Ambit processes and stochastic partial differential equations

37 Pages·2010·0.32 MB·English

CREATES Research Paper 2010-17

Ambit processes and stochastic partial
differential equations

Ole E. Barndorff–Nielsen, Fred Espen Benth
and Almut E. D. Veraart

School of Economics and Management
Aarhus University
Bartholins Allé 10, Building 1322, DK-8000 Aarhus C
Denmark
Ambit processes and stochastic partial differential equations
OleE. Barndorff–Nielsen
ThieleCenter, DepartmentofMathematicalSciences
and
CREATES, School ofEconomicsand Management
AarhusUniversity
Ny Munkegade118
DK–8000AarhusC, Denmark
[email protected]
Fred EspenBenth
Centre ofMathematicsforApplications
UniversityofOslo
P.O. Box 1053,Blindern
N–0316Oslo, Norway
and
FacultyofEconomics
UniversityofAgder
Serviceboks 422
N–4604Kristiansand,Norway
[email protected]
AlmutE. D. Veraart*
CREATES, School ofEconomicsand Management
AarhusUniversity
DK–8000Aarhus, C, Denmark
[email protected]
Abstract
Ambitprocessesaregeneralstochasticprocessesbasedonstochasticintegralswithrespectto
Le´vybases. Duetotheirflexiblestructure,theyhavegreatpotentialforprovidingrealisticmodels
for various applications such as in turbulence and finance. This papers studies the connection
betweenambitprocessesandsolutionstostochasticpartialdifferentialequations. Weinvestigate
this relationship from two angles: from the Walsh theory of martingalemeasures and from the
viewpointoftheLe´vynoiseanalysis.
Keywords: Ambitprocesses,stochasticpartialdifferentialequations, Le´vybases,Le´vynoise,Walsh
theoryofmartingalemeasures, turbulence, finance.
JELcodes: C0,C1,C5.
1
1 INTRODUCTION
1 Introduction
In physics, partial differential equations (PDEs) give a dynamic way to describe how phenomena
in nature evolve over time and space. For instance, the classical heat equation of Einstein gives a
dynamic model for how heat diffuses in amedium. Stochastic partial differential equations (SPDEs)
addrandomness tosuchevolution equations, wherethenoise source maycomefrom uncertainties in
measurements, non–explainable effectsandturbulent phenomena. Thenoiseisusually modelledasa
random field in time and space, also called white noise or, more generally, Le´vy noise. We shall be
mostlyconcerned withparabolic PDEsinthispaper.
AmbitprocesseshavebeenproposedandintroducedbyBarndorff–Nielsen&Schmiegelandhave
thereafter been applied in various areas such as turbulence modelling (see e.g. [6; 13]), in medical
context in form of describing tumor growth ([12]), and more recently for modelling energy markets
([4;5]).
Thesolution ofaparabolic differential equation isoftenrepresented asanintegraloveraGreen’s
function (the fundamental solution of the PDE) convoluted with some initial condition. Such rep-
resentations look very similar to the definition of stationary ambit processes of [13]. The Green’s
function representation is an explicit solution as long as the Green’s function is known, where the
deterministic space–time dynamics of the phenomena in question is encapsulated in the form of this
function. Itiscloselylinkedtodensity functions ofstochastic diffusionprocesses.
Introducing noise leads to complications of interpreting in what sense we have a solution. This
requires a theory for stochastic integration in time and space, such as proposed in Walsh [46]. It
turnsout, thatsolutions ofparabolic equations withanadditive source ofnoisecanberepresented as
the stochastic convolution of the Green’s function and the initial value, where the integration is with
respecttotherandom field. WepresentthetheoryofWalsh[46]andlinkittoambitprocesses.
Whenhavingastochastic sourceterm,onemayhavesolutionsbeingsingular. Thisisthestarting
point for applying white noise analysis (WNA) or, more generally, Le´vy noise analysis (LNA) to
analyse SPDEs. We discuss the theory of LNA and link it to ambit processes. Here we will also
includediscussions ofSPDEsandhowtheyarerelatedtoambitprocesses.
