Ambit fields: a stochastic modelling approach

ABSTRACT

This paper reviews some recent developments in the field of ambit stochastics with focus on modelling applications. The great flexibility and mathematical tractability of ambit fields is illustrated by a wide range of applications, covering such diverse areas as turbulence, tumour growth and finance.

KEYWORDS:

• Stochastic modelling
• turbulence physics
• financial mathematics
• biological processes

1 Introduction

Historically, the term ambit field or ambit stochastics has been coined for the first time in [1] in connection with modelling of some of the statistical aspects of turbulent flows. The term ambit refers to a region of influence in space-time that is at most able to influence a given quantity at a certain location in space-time. This region is called the associated ambit set. Ambit fields are then constructed as integrals over these ambit sets with respect to a suitable random measure and also including a possible additional source of randomness termed the volatility or intermittency field.

From the modelling point of view, a main goal of ambit stochastics is to explicitly construct random fields that have some given statistical properties. The present review discusses this approach, focussing on the rich potential of ambit fields to model a wide range of phenomena that include turbulence, finance and growth modelling and some more general questions of modelling random vector fields. Besides its relevance for applications, ambit stochastics developed into a very active research area from the purely mathematical side. However, here we will only present the mathematics that are necessary for the applications. A comprehensive account of the more theoretical issues of ambit stochastics may be found in [2]. Chapter 2 in [2] also provides a discussion of some aspects of the numerical simulation of ambit fields (see also [3]). Related topics on efficiency of numerical recipes are further discussed in [4,5].

This paper is organized as follows. In Section 2, ambit fields are defined together with the basic notion of a Lévy basis. More mathematical details are provided in the subsequent Sections, whenever needed for applications. A first of these applications is the modelling of continuous cascade processes in Section 3.1, in particular the modelling of the turbulent energy dissipation which serves as a prototype of such a cascading structure. Section 3.2 discusses, in more general terms, the application of ambit stochastics to the modelling of space-time symmetric fields. Based on some of the results presented in Section 3.1, we illustrate in Section 3.3 how cascade models can be used to construct realistic time series for turbulent velocities. A generalization from the time domain to the spatial domain is presented in Section 3.4 which focuses on the construction of random vector fields with a prescribed correlation structure. Sections 3.5 and 3.6 briefly discuss some of the applications of ambit fields to the modelling of financial markets. As the last application, growth modelling is briefly presented in Section 3.7. Finally, in Section 4, the relation between ambit fields and stochastic partial differential equations is briefly outlined. Section 5 concludes.

The topics presented in this review are chosen to give the reader an impression of the great potential of ambit fields to model a wide range of natural phenomena. A guidance in the selection of the topics is the authors own interest which necessarily implies that this overview is incomplete.

2 Background

This Section provides definitions that are basic for the construction of ambit fields. The main ingredients are Lévy bases and the associated spot variables which are essential for all applications discussed in Section 3. The notion of an ambit field and that of exponentiated ambit fields is discussed in a broad sense. Specific properties of specific realizations are provided together with the applications in Section 3.

2.1 Lévy basis

The basic notion for the definition of ambit fields is that of a Lévy basis on ${\mathbb{R}}^k$. A family of random vectors $\left\{L(A):A\in B_b(S)\right\}$ in ${\mathbb{R}}^d$ is called an ${\mathbb{R}}^d$-valued Lévy basis on $S$ if

1. the law of $L(A)$ is infinitely divisible for all $A\in B_b(S)$,

2. $L(A_1),\dots ,L(A_n)$ are independent for disjoint subsets $A_1,\dots ,A_n\in B_b(S)$,

3. for disjoint subsets $A_1,A_2,\dots \in B_b(S)$ with ${\cup }_{i=1}^{\mathrm{\infty }}{A}_i\in B_b(S)$, we have:

   $L\left({\cup }_{i=1}^{\mathrm{\infty }}{A}_i\right)=\sum _{i=1}^{\mathrm{\infty }}L(A_i)a.s.$

Here, $B_b(S)$ denotes the bounded Borel sets of $S\subseteq {\mathbb{R}}^k$.

For each $s\in S$, there is an associated ${\mathbb{R}}^d$-valued random variable $L^{\prime }(s)$, called the spot variable, such that the cumulant-generating function of $L(A)$, $A\in B_b(S)$ can be expressed as:

(1) $C(z,L(A))=logE\left[e^{⟨z,L(A)⟩}\right]={\int }_{A}C(z,L^{\prime }(s))\lambda (\mathrm{d}s)$(1)

where $\lambda$ is the control measure of the Lévy basis (see [5] and references therein for more details). The Lévy basis is called homogeneous if the distribution of $L^{\prime }(s)$ does not depend on $s$ and if $\lambda$ is the Lebesgue measure on $S$.

For a homogeneous Lévy basis, it follows that:

(2) $C\left(z,{\int }_{A}f\sigma L(\mathrm{d}s)\right)=logE\left[exp\left({\int }_{A}C\left({\left(f(s)\sigma (s)\right)}^{T}z,L^{\prime }(s)\right)\mathrm{d}s\right)\right]$(2)

for a field $\sigma$ on $S$, independent of the Lévy basis $L$ and a deterministic function $f$ [6].

The starting point for the concept of integration in (2) is that of integration of deterministic functions $f$ defined as the limit in probability

${\int }_{A}f(s)L(\mathrm{d}s)=\underset{j\to \mathrm{\infty }}{lim}{\int }_A{f}_j(s)L(\mathrm{d}s),$

where $(f_j{)}_{j=1}^{\mathrm{\infty }}$ is a sequence of simple functions (linear combinations of indicator functions) such that $f_j\to f$ $\lambda$-a.s. and $({\int }_A{f}_{j}L(\mathrm{d}s){)}_{j=1}^{\mathrm{\infty }}$ converges in probability. For a field $\sigma$ that is independent of $L$, ${\int }_{S}f\sigma L(\mathrm{d}s)$ is then defined conditional on $\sigma$, assuming that $f\sigma$ is a.s. $L$-integrable. Theorem 2.7 in [6] provides necessary and sufficient conditions for $f$ being $L$-integrable in terms of the characteristics of $L$. There it is also shown that the space of integrable functions is a Musielak–Orlicz modular space.

