step size . 10

1 Figure 3. Normal inverse Gaussian-like processes with bðÞx : ~{ arctanðÞ x . The process features mean reversion to 0. doi:10.1371/journal.pone.0015102.g003 p

PLoS ONE | www.plosone.org 6 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling

{ 1 2 Figure 4. Meixner-like process with axðÞ: ~1z10e 25{x 1ðÞ{5,5 ðÞx . The process moves with bigger steps around the origin than for larger (and smaller) values. In fact by the choice of x.a(x) the rate of the exponential decay of the transition density is reduced around the origin. doi:10.1371/journal.pone.0015102.g004

Meixner-like process The chosen functions satisfy the existence conditions from The characteristic function of a Meixner process Lt is given by above. Furthermore the function x.a(x) yields that the process moves with bigger steps around the origin, to be precise: the Meixner

0 12rt distribution has semiheavy tails [38] and the parameter a b determines the rate of the exponential decay factor for the B cos C EeijLt ~@ 2 A expðÞimtj density. The effect on the sample path can be observed in Figure 4. aj{ib cosh 2 Conclusion where a,rw0,{pvbvp,m [ R. Details can be found in [37]. Using the presented mixture approach one can easily construct A family of Meixner processes which corresponds to a Feller Feller models based on given Le´vy models. In these cases the process can be constructed by substituting the parameters existence of the process is granted. a,b,r,m by arbitrary often differentiable bounded functions Furthermore the presented approximation is a very intuitive a(x),b(x),r(x),m(x) with bounded derivatives. The functions way to generate the sample path of a Feller processes. Obviously have to be bounded away from the critical values, i.e. the method requires that one can simulate the increments of the 0va0ƒa(x),0vr0ƒr(x) and {pvb{vb(x)vbzvp for some corresponding Le´vy processes. But for Le´vy processes used in fixed r0,a0,b{,bz. For further details see [17]. applications, especially together with Monte Carlo techniques, this Figure 4 shows some samples of the Meixner-like process on poses no new restriction. 1 Thus all necessary tools are available to use Feller processes as ½0,100 with time-step size and 10 models for a wide range of applications.

b(x) : ~m(x) : ~0, r(x) : ~1 8 Materials and Methods < { 1 z 25{x2 v The simulations where done in R [36] and the source ~ 1 10e ,ifjxj 5 and axðÞ: : : code of the figures can be found as supporting information 1 , otherwise (Appendix S1).

PLoS ONE | www.plosone.org 7 December 2010 | Volume 5 | Issue 12 | e15102 Feller Processes: The Next Generation in Modeling

Supporting Information Author Contributions Appendix S1 Source code of the figures. Contributed reagents/materials/analysis tools: BB. Wrote the paper: BB. (TXT)