Notethatambitprocessesmayprovideastatisticalapproachtomodelphysicalprocessesinnature
farsimplerthanSPDEs,sincetheyprovideawaytospecifydirectlythemodelbasedonaprobabilistic
understanding ofthephenomena inquestion. Theyalsogiveaframeworkforextending thesolutions
of SPDEs. In order to have a solution in the sense of Walsh, often strong integrability conditions
are imposed. The ambit processes are well–defined under very weak conditions of integrability, and
therebywemayextendthesolutionsofcertainequationstoincludefarmoregeneralinitialconditions,
say,ormoregeneraltypesofnoise.
The main issue of this paper is to relate the use of the building stone in ambit processes, Le´vy
bases, to the language of Walsh and the theory of LNA. The latter talks about processes being the
derivativesofLevyprocesses, whileWalshtalksaboutrandommeasuresandtheirderivatives.
The outline for the remaining part of the paper is as follows. In Section 2 the concepts of ambit
fields and processes are outlined, and the important special case of spatial dimension 0 is treated in
somedetail; inthat case the ambitprocesses are referred toasLe´vysemistationary ( )processes
LSS
or, intheGaussian case, asBrowniansemistationary ( )processes. Inparticular, anindication of
BSS
the theory and use of multipower variations for inference on the volatility process is given. Section
2 concludes by a brief discussion of some applications to turbulence and energy markets. Section 3
connectstheideaofLe´vybasestothetheoryofrandom fieldsduetoWalsh. Weshowhow,subjectto
anL2 restriction andbased onthetheory ofHilbertspacerandom fields,itispossible todefineLe´vy
noiseforLe´vybases, andtheassociated integration theoryisdiscussed. Finally,someapplications to
2
2 AMBITPROCESSES
SPDEsandtheir relation toambitprocesses areconsidered. Section 4links the theory ofLe´vynoise
analysisforLe´vyprocesses, asdevelopedinHolden,Øksendal,UbøeandZhang[31],tothatofLe´vy
bases and ambit processes, and discusses SPDEs in that context. The concluding Section 5 briefly
bringsthevariousstrandstogether.
2 Ambit processes
2.1 Background
Thegeneral background setting for theconcept ofambitprocesses consists ofastochastic fieldY =
Y (x) in space–time R, a curve τ (θ) = (x(θ),t(θ)) in R, and the values X =
t θ
{ } X × X ×
Y (x(θ)) of the field along the curve, the focus being on the dynamic properties of the stochastic
t(θ)
processX = X . Herethespace isoften,butnotnecessarily, takenasRd ford = 1,2or3. The
θ
{ } X
stochastic field is supposed to be generated by innovations in space–time and the values Y (x) are
t
assumed todepend only oninnovations that occur prior toorattime t. Moreprecisely, ateach point
(x,t) only the innovations in some subset A (x) of R (where R = ( ,t]) are influencing
t t t
X × −∞
thevalueofY (x),andwerefertoA (x)astheambitset,associated to(x,t),andtoY andX asan
t t
ambitfield andanambitprocess, respectively; seeFigure1foranillustration.
X (x(θ),t(θ))
θ
@
A (x(θ))
t(θ)
(cid:0)
(cid:0)
(cid:0)
x
Figure 1: Example ofanambit process X along the curve (x(θ),t(θ)), where theambit set isgiven
θ
byA (x(θ)).
t(θ)
Obviously, without further structure nothing interesting can be said about the field Y and the
process X, and we shall specify such structure in mathematical detail in a moment. But in verbal
terms,Y (x)willbedefinedintheformofastochastic integralplusasmoothterm,andtheintegrand
t
inthestochasticintegralwillconsistofadeterministickerneltimesapositiverandomvariatewhichis
takentoembodythevolatility orintermittency ofthefieldY. Weshallmostlyconsiderspecifications
underwhichY (x)isstationary intimeforeachfixedx.
t
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2 AMBITPROCESSES
The volatility field, denoted by σ, is given also as an ambit field, and a central issue is what can
belearnedaboutσ fromobservation ofY orX.
Note that, in general, ambit processes are not semimartingales. Many of the standard tools from
semimartingale theoryarethereforenotapplicable andalternativemethodsarerequired.
Themore precise mathematical specification of whatis meant generally by ambit fieldsand pro-
cesses is given in Section 2.2. In Sections 2.3, 2.4 and 2.5 we focus on the null–spatial case, i.e.
where consists of a single point. There the concept of ambit processes specialises to that of Le´vy
X
and Brownian semistationary processes ( and processes). Already in that setting there are
LSS BSS
many interesting questions of a nonstandard character. These have important analogues in the gen-
uinelytempo–spatial case.