2.2 Ambit fields

Now, let $A(s)\subset S$ and $D(s)\subset S$ be subsets of $S$, labelled by $s\in S$. We may think of $A(s)$ and $D(s)$ as being attached to the points $s$ and we call these sets the (associated) ambit sets (attached to $s$ ). An ${\mathbb{R}}^d$-valued ambit field is then defined as:

(3) $Y(s)={\int }_{A(s)}g(s,z)\sigma (z)L(\mathrm{d}z)+{\int }_{D(s)}q(s,z)a(z)\mathrm{d}z.$(3)

Here, $g$ and $q$ are deterministic functions and $\sigma ,a$ are stochastic fields such that $g\sigma L$ and $qa$ have values in ${\mathbb{R}}^d$. The integral may be defined in the sense of integration with respect to an independently scattered random measure [6], and we assume that $g,q,\sigma ,a$ are suitable for the integral to exist.

In what follows, we restrict attention to the case where $S\subset \mathbb{R}\times {\mathbb{R}}^k$ and interpret $(t,x)\in S$ as being composed of time $t$ and spatial position $x$. Furthermore, we will mostly concentrate on stationary (translational invariant in time) and homogeneous (translational invariant in space) ambit fields of the form

(4) $Y_t(x)={\int }_{A(t,x)}g(t-s;x-\xi ){\sigma }_s(\xi )L(\mathrm{d}\xi ,\mathrm{d}s)+{\int }_{D(t,x)}q(t-s;x-\xi )a_s(\xi )\mathrm{d}\xi \mathrm{d}s,$(4)

where the fields $\sigma ,a$ and the ambit sets $A(t,x),D(t,x)$ are stationary and homogeneous, $A(t,x)=(t,x)+A$, $D(t,x)=(t,x)+D$ and the Lévy basis $L$ is assumed to be homogeneous. To include causality, we also require that $A,D\subset [-\mathrm{\infty },0]\times {\mathbb{R}}^k$. Assuming that $\sigma$ has finite second moments, then the stochastic integral in (4) exists whenever $g$ is absolutely-integrable and square-integrable over $A(t,x)$.

Of particular interest for the modelling of time series of scalar fields are Brownian semistationary (BSS) processes [7]. BSS processes are one-dimensional ambit fields of the form:

(5) $Z_t=\mu +{\int }_{-\mathrm{\infty }}^{t}g(t-s){\sigma }_s\mathrm{d}{W}_s+{\int }_{-\mathrm{\infty }}^{t}q(t-s)a_s\mathrm{d}s,t\in \mathbb{R}$(5)

where $\mu$ is a constant, $(W_t{)}_{t\in \mathbb{R}}$ is a standard Brownian motion on $\mathbb{R}$, $g$ and $q$ are nonnegative deterministic functions on ${\mathbb{R}}_{+}$ and $({\sigma }_t{)}_{t\in \mathbb{R}}$ and $(a_t{)}_{t\in \mathbb{R}}$ are cadlag processes. When $(\sigma ,a)$ is stationary and independent of $W$, then $Z$ is stationary. Replacing $W$ by a general Lévy process leads to the notion of a Lévy semistationary process which is of particular interest in financial applications [8].

2.3 Exponentiated ambit fields

Exponentiated ambit fields provide a useful and flexible modelling framework for fields of a multiplicative character. In its simplest version, an exponentiated ambit field is defined as:

(6) $e_t(x)=exp\left\{{\int }_{A(t,x)}h(x,t;\xi ,s)L(\mathrm{d}\xi ,\mathrm{d}s)\right\},$(6)

where $L$ is a $\mathbb{R}$-valued homogeneous Lévy basis with spot variable $L^{\prime }$ and $A(t,x)=(t,x)+A$ for some bounded set $A\in B_b({\mathbb{R}}^2)$. We also assume that the $\mathbb{R}$-valued deterministic kernel (suitable for the integral to exist) fulfils $h(x,t;\xi ,s)=h(x-\xi ,t-s)$. Thus, $e$ is stationary and homogeneous.

The discussion below is based on the fundamental relation (see also (1)):

(7) $\mathrm{E}\left\{exp\left\{{\int }_{A}h(a)L(\mathrm{d}a)\right\}\right\}=exp\left\{{\int }_{A}K[h(a)]\mathrm{d}a\right\},$(7)

where $K$ denotes the cumulant-generating function of the spot variable $L^{\prime }$ (provided it exists), following the relation:

(8) $log\mathrm{E}\left\{exp\left\{s\mathrm{L}(\mathrm{d}a)\right\}\right\}=K[s]\mathrm{d}a.$(8)

To ensure causality, the ambit set $A(t,x)$ is assumed to only extend to the past, before time $t$. Furthermore, it is assumed that there is a finite decorrelation time $T_0$ and that the ambit set is symmetrically bounded by a decreasing function $r\ge 0$

(9) $A(t,x)=(t,x)+A=(t,x)+\left\{(s,\xi ):-T_0\le s\le 0,-r(s+T_0)\le \xi \le r(s+T_0)\right\}.$(9)

The correlation structure of the field $e$ can conveniently be characterized by so-called correlators defined as:

(10) $c_{n_1,\dots ,n_p}(t_1,x_1;\dots ;t_p,x_p)=\frac{\mathrm{E}\left\{e_{t_1}{(x_1)}^{n_1}{e}_{t_2}{(x_2)}^{n_2}\cdots e_{t_p}{(x_p)}^{n_p}\right\}}{\mathrm{E}\left\{e_{t_1}{(x_1)}^{n_1}\right\}\cdots \mathrm{E}\left\{e_{t_p}{(x_p)}^{n_p}\right\}}.$(10)