References 1. Bo¨ttcher B, Schilling R (2009) Approximation of Feller processes by Markov 19. Hoh W (1998) Pseudo differential operators generating Markov processes. chains with Le´vy increments. Stoch Dyn 9: 71–80. Habilitationsschrift. Universita¨t Bielefeld. 2. Schilling R, Schnurr A (2010) The Symbol Associated with the Solution of a 20. Bo¨ttcher B (2008) Construction of time inhomogeneous Markov processes via Stochastic Differential Equation. Electron J Probab 15: 1369–1393. evolution equations using pseudo-differential operators. J Lon Math Soc 78: 3. Janssen H, Oerding K, van Wijland F, Hilhorst H (1999) Le´vy-flight spreading 605–621. of epidemic processes leading to percolating clusters. Eur Phys J B 7: 137–145. 21. Bo¨ttcher B (2005) A parametrix construction for the fundamental solution of the 4. Doney RA, Maller RA (2005) Crame´r’s Estimate for a Reflected Le´vy Process. evolution equation associated with a pseudo-differential operator generating a Ann Appl Probab 15: 1445–1450. Markov process. Math Nachr 278: 1235–1241. 5. Reynolds AM FM (2007) Free-flight odor tracking in drosophila is consistent 22. Kochubei AN (1989) Parabolic pseudodifferential equations, hypersingular with an optimal intermittent scale-free search. 2(4): e354. doi:. PLoS ONE 2: integrals and Markov processes. Math USSR Izvestija 33: 233–259. e354. 23. Kolokoltsov VN (2000) Symmetric stable laws and stable-like jump-diffusions. 6. Woyczyn´ski WA (2001) Le´vy processes in the physical sciences. In: Le´vy Process- Proc London Math Soc 80: 725–768. Theory and Applications, Birkha¨user. pp 241–266. 24. Stroock D (1975) Diffusion processes associated with levy generators. Probab 7. Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman & Theory Relat Fields 32: 209–244. Hall/CRC. 25. Ikeda N, Watanabe S (1989) Stochastic Differential Equations and Diffusion 8. Zhang Y, Benson D, Meerschaert M, LaBolle EM (2007) Space-fractional Processes, volume 24 of Math. Library North-Holland, 2nd edition. advection-dispersion equations with variable parameters: Diverse formulas, 26. Jacod J, Shiryaev AN (2002) Limit Theorems for Stochastic Processes. Springer, numerical solutions, and application to the MADE-site data. Water Resources 2nd edition. Research 43: W05439. 27. Stroock DW (2003) Markov Processes from K. Itoˆ’s Perspective. Princeton 9. Reeves D, Benson D, Meerschaert M, Scheffler H (2008) Transport of University Press. Conservative Solutes in Simulated Fracture Networks 2. Ensemble Solute 28. Ma ZM, Ro¨ckner M, Zhang TS (1998) Approximation of arbitrary Dirichlet Transport and the Correspondence to Operator-Stable Limit Distributions. processes by Markov chains. Ann Inst Henri Poincare 34: 1–22. Water Resources Research 44: W05410. 29. Sato K (1999) Le´vy Processes and Infinitely Divisible Distributions. Cambridge 10. Ditlevsen PD, Svensmark H, Johnsen S (1996) Contrasting atmospheric and University Press. climate dynamics of the last-glacial and Holocene periods. Nature 379: 810–812. 30. Chernoff P (1974) Product Semigroups, Nonlinear Semigroups and Addition of 11. Gonchar VY, Chechkin AV, Sorokovoi EL, Chechkin VV, Grigoreva LI, et al. unbounded Operators. Number 140 in Mem. AMS. (2003) Stable Le´vy distributions of the density and potential fluctuations in the 31. Jacob N, Potrykus A (2005) Roth’s method applied to some pseudo-differential edge plasma of the U-3M torsatron. Plasma Physics Reports 29: 380–390. operators with bounded symbols. A case study. Rend Cir Mat Palermo (Ser II) 12. Meerschaert MM, Benson DA, Scheffler HP, Becker-Kern P (2002) Governing 76: 45–57. equations and solutions of anomalous random walk limits. Physical Review E 66: 32. Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest 060102. rates. Econometrica 53: 385–408. 13. Scalas E, Gorenflo R, Mainardi F (2004) Uncoupled continuous-time random 33. Feller W (1951) Diffusion processes in genetics. In: Neyman J, ed. Proceedings of walks: Solution and limiting behavior of the master equation. Physical Review E the Second Berkeley Symposium on Mathematical and Probability 69: 011107. University of California Press. pp 227–246. 14. Baeumer B, Meerschaert M, Mortensen J (2005) Space-time fractional derivative 34. Protter P, Talay D (1997) The Euler scheme for Le´vy driven stochastic operators. Proc Amer Math Soc 133: 2273–2282. differential equations. Ann Probab 25: 393–423. 15. Barndorff-Nielsen O, Levendorskilˇ S (2001) Feller processes of normal inverse 35. Bo¨ttcher B, Schnurr A (2010) The Euler scheme for Feller processes. URL Gaussian type. Quant Finance 1: 318–331. http://arxiv.org/abs/0911.5245. Accessed 2010 Nov 11. 16. Bass R (1988) Uniqueness in law for pure jump Markov processes. Probab 36. R Development Core Team (2010) R: A language and environment for Theory and Relat Fields 79: 271–287. statistical computing. URL http://www.R-project.org. Accessed 2010 Nov 11. 17. Bo¨ttcher B, Jacob N (2003) Remarks on Meixner-type processes. Probabilistic ISBN 3-900051-07-0. Methods in Fluids (eds IM Davies et al). pp 35–47. 37. Grigelionis B (1999) Processes of Meixner type. Lithuanian Math J 39: 33–41. 18. Jacob N (2001) Pseudo-differential operators and Markov processes, volume 38. Grigelionis B (2001) Generalized z-distributions and related stochastic processes. I-III. Imperial College Press. Lithuanian Math J 41: 239–251.

PLoS ONE | www.plosone.org 8 December 2010 | Volume 5 | Issue 12 | e15102