Asforsemimartingales,thequestionsofexistenceandpropertiesofquadraticvariations,andmore
generally multipower variations, areofcentral importance inthestudy ofambit fieldsandprocesses,
in particular as these objects relate to the volatility/intermittency. We will review the main results in
thatcontextinSection2.6andreferto[17],[8]and[9]formoredetails.
Section2.7containssomeapplicationsofambitprocessestoturbulence(Section2.7.1)andenergy
finance(Section2.7.2),respectively.
2.2 Ambitfields andprocesses
Generallywethinkofambitfieldsasbeingoftheform
Y (x) = µ + g(ξ,s;x,t)σ (ξ)L(dξ,ds) + q(ξ,s;x,t)a (ξ)dξds. (1)
t s s
ZAt(x) ZDt(x)
whereA (x),andD (x)areambitsets,gandqaredeterministicfunction, σ 0isastochasticfield
t t
≥
referredtoastheintermittency orvolatility, andLisaLe´vybasis,definedasfollows(see[20],[36]):
Let (Rk)betheBorelsetsofRk anddenote (S)thebounded BorelsetsofS (Rk).
b
B B ∈B
Definition 1. A family Λ(A) : A (S) of random vectors in Rd is called an Rd–valued Le´vy
b
{ ∈ B }
basisonS ifthefollowingthreeproperties aresatisfied:
1. ThelawofΛ(A)isinfinitely divisible forallA (S).
b
∈ B
2. IfA ,...,A aredisjointsubsets in (S),thenΛ(A ),...,Λ(A )areindependent.
1 n b 1 n
B
3. IfA ,A ,...aredisjointsubsets in (S)with ∞ A (S),then
1 2 Bb i=1 i ∈ Bb
∞ ∞S
Λ A = Λ(A ),a.s.,
i i
!
i=1 i=1
[ X
wheretheconvergence ontherighthandsideisa.s..
Conditions (2) and (3) define an independently scattered random measure. Note that we use Λ
when we refer to a general Le´vy basis, and when we have separated out time as one dimension, we
talk of Le´vy bases defined on S = R and weindicate integration with respect to such bases by
X ×
L(dξ,ds).
Inference on the volatility/intermittency field σ is a focal point for the study of ambit processes
and fields. Often the volatility field (or the logarithmic volatility field) will itself be defined as an
ambitfieldthrough
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2 AMBITPROCESSES
σ2(x) = h(ξ,s;x,t)L(dξ,ds), (2)
t
ZCt(x)
with h a positive function, C (x) some ambit set and where L is a nonnegative non–Gaussian Le´vy
t
basis.
At the present level of generality we take the integrals in (1) to be defined in the sense of inde-
pendentlyscatteredrandommeasures,cf. [38],assumingthatg,σ,qandaaresufficientlyregularfor
the integrals to exist. However, in more concrete cases it is often of interest to establish whether the
definition of the integrals can besharpened toa moredynamical version, for instance in thesense of
Itoˆ–type integrals. Wereturntothisquestion later, seeinparticular Sections 3.4and4.
Of particular interest are ambit processes that are stationary in time and nonanticipative. More
specifically, theymaybederivedfromambitfieldsY oftheform
Y (x) = µ + g(ξ,t s;x)σ (ξ)L(dξ,ds) + q(ξ,t s;x)a (ξ)dξds. (3)
t s s
− −
ZAt(x) ZDt(x)
Herethe ambit sets A (x), and D (x)are taken tobe homogeneous and nonanticipative, i.e. A (x)
t t t
isoftheformA (x)= A+(x,t)whereAonlyinvolvesnegativetimecoordinates, andsimilarlyfor
t
D (x). Further,weassumethatg(ξ,τ;x) = 0andq(ξ,τ;x) = 0forallτ < 0.
t
Remark Recallfrom[12;36]thateveryLe´vybasisLexhibitsaLe´vy–Itoˆ decomposition. LetN de-
notethePoissonbasisassociatedwiththeLevybasisLthroughsuchadecompositionandletν denote
theintensitymeasureofN. Clearly,wehaveE(N(dx;dξ,ds)) = ν(dx;dξ,ds). Inthefollowing,we
areinterestedinhomogeneous Le´vybases,i.e.Le´vybaseswhichsatisfyν(dx;dξ,ds) = ν(dx;dξ)ds
forameasureν.
e
Remark Many prominent tempo–spatial models are constructed from an ordinary, partial or frac-
e
tional differential equation by adding a noise term, for instance in the form of white noise, to the
equation. Thesolutiontotheequationthenbeingoftenrepresentable asanintegralwithrespecttothe
noiseoftheGreen’sfunctionoftheoriginaldeterministicdifferential equation(see[3;24]). Thusthe
solutionistakingtheformofanambitprocess. Forsomeexampleswithdiscussion, seeSections 3.5
and4.2.