Of particular interest for applications is the case $h\equiv 1$. In this case, it follows immediately from (7) that:

$c_{n_1,n_2}(t_1,x_1;t_2,x_2)=exp\left\{\bar{K}[n_1,n_2]{\int }_{A(t_1,x_1){\cap }^A(t_2,x_2)}\mathrm{d}x\mathrm{d}t\right\}$

$=exp\left\{\bar{K}[n_1,n_2]\mathrm{V}\mathrm{o}\mathrm{l}(A(t_1,x_1){\cap }^A(t_2,x_2))\right\}$

where $\mathrm{V}\mathrm{o}\mathrm{l}$ denotes the Euclidean volume and where $\bar{K}[n_1,n_2]=K[n_1+n_2]-K[n_1]-K[n_2]$. The important fact here is the factorization into the volume of the overlap of the associated ambit sets and a factor $\bar{K}$ depending only on the spot variable $L^{\prime }$. Thus, we get the so-called self-scaling relation between correlators of different orders [9]:

(11) $c_{n_1,n_2}(t_1,x_1;t_2,x_2)=c_{m_1,m_2}(t_1,x_1;t_2,x_2{)}^{k[m_1,m_2;n_1,n_2]}$(11)

with

(12) $k[m_1,m_2;n_1,n_2]=\frac{\bar{K}[n_1,n_2]}{\bar{K}[m_1,m_2]}.$(12)

It also follows that the boundary function $r$ satisfies

(13) $\frac{\mathrm{\partial }}{\mathrm{\partial }l}log{c}_{n_1,n_2}(t,x;t,x+l)=-\bar{K}[n_1,n_2]r^{(-1)}\left(\frac{l}{2}\right)$(13)

and

(14) $\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{l}^2}log{c}_{n_1,n_2}(t,x;t,x+l)=-\bar{K}[n_1,n_2]\frac{\mathrm{\partial }}{\mathrm{\partial }l}{r}^{(-1)}\left(\frac{l}{2}\right)$(14)

where $r^{(-1)}$ denotes the inverse of $r$. Thus, the model (6) together with a finite ambit set of the form (9) bounded by a decreasing function $r$ and a constant kernel $h$ allows to reproduce any spatial correlations that fulfil:

(15) $\frac{\mathrm{\partial }}{\mathrm{\partial }l}log{c}_{n_1,n_2}(t,x;t,x+l)<0,\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{l}^2}log{c}_{n_1,n_2}(t,x;t,x+l)>0.$(15)

3 Applications

The applications discussed below comprise various areas of natural science ranging from turbulence to tumour growth and finance. However, it is important to note that ambit fields are not restricted to be applied only to these topics. In fact, the flexibility of ambit fields opens the door to a much broader applicability.

3.1 Cascade processes

Geometric cascade processes play a fundamental role for modelling strongly intermittent fluctuations, long-range correlations, multi-scale structuring and self-similarity. A prototype for the observation of such properties is the energy dissipation in a turbulent flow [10]. The small-scale intermittency of the energy dissipation is usually expressed in terms of multifractal scaling of inertial range statistics. Discrete geometrical cascade processes are maybe the simplest construction of such multifractal fields. Various generalizations of purely spatial and discrete processes towards continuous cascade processes in time and/or space formulated in terms of integrals of an uncorrelated noise field over an associated ambit set have been proposed in the literature [9,11–21]. Ambit fields provide a unifying framework that captures the main characteristics of continuous cascades in space and time.

The following discussion focusses on the modelling of the statistical properties of the turbulent energy dissipation ${ϵ}_t(x)$ in one-dimensional space $x\in \mathbb{R}$. In homogeneous, stationary and isotropic flow regimes and under the assumption of Taylor’s Frozen Flow hypothesis (TFFH) [22] (see Section 3.2), the statistics of the energy dissipation in terms of scaling relations at a fixed position in space are believed to be the same as the statistics at a fixed instant in time. These scaling relations refer to the behaviour of correlators and to the behaviour of the coarse-grained field amplitudes.

The basic observation is that two-point correlators follow (approximate) scaling relations with scaling exponents $\tau$ for a certain scaling range $I_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}}=[l_{\mathrm{m}\mathrm{i}\mathrm{n}},l_{\mathrm{m}\mathrm{a}\mathrm{x}}]$ with $0<l_{\mathrm{m}\mathrm{i}\mathrm{n}}<l_{\mathrm{m}\mathrm{a}\mathrm{x}}$

(16) $c_{n_1,n_2}(t,x;t,x+l)\propto l^{-\tau (n_1,n_2)},l\in I_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}}.$(16)

Such a scaling relation trivially implies self-scaling of correlators. However, for the energy dissipation, this self-scaling property is observed for a much extended scaling range, exceeding the scales $I_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}}$ [9].

Another important observation is linked to classical multifractality in the sense of scaling of coarse-grained moments [10,12]:

(17) $M_n(t,\sigma ,l)=\mathrm{E}\left\{{\left(\frac{1}{l}{\int }_{\sigma -l/2}^{\sigma +l/2}{\epsilon }_t(\xi )\mathrm{d}\xi \right)}^n\right\}\propto l^{-\mu (n)}$(17)

for $l_{\mathrm{m}\mathrm{i}\mathrm{n}}\ll l\le l_{\mathrm{m}\mathrm{a}\mathrm{x}}$.

To account for these scaling laws, the exponentiated ambit field

(18) ${ϵ}_t(x)=exp\left\{{\int }_{A(t,x)}L(\mathrm{d}\xi ,\mathrm{d}s)\right\}$(18)

is proposed in [21], where $L$ is a homogeneous Lévy basis with a normal inverse Gaussian spot variable and where the finite ambit set of the type (9) is bounded by:

(19) $r(t)={\left(\frac{1-{(t/T_0)}^{\theta }}{1+{(kt/T_0)}^{\theta }}\right)}^{1/\theta },0\le t\le T_0$(19)

where $\theta ,k,T_0$ are tunable parameters.

The model (18) has been applied in [21] to model the energy dissipation for various turbulent data sets from a helium jet experiment. All model parameters were estimated from data. It is shown that this set-up reproduces the approximate scaling relations for correlators and the empirically observed normal inverse Gaussian distribution for $log\epsilon$. It is important to note that the special choice of $r$ in (19) implies that the correlations at a fixed position in space, as a function of time, are identical to those at a fixed instant in time, as a function of the spatial distance (see also Section 3.2).

Within the modelling framework (18), it can be shown that all multi-point correlators can be expressed solely in terms of two-point correlators (in fact this is not restricted to the special choice of $r$ in (19)) [12,13]. Restricting to the case where two-point correlators show a strict scaling behaviour, by properly choosing the ambit set [12], we get for $x_1<x_2<\dots <x_n$ and $x_{i+1}-x_i\in I_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}}$, $i=1,\dots ,n-1$

(20) $\mathrm{E}\left\{{ϵ}_t{(x_1)}^{m_1}\cdots {\epsilon }_t{(x_n)}^{m_n}\right\}$(20)

$\propto \left(\prod _{i=1}^{n-1}{\left(x_{i+1}-x_i\right)}^{-\tau (m_i,m_{i+1})}\right)\prod _{j=2}^{n-1}\prod _{l=j+1}^n{\left(x_l-x_{l-j}\right)}^{-\xi (m_{l-j},\dots ,m_l)},$

where

$\xi (m_{l-j},\dots ,m_l)=\tau (m_{l-j}+\dots +m_{l-1},m_l)-\tau (m_{l-j+1}+\dots +m_{l-1},m_l).$

Equation (20) expresses the fact that $n$-point correlations factorize into contributions arising at the smallest scales $x_{i+1}-x_i$, at next-to-smallest scales $x_{i+2}-x_i$, and so on up to the largest scale, $x_n-x_1$.

For $n=3$, the relation (20) has been verified for turbulent data in [13]. Based on (20), it can be shown that (17) holds where

$\mu (n)=\frac{\tau (2)}{K[2]-2K[1]}\left(K[n]-nK[1]\right).$

3.2 Space-time symmetry

Ambit fields allow to construct spatio-temporal stochastic fields that obey exact symmetry in space and time [23], i.e. the field amplitude considered as a stochastic process in time at a fixed position in space is identical, as a stochastic process, to the field amplitude considered as a stochastic process in space at a fixed time point. Such fields are of particular importance in connection to turbulence modelling and the TFFH [22] .

The application of TFFH is a standard tool in turbulence studies where it is used to interpret time series as spatial recordings, see e.g. [10]. The basic idea of converting spatial scales to temporal scales has also been applied to a great variety of other phenomena in natural sciences ranging from rain field measurements and modelling to the interpretation of galactic turbulence [24–28].

A basic example of a space-time symmetric field within the ambit framework is discussed in [23], where the spatio-temporal field amplitude $Y_t(x)$ is defined as:

(21) $Y_t(x)=\mu +{\int }_{{\mathbb{R}}^2}g(t-s,x-\xi ){\sigma }_s(\xi )W(\mathrm{d}s,\mathrm{d}\xi ),$(21)

and where $t$ denotes time, $x$ denotes position in one-dimensional space, $\mu =\mathrm{E}\left\{Y_t(x)\right\}$ is the mean amplitude, $g$ is a deterministic kernel (suitable for the integral to exist) and $W$ is a homogeneous Lévy basis with standard Gaussian spot variable. The stochastic field $\sigma$ is assumed to be independent of $W$, stationary in time and homogeneous in its spatial variable which implies stationarity and homogeneity for $Y$. The field $\sigma$ allows the distribution of $Y$ and that of its increments to be strongly non-Gaussian with intermittent amplitudes, as has been discussed in [1,29].

The model ingredients $g$ and $\sigma$ can be specified such that $Y$ behaves for fixed $t$ as a process in $x$ stochastically identical to $Y$ for a fixed $x$ as a process of $t$, i.e $Y_t(0)$ is stochastically identical (as a stochastic process) to $Y_0(t)$. This can be achieved by assuming a symmetric kernel:

(22) $g(t,x)=g(x,t)$(22)

and by the requirement that ${\sigma }_t(0)$ and ${\sigma }_0(t)$ are stochastically identical processes. An explicit example is given by an exponentiated ambit field (see Section 3.1):

(23) ${\sigma }_t(x)=exp\left\{L(A(t,x))\right\},$(23)

where $A(t,x)=(t,x)+A_0$ is translational invariant and $L$ is a homogeneous Lévy basis. The assumption of a symmetric ambit set $A_0$ with respect to the identity line $x=t$ implies that

(24) $A(0,0)\cap A(0,x)=A(0,0)\cap A(t,0)$(24)

for all $x=t$ and consequently the processes ${\sigma }_t(0)$ and ${\sigma }_0(x)$ are stochastically identical (see also Section 3.1) implying that $Y_t(0)$ and $Y_0(t)$ are also stochastically identical.

Causality, in the sense that present field amplitudes only depend on past innovations, can easily be incorporated by the requirement $g(t,x)=0$ for $t\le 0$ (implying that $g(t,x)=0$ for $x\le 0$) and $A(t,x)\subset [-\mathrm{\infty },t]\times \mathbb{R}$.