Note that, in general, ambit processes involve time varying ambit sets and allow for a stochastic
volatility factor. Such stochastic volatility is important in many areas in science, not only in the
contextsofturbulence andfinancewhichareinfocusinthispaper.
ForunderstandingthenatureofambitprocessesX = Y (x(θ)),andasasteptowardshandling
θ t(θ)
questions of inference on σ, it is useful to discuss the cores of Y and X. With the ambit field given
by(1),thecoresY andX ofY andX aredefined,respectively, by
◦ ◦
Y (x)= g(ξ,s;x,t)L(dξ,ds),
◦t
ZAt(x)
and
X = g(ξ,s;τ (θ))L(dξ,ds),
◦θ
ZA(θ)
where,asabove,τ (θ)= (x(θ),t(θ))andwherewehaveusedA(θ)asashorthandforA (x(θ)).
t(θ)
IncasetheLe´vybasisListheWienerbasisW wespeakofaGaussiancore.
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2 AMBITPROCESSES
Remark A class of processes having some properties common with one–dimensional ambit pro-
cesses is studied in [44] under the name mixed moving averages. More precisely the authors study
processes X = (Xt)t∈R oftheform
X = f(x,t s)Λ(dx,ds), (4)
t
ZX×R −
where is a non–empty set and Λ is a symmetric α–stable (SαS) random measure on R with
X X ×
Le´vymeasureν leb,wherelebistheLebesguemeasureandν isaσ–finitemeasureon . Notethat
× X
suchprocessesarealwaysstationary. IntheSαSnon–Gaussiancase,theyshowthatthisisthesmallest
class containing all superpositions and weak limits of ordinary SαS moving averages. Furthermore,
Rosinski [39] has obtained a Wold–Karhunen type decomposition of stationary SαS non–Gaussian
processes in which mixed moving averages play a role similar to ordinary moving averages in the
Gaussiancase. Andin[40]thistypeofresultisextended toabroadrangeofnon–Gaussian infinitely
divisible processes.
2.3 Null–spatialcase: Le´vy Semistationary Processes ( )
LSS
When the space consists of asingle point (or wejust consider Y (x)of (1)in its dependence on t
t
X
keeping xfixed) the concept of ambit processes specialises to that of Le´vy Semistationary Processes
( ),introduced in[5],whichareprocesses Y = Y oftheform
LSS { t}t∈R
t t
Y = µ+ g(t s)σ dL + q(t s)a ds, (5)
t s s s
− −
Z−∞ Z−∞
whereµisaconstant,LisaLe´vyprocess,gandqarenonnegativedeterministicfunctionsonR,with
g(t) = q(t)= 0fort 0,andσ andaareca`dla`gprocesses. Whenσandaarestationary, aswewill
≤
require henceforth, then so is Y. Hence the name Le´vy semistationary processes. It is convenient to
indicate theformulaforY as
Y = µ+g σ L+q a leb, (6)
∗ • ∗ •
wherelebdenotesLebesguemeasure.
Generally we have taken the stochastic integrals as defined in the sense of [38]. However, in the
present case, of processes, onemaydefinetheintegrals intheItoˆ sense, relative tothefiltration
LSS
L generated by the increments L L , < s t < . Here we adopt the latter definition,
t s
F − −∞ ≤ ∞
notingthatthetwoversions agreewithrespect toallfinitedimensional distributions.
WhenL = B informula(5)forastandardBrownianmotionB,thenY specialisestoaBrownian
Semistationary Process( ),introduced in[17]. TheGaussiancoreofa processis
BSS BSS
t
Y = g(t s)dB . (7)
◦t s
−
Z−∞
We consider the processes to be the natural analogue, for stationarity related processes, of
BSS
theclass ofBrowniansemimartingales
BSM
t t
Y = σ dB + a ds.
t s s s
Z0 Z0
Alreadyinthisnull–spatial casethequestionofdrawinginference onσ2 ishighlynontrivial. The
maintoolismultipowervariation, see[8]and[9].