The above modelling framework can straightforwardly be extended to $k$ spatial dimensions:

(25) $Y_t(x)=\mu +{\int }_{{\mathbb{R}}^{k+1}}g(t-s,x-\xi ){\sigma }_s(\xi )W(\mathrm{d}s,\mathrm{d}\xi ),$(25)

where $W$ is a Gaussian Lévy basis in $k+1$ dimensions. The intermittency field can be defined as in (23) with $A\subset {\mathbb{R}}^{k+1}$. Let $x=(x_1,x_2,\dots ,x_k)$ denote a vector in ${\mathbb{R}}^k$ and let $\bar{x}=(0,x_2,\dots ,x_k)$. Assuming $A_0$ to be symmetric with respect to the identity plane $x_1=t$ gives

(26) $A(0,x)\cap A(0,\bar{x})=A(0,\bar{x})\cap A(t,\bar{x}),$(26)

for all $x_1=t$ which implies that the processes ${\sigma }_0(x)$ and ${\sigma }_t(\bar{x})$ are stochastically identical. From the assumption

(27) $g(t,x_1,x_2,\dots ,x_k)=g(x_1,t,x_2,\dots ,x_n),$(27)

it follows that $Y_t(\bar{x})$ and $Y_0(x)$ are also stochastically identical.

3.3 Turbulent velocity time series

In [7,29,30], BSS processes (5) have been proposed as a class of stochastic models for time series of the turbulent velocity field. Here, stochastic modelling of the velocity field is understood as an explicit stochastic approach in contrast to an implicit set-up in terms of governing equations and/or in terms of related quantities like velocity increments or velocity derivatives. The BSS approach allows for analytic calculations and identification of the parameters of the model with physical quantities.

In [30], the important case of the BSS process (5) being a semimartingale is discussed and its relevance for turbulent time series is illustrated by a detailed analysis of experimental data. For the semimartingale property, it is sufficient to require that $\sigma$ and $a$ have finite second moments, $g,q\in L^1({\mathbb{R}}_{+}){\cap }^{L^2}({\mathbb{R}}_{+})$, $g^{\prime }\in L^2({\mathbb{R}}_{+})$ and $g(0+)<\mathrm{\infty }$ (see [30]). These properties are assumed to hold for the model (28) below.

The semimartingale model proposed in [30] for turbulent velocity time series $(v_t{)}_{t\in \mathbb{R}}$ is a special case of (5) and given by:

(28) $v_t=v_t(g,\sigma ,\beta )={\int }_{-\mathrm{\infty }}^{t}g(t-s){\sigma }_s\mathrm{d}{W}_s+\beta {\int }_{-\mathrm{\infty }}^{t}g(t-s){\sigma }_s^2\mathrm{d}s$(28)

where $\sigma$ is a stationary process independent of the standard Brownian motion $W$ with $\mathrm{E}\left\{{\sigma }^6\right\}<\mathrm{\infty }$ (needed for estimation purposes), and $\beta$ is a constant.

For any semimartingale $X$, the limit

(29) ${\left[X\right]}_t=\underset{n\to \mathrm{\infty }}{lim}\sum _{j=1}^n{\left(X_{j\frac{t}{n}}-X_{\left(j-1\right)\frac{t}{n}}\right)}^2$(29)

exists as a limit in probability. The derived process $\left[X\right]$ expresses the cumulative quadratic variation exhibited by $X$ and is called quadratic variation. For the semimartingale (28), we get

${\left(\mathrm{d}{v}_t\right)}^2=g^2(0+){\sigma }_t^2\mathrm{d}t$

and

(30) ${\left[v\right]}_t={\int }_0^t{\left(\mathrm{d}{v}_s\right)}^2=g^2(0+){\int }_0^t{\sigma }_s^2\mathrm{d}s.$(30)

In this setting, the quantity ${\left(\mathrm{d}{v}_t\right)}^2/\mathrm{d}t$ is the natural analogue of the squared first-order derivative of $v$ which in the classical formulation is taken to express the energy dissipation in turbulent flows obtained from velocity time series. Consequently, the quadratic variation $\left[v\right]$ is the stochastic analogue of the integrated energy dissipation and ${\sigma }^2$ can be identified with the energy dissipation. Following the discussion in Section 3.1, it is natural to assume that ${\sigma }^2$ is a cascade process of the form (18).

The parameters for the cascade process ${\sigma }^2$ are estimated in [21] from turbulent data using the marginal law, the correlator $c_{1,1}$ (10) and the observed self-scaling property (see Sections 2.3 and 3.1).The remaining parameters of the model (28) can be estimated from second- and third-order moments of velocity increments. In particular, the empirically observed spectral density function can, to very high accuracy, be reproduced by assuming a shifted convolution of gamma functions for the kernel $g$. The constant $\beta$ can be estimated in an iterative way from second- and third-order moments of velocity increments. Following this procedure, the model (28) is able to reproduce the main stylized features of turbulent velocity time series. The model correctly predicts higher order moments, the evolution of the density of velocity increments across scales and the main statistical properties of the so-called Kolmogorov variable which relates velocity increments to the coarse-grained energy dissipation process [10,29]. This is achieved without any tuneable parameters left.

3.4 Vector ambit fields

The applications discussed in the previous Sections focus on the construction of scalar fields. An application of ambit fields to construct vector fields with a prescribed correlation structure is outlined in [5]. The construction is based on using integration of deterministic, matrix-valued functions with respect to vector-valued, volatility modulated Lévy bases. The model is applied to reproduce the Shkarofsky correlation family (and a modification of it) which is shown to accurately fit the empirically observed spectral density function of the turbulent velocity in the atmospheric boundary layer. Furthermore, a simple algorithm is derived to decompose the simulation problem into computationally tractable subproblems. The procedure described there can be viewed as a more flexible generalization of the so-called Mann model [31] that is widely applied in the wind energy industry. There, the turbulent velocity field at a fixed instant in time is modelled as a homogeneous Gaussian field.

A ${\mathbb{R}}^m$-valued ambit field $X=\{X(x)|x\in \mathrm{\Xi }\}$ on $\mathrm{\Xi }\subset {\mathrm{R}}^n$ is defined as:

(31) $X(x)={\int }_{S}f(x,s)\sigma (s)L(\mathrm{d}s)$(31)

where $S\in B({\mathbb{R}}^n)$, $L$ is a ${\mathbb{R}}^d$-valued Lévy basis on $S$, $\sigma$ is a $\mathbb{R}$-valued field on $S$, independent of the Lévy basis, and $f:\mathrm{\Xi }\times S\to M_{m,d}$ is a function such that $x\mapsto f(x,s)\sigma (s)$ is almost surely $L$-integrable for every $x\in \mathrm{\Xi }$. Here, $M_{m,d}$ denotes the set of $m\times d$ matrices with real entries.

For applications to turbulence modelling, one can restrict the discussion to the special case where $\mathrm{\Xi }=S={\mathbb{R}}^n$, $L$ is a homogeneous ${\mathbb{R}}^d$-valued Lévy basis on ${\mathbb{R}}^n$ with spot variable $L^{\prime }$, $\sigma$ is an $\mathbb{R}$-valued stationary field on ${\mathbb{R}}^n$, independent of $L$, and $f:{\mathbb{R}}^n\to M_{m,d}$ is such that $f\sigma$ is almost surely $L$-integrable. In this case, the simplified field is defined as:

(32) $X(x)={\int }_{{\mathbb{R}}^n}f(x-y)\sigma (y)L(\mathrm{d}y).$(32)

It follows that, assuming $\mathrm{E}\left\{{\sigma }^2\right\}=1$,

$R(x)=\mathrm{c}\mathrm{o}\mathrm{v}\left(X(x),X(0)\right)={\int }_{{\mathbb{R}}^n}f(x-y)\mathrm{c}\mathrm{o}\mathrm{v}\left(L^{\prime }(y),L^{\prime }(y)\right)f(-y{)}^T\mathrm{d}y,$

independent of the volatility modulation $\sigma$.

Without loss of generality, one may assume that $\mathrm{c}\mathrm{o}\mathrm{v}(L^{\prime },L^{\prime })=I$ which gives

$F[R]=(2\pi {)}^{n}F[f]F[f{]}^{\ast },$

where $F$ denotes the Fourier transform and $\ast$ the complex conjugate. Then, for every $y$ , there is a $m\times m$ matrix with complex entries $Q(y)$ such that

$F[R](y)=Q(y)Q(y{)}^{\ast }.$

Assuming that the inverse Fourier transform of $Q$ exists, and that the real part of $Q$ is an even function and the imaginary part of $Q$ is an odd function, then

$f=(2\pi {)}^{-n/2}{F}^{-1}[Q]$

gives the desired covariance:

$R(x)={\int }_{{\mathbb{R}}^n}f(x-y)f(-y{)}^T\mathrm{d}y.$

An application where this method works is given by the Shkarofsky correlation function

$\rho (x)=\frac{{\tilde{K}}_{\nu }(\kappa \sqrt{||x{||}^2+{\lambda }^2})}{{\tilde{K}}_{\nu }(\kappa \lambda )},$

which includes the Matérn correlation family

$\rho (x)=\frac{2^{1-\nu }}{\mathrm{\Gamma }(\nu )}{\tilde{K}}_{\nu }(\kappa ||x||)$

as a special case for $\lambda \to 0$. Here, ${\tilde{K}}_{\nu }(z)=z^{\nu }{K}_{\nu }(z)$ with $K$ being the modified Bessel function of the third kind.

The Shkarofsky correlation family accurately fits the correlation observed for time series obtained in the atmospheric boundary layer which can be translated to spatial correlations using TFFH (see Section 3.2). This has been done in [5] which allows to estimate the kernel of the model (32). Simulations then show that (32) reasonably well reproduces other statistical features (not used for the estimation), for instance, the evolution of the densities of velocity increments across scales. This evolution from heavy tails at small scales towards approximate Gaussian distribution at large scales is recovered by using a normal inverse Gaussian spot variable (see also [32]).

3.5 Electricity markets

Ambit fields in the form of BSS processes have also been used to model financial data. In [8], the empirically observed generalized hyperbolic marginal law of deseasonalized energy spot prices is modelled as:

$Y_t=\mu +c{\int }_{-\mathrm{\infty }}^{t}g(t-s){\sigma }_s\mathrm{d}{W}_s+\gamma {\int }_{-\mathrm{\infty }}^{t}q(t-s){\sigma }_s^2\mathrm{d}s,$

where $g$ and $q$ are deterministic kernels (suitable for the integrals to exist), $W$ denotes standard Brownian motion assumed to be independent of the stationary volatility process $\sigma$ and $\mu ,c,\gamma$ are real constants.

Of particular interest for the application are gamma-type kernel functions of the form

$g(t)=\frac{{\bar{\lambda }}^{\bar{\nu }-1/2}}{\mathrm{\Gamma }{(2\bar{\nu }-1)}^{1/2}}{t}^{\bar{\nu }-1}exp\left(-\frac{\bar{\lambda }}{2}t\right).$

For $\gamma =0$ and $\bar{\nu }>\frac{1}{2}$, the implied autocorrelation function is given as:

$Cor(Y_t,Y_{t+h})=\frac{1}{2^{\bar{\nu }-3/2}\mathrm{\Gamma }(\bar{\nu }-1/2)}{\bar{K}}_{\bar{\nu }-1/2}\left(\frac{\bar{\lambda }h}{2}\right),h>0,$

where ${\bar{K}}_{\bar{\nu }}(x)=x^{\bar{\nu }}{K}_{\bar{\nu }}(x)$ and $K_{\bar{\nu }}$ denotes the modified Bessel function of the third kind.

Forward prices based on the above spot price are further discussed in [2], where it is also pointed out that the underlying spot price is not a semimartingale which is concluded from the estimated value $\widehat{\stackrel{ˉ}{\nu }}=1.135$ of $\bar{\nu }$.