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2 AMBITPROCESSES
2.4 Keyexamplefora process
BSS
Anexampleofparticular interestinthecontextof processes iswhere
BSS
g(t) = tν−1e−λt, for t (0, ), (8)
∈ ∞
forsomeλ> 0andwithν > 1. Thelatterconditionisneededtoensuretheexistenceofthestochastic
2
integralin(7).
Remark For the key example (8) the derivative g′ of g is not square integrable if 1 < ν < 1 or
2
1 < ν 3; hence, in these cases Y is not a semimartingale. For 1 < ν < 1 we have g(0+) =
≤ 2 2 ∞
while g(0+) = 0 when 1 < ν 3. These two cases are radically different in nature. Of course,
· ≤ 2
for ν = 1 the process Y = g( s)σ B(ds) is simply a modulated version of the Gaussian
−∞ · − s
Ornstein–Uhlenbeck process, and in particular, a semimartingale. Note also that when ν > 3 then
R 2
Y is of finite variation and hence, trivially, a semimartingale. To summarise, the nonsemimartingale
casesareν 1,1) (1, 3 .
∈ 2 ∪ 2
(cid:0) (cid:3)
2.5 Generality of
BSS
Asamodelling frameworkforcontinuous timestationary processes thespecification (6)isquitegen-
eral. Infact,thecontinuoustimeWold–Karhunendecompositionsaysthatanysecondorderstationary
stochastic process,possiblycomplexvalued,ofmean0andcontinuous inquadratic meancanberep-
resented as
t
Z = φ(t s)dΞ +V , (9)
t s t
−
Z−∞
wherethedeterministic functionφisaningeneralcomplex,deterministic squareintegrablefunction,
the process Ξ has orthogonal increments with E dΞ 2 = ̟dt for some constant ̟ > 0and the
t
| |
process V is nonregular (i.e. its future values cannbe predoicted, in the L2 sense, by linear operations
onpastvalueswithouterror).
Under the further condition that t∈Rsp Zs :s t = 0 , the function φ is real and uniquely
∩ { ≤ } { }
determined up to a real constant of proportionality; and the same is therefore true of Ξ (up to an
additiveconstant).
In particular, if dΞ = σ dB with σ and B as in (6), then Ξ is of the above type with ̟ =
s s s
E σ2 .
0
(cid:8) (cid:9)
2.6 Multipower variations
One of the interesting aspects in the context of models is the question on how to estimate the
BSS
stochastic volatility σ and how to make inference on it. A key tool for tackling this question is a
statistic calledrealised varianceand,moregenerally, realisedmultipowervariation.
Arealisedmultipowervariationofastochastic processX isanobjectofthetype
[nt]−k+1 k
∆n X pj, (10)
| i+j−1 |
i=1 j=1
X Y
where ∆niX = Xni −Xi−n1 and p1,...,pk ≥ 0. I.e. it is assumed that the process X = (Xt)t≥0 is
observed at times iδ, where δ = 1 and i = 0,1,...,[nt]. These concepts have been developed in
n
7
2 AMBITPROCESSES
the context of financial times series, seee.g. [10; 11; 18;19; 21]forresults inaframework based on
Brownian semimartingales. In the presence of jumps, these quantities have been studied by [32; 33]
and [45]. A detailed survey on this aspect is also given by [2]. However, in the non–semimartingale
setup theunderlying theory ismuchmore involved. Wejust sketch the mainresults here briefly and
referto[17],[8]and[9]formoredetails.
Considerafilteredprobabilityspace(Ω, ,( ) ,P),assumingtheexistencethereonofa
t t≥0
F F BSS
processY definedasin(5),whereL = B isastandardBrownianmotion. LetGdenotetheGaussian
coreofY asdefinedin(7),i.e.
t
G =Y = g(t s)dB ,
t ◦t s
−
Z−∞
and let bethe σ–algebra generated by G. Thecorrelation function of the increments of Gis given
G
by
∆nG ∆n G R¯(j+1) 2R¯(j)+R¯(j−1)
r (j) = cov 1 , 1+j = n − n n .
n τ τ 2τ2
(cid:18) n n (cid:19) n
Next, we introduce a class of measures that is crucial for establishing an asymptotic theory for
realisedmultipowervariations. Wedefine
(g(x δ) g(x))2dx
π (A) = A − − , y 0,
δ ∞(g(x δ) g(x))2dx ≥
R0 − −
andwefurthersetπ (x) = π ( y :Ry > x ). Notethatπ isaprobability measureonR .