The direct modelling of energy forward and future prices by applying ambit fields on ${\mathbb{R}}^2$ is discussed in [2,33,34]. Let $F(t,\bar{T})$ denote the forward price where $t\ge 0$ denotes current time and $\bar{T}\ge 0$ denotes the time to maturity. The ambit model for $F$ is of exponentiated type and involves ambit sets of the form $(-\mathrm{\infty },t]\times [0,\mathrm{\infty })$. A multivariate version of the ambit model then allows to discuss spread option prices. These ambit fields can be viewed as infinite factor models with a continuum of state variables in a parsimonious representation.

3.6 Count data

In [3], a continuous-time modelling framework for multivariate time series of counts with infinitely divisible marginal distributions is discussed. The model is a particular type of ambit field – called a trawl process [35,36] – that allows to model serial correlations, cross-sectional dependence and long or short memory independently. In [3], the methodology is applied to high-frequency financial data, but, as pointed out in [3], may also find application within the area of medical science, epidemiology and meteorology.

The term count-data refers to data that are non-negative and integer-valued. The term trawl refers to any Borel set $A\subset (-\mathrm{\infty },0]\times \mathbb{R}$ with finite Lebesgue measure. The associated translational invariant ambit set for the integer-valued ambit field is defined as $A_t=(t,0)+A$ (see also Section 3.1) with

$A=\left\{(s,x):s\le 0,0\le x\le r(s)\right\}$

bounded by $r:(-\mathrm{\infty },0]\mapsto \mathbb{R}$. In case $r$ is monotonically non-decreasing, $A$ is called a monotonic trawl. A stationary integer-valued ambit field $(Y_t{)}_{t\ge 0}$ is then defined as:

$Y_t=L(A_t)={\int }_{A_t}L(\mathrm{d}s,\mathrm{d}x),$

where $L$ is a integer-valued, homogeneous Lévy basis on ${\mathbb{R}}^2$.

Such trawl processes can straightforwardly be generalized to $n$-dimensional stationary integer-valued trawl processes $(Y_t{)}_{t\ge 0}$ by $Y_t=(L^{(1)}(A_t^{(1)}),\dots ,L^{(n)}(A_t^{(n)}))$ where $L=(L^{(1)},\dots ,L^{(n)})$ is a $n$-dimensional integer-valued, homogeneous Lévy basis. Such a Lévy basis can be defined in terms of a Poisson random measure. Let $N$ be a positive integer-valued homogeneous Poisson random measure on ${\mathbb{R}}^n\times \mathbb{R}$ (i.e. its Lévy measure in concentrated on $\mathbb{N}$ ). One defines a ${\mathbb{N}}^n$-valued, homogeneous Lévy basis on ${\mathbb{R}}^2$ as:

$L(\mathrm{d}s,\mathrm{d}x)=(L^{(1)}(\mathrm{d}s,\mathrm{d}x),\dots ,L^{(n)}(\mathrm{d}s,\mathrm{d}x))={\int }_{{\mathbb{R}}^n}y\mathrm{N}(\mathrm{d}y,\mathrm{d}x,\mathrm{d}s).$

These integer-valued trawl processes are stationary and infinitely divisible and allow to calculate analytically key statistical features like characteristic function and covariances which give great flexibility and tractability for modelling purposes. A methodology for simulation and inference is also provided in [3], which is applied to a model for the number of order submissions and cancellations in a limit order book.

3.7 Growth modelling

The ambit modelling framework has also been applied to model growth processes [37–39], in particular to the modelling of the statistical properties of the star-shaped approximation of in vitro tumour profiles [39]. There, the main focus is on two-point statistics of the radii of the tumour as a function of time and direction. It turns out that two-point correlators of the tumour profiles are well approximated by a cosine law and display a pronounced self-scaling behaviour of correlators of different orders which is also observed for the statistics of the energy dissipation in a turbulent flow (see Sections 2.3 and 3.1). This similarity is also displayed in the similar structure of the model used to reproduce this striking statistical feature.

The original data in the study of tumour profiles are snapshots of a growing brain tumour in vitro, confined to grow mainly on a plate surface. Thus, the tumour can be considered to be two dimensional. As a working definition of the boundary, one considers the star-shaped approximation of the profile defined in terms of the radial function

$R_t(\varphi )=\mathrm{m}\mathrm{a}\mathrm{x}\left\{R:c_0+R{\vec{e}}_{\varphi }\in Y_t\right\},$

where $Y_t$ denotes the domain occupied by the tumour at time $t$, $c_0$ denotes its centre of mass at time $t=0$ and ${\vec{e}}_{\varphi }$ is the unit vector in direction $\varphi \in [0,2\pi ]$. The star-shaped profiles show structures a very different scales with localized bursts of different sizes. The growth is unrestricted in the experiments which implies that the profiles may be considered statistically isotropic.

For the modelling, it is convenient to consider the normalized radial function

$r_t(\varphi )=\frac{R_t(\varphi )}{\mathrm{E}\left\{R_t(\varphi )\right\}}.$

The spatial variation of the normalized radial function is characterized by two-point correlators of order $(n_1,n_2)$

$c_{n_1,n_2}(t,\mathrm{\Delta }\varphi )=\frac{\mathrm{E}\left\{r_t{(\varphi )}^{n_1}{r}_t{(\varphi +\mathrm{\Delta }\varphi )}^{n_2}\right\}}{\mathrm{E}\left\{r_t{(\varphi )}^{n_1}\right\}\mathrm{E}\left\{r_t{(\varphi +\mathrm{\Delta }\varphi )}^{n_2}\right\}}.$

These correlators display two different regimes. For angular distances $\mathrm{\Delta }\varphi >{\varphi }_0(t)$ , it seems reasonable to fit a cosine behaviour

$log{c}_{n_1,n_2}(t,\mathrm{\Delta }\varphi )=b_{n_1,n_2}(t)cos(\mathrm{\Delta }\varphi ),\mathrm{\Delta }\varphi >{\varphi }_0(t),$

with a time-dependent amplitude $b_{n_1,n_2}(t)$ and a critical angle ${\varphi }_0$ that does not depend on the order $(n_1,n_2)$. For the very small angular distances, this cosine law does not hold and a clear behaviour cannot be identified from the data.