δ δ δ +
{ }
Weareinterested intheasymptotic behaviour ofthenormalised multipowervariations
[nt]−k+1 k
1
V¯(Y,p ,...,p )n = ∆n Y pj,
1 k t nτp+ | i+j−1 |
n
i=1 j=1
X Y
wherep = k p andτ2 = R¯(1/n)withR¯(t) = E[G G 2],t 0.
+ j=1 j n | s+t− s| ≥
Inordertoestablishaweaklawoflargenumbers, oneneedsthefollowingassumption.
P
(LLN):Thereexistsasequence r(j)with
n−1
1
r2(j) r(j), r(j) 0.
n ≤ n →
j=1
X
Moreover, itholdsthat
lim π (ε) = 0,
δ
n→∞
foranyε > 0.
Thenthelawoflargenumbersisgivenbythefollowingproposition.
Proposition 1. Assumethatthecondition (LLN)holdsforY = g σ W +q a leb. Define
∗ • ∗ •
∆nG p1 ∆nG pk
ρ(n) = E 1 k .
p1,...,pk τ ··· τ
(cid:20)(cid:12) n (cid:12) (cid:12) n (cid:12) (cid:21)
(cid:12) (cid:12) (cid:12) (cid:12)
Thenwehave (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) t(cid:12) (cid:12)
V¯(Y,p ,...,p )n ρ(n) σ p+ds ucp 0,
1 k t − p1,...,pk | s| −→
Z0
wheretheconvergence isuniform oncompactsinprobability (ucp).
8
2 AMBITPROCESSES
Furthermore, foracentrallimittheorem, oneneedsthefollowingassumption.
(CLT):Assumption(LLN)holds,and
r (j) ρ(j), j 0,
n
→ ≥
where ρ(j) is the correlation function of some stationary centered discrete time Gaussian process
(Q ) withE[Q2] = 1(asbefore). Moreover,foranyj,n 1,thereexistsasequence r(j)with
i i≥1 i ≥
∞
r2(j) r(j), r(j) < .
n ≤ ∞
j=1
X
Finally,thetailmassfunction πn isassumedtosatisfyanadditional mildcondition.
Now,wecanformulateajointcentral limittheorem forafamily
(V¯(Y,pj,...,pj)n) ofmultipowervariations asfollows.
1 k t 1≤j≤d
Proposition 2. Assume that the process σ is –measurable and the condition (CLT) holds. Then
G
weobtainthestableconvergence
t t
√n(cid:16)V¯(Y,pj1,...,pjk)nt −ρ(pnj1),...,pjkZ0 |σs|pj+ds(cid:17)1≤j≤d G−−→stZ0 Zs1/2dBs,
whereBisad–dimensionalBrownianmotionthatisdefinedonanextensionofthefilteredprobability
space(Ω, ,( ) ,P)andisindependent of ,andZ isad d–dimensional processgivenby
t t≥0
F F F ×
Zsij = βij|σs|pi++pj+ , 1≤ i,j ≤ d,
wherethed dmatrixβ isdefinedasin[8].
×
Notethatinordertoobtainanasymptotic limittheory forawiderange ofmultipowervariations,
one is forced to consider also multipower variations of second order differences. (For Brownian
semimartingales passing to second order differences would make no essential change in the limit
theory.) Multipower variations based on second order differences are quantities having the same
formas(10)butusing
3nX = X 2X +X ,
j jδ − (j−1)δ (j−2)δ
instead of ∆nX. However, we shall not dwell on this aspect here, but refer to [7; 9] for discussions,
j
detailed resultsandapplications.
2.7 Applications toturbulence andfinance
After having introduced the concept of ambit fields and ambit processes, we turn our attention to
applications ofsuchprocesses inturbulence andinfinance.
2.7.1 Tempo–spatialsettingsinturbulence
The idea of ambit processes arose out of a project aimed at establishing realistic stochastic models
of the velocity fields in stationary turbulent regimes (cf. [6; 12] and also [13–17]). In turbulence the
basicnotionofintermittencyreferstothefactthattheenergyinaturbulentfieldisunevenlydistributed
in space and time, and the paper [12] introduced stochastic models for turbulent intermittency (also
9