The empirical structure of the two-point correlators for $\varphi >{\varphi }_0(t)$ immediately implies self-scaling of correlators. This similarity to the behaviour of the energy dissipation in a turbulent flow is the basic motivation for the model selection. A non-stationary modification of the exponentiated model used for the turbulent energy dissipation (see Section 3.1) that reproduces the empirical correlators is given as:

$r_t(\varphi )=exp\{a(t){\int }_{t-T_0(t)}^{t-t_0(t)}{\int }_{\varphi -\pi }^{\varphi +\pi }cos(\varphi -z)L(\mathrm{d}z,\mathrm{d}s)$

$+h(t){\int }_{t-t_0(t)}^t{\int }_{\varphi -g_t(s)}^{\varphi +g_t(s)}L(\mathrm{d}z,\mathrm{d}s)\}.$

Here, $L$ is a normal Lévy basis on $\mathbb{R}\times [0,2\pi ]$ (defined in a cyclic way) and the deterministic functions $a(t)$ and $h(t)$ account for possible time variations of the tumour dynamics and $T_0(t)$ is a time-dependent decorrelation time. The first ambit set is assumed to be a rectangle for convenience. The data do not allow to specify this ambit set in a more specific way. The second ambit set is bounded by a deterministic function $g_t$. Simulations with a triangular ambit set show good correspondence to the experimental tumour profiles.

4 Stochastic partial differential equations

For parabolic differential equations, solutions are often expressed as integrals over a Green’s function convoluted with initial conditions. Such solutions are very similar to some versions of ambit fields. In this respect, it is important to note that ambit fields are stochastic models of natural phenomena that are, in many respects, much simpler than differential equations due to their inherent ability to model specific properties of interest explicitly. Another important point is that ambit fields may provide a way to extent and generalize solutions of stochastic partial differential equations (SPDE). In general, solutions to SPDEs require strong integrability conditions which are much weaker in the framework of ambit fields. This might allow to tackle more general types of driving noise of the underlying SPDE and might also allow to discuss more general initial conditions.

A key example discussed in [40] concerns the nonlinear parabolic SPDE

$\frac{\mathrm{\partial }v}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}-v+f(t,v)W^{\prime },t>0,0<x<K,$

$\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(t,0)=\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(t,K)=0,t>0,$

$v(0,x)=v_0(x),0<x<K,$

where $K>0$ is a constant, $f$ is Lipschitz continuous in $x$ of at most linear growth, and where $v_0$ is $F_0$ measurable with $\mathrm{E}[v_0^2(x)]$ bounded. Furthermore, $W^{\prime }$ denotes white noise in the sense of Walsh [41].

A weak solution (in the sense of Walsh [41]) for the case $f=1$ is then given as:

$v(t,x)={\int }_0^K{G}_t(x,y)v_0(y)\mathrm{d}y+{\int }_0^t{\int }_0^K{G}_{t-s}(x,y)W(\mathrm{d}y,\mathrm{d}s),$

where

$G_t(x,y)=\frac{e^{-t}}{\sqrt{4\pi t}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}exp\left(-\frac{{(y-x-2nK)}^2}{4t}\right)+exp\left(-\frac{{(y+x-2nK)}^2}{4t}\right).$

A generalization of this solution in the framework of ambit stochastics is provided by:

$v(t,x)={\int }_0^K{G}_t(x,y)v_0(y)\mathrm{d}y+{\int }_0^t{\int }_0^K{G}_{t-s}(x,y)L(\mathrm{d}y,\mathrm{d}s),$

where a more general Lévy basis is substituted for the white noise $W$. An even more general set-up is generated by including a random field $\sigma (t,x)$ to finally obtain:

$v(t,x)={\int }_0^K{G}_t(x,y)v_0(y)\mathrm{d}y+{\int }_0^t{\int }_0^K{G}_{t-s}(x,y)\sigma (t,x)\mathrm{L}(\mathrm{d}y,\mathrm{d}s),$

subject to some regularity conditions. This latter ambit field constitutes a mild solution (in the sense of [42]) to the parabolic SPDE

$\frac{\mathrm{\partial }v}{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}-v+\sigma (t,v)L^{\prime },t>0,0<x<K,$

$\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(t,0)=\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(t,K)=0,t>0,$

$v(0,x)=v_0(x),0<x<K,$

where $L^{\prime }$ denotes the noise of the Lévy basis $L$. The noise $L^{\prime }$ extends the white noise concept and is rigorously defined in [42].

5 Concluding remarks

Ambit fields provide a very flexible and, to a large extent, mathematically tractable modelling framework that can be applied to a variety of situations where spatio-temporal dynamics are to be modelled. It is important to note that ambit fields, in general, are not semimartingales which imply that many standard tools from semimartingale theory are not applicable or need to be adapted. A comprehensive account of these more theoretical issues may be found in [2].

The applications discussed here state the models explicitly. This allows to choose the ingredients according to the statistical properties at hand. In a somehow opposite manner, such empirical properties may also be modelled by stating a SPDE, constructed such that the main mechanism, believed to imply the observed dynamics, are included. Ambit fields and SPDEs can thus be considered as tackling a given modelling problem from opposite sides. However, these two approaches may intimately be connected to each other as is discussed in [40]. This connection is revealed in [40] in relation to the Walsh theory of martingale measures and in relation to Lévy noise analysis.

Disclosure statement

No potential conflict of interest was reported by the author.