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In [[mathematics]], an '''ambit field''' is a ''d''-dimensional [[random field]] describing the stochastic properties of a given system. The input is in general a ''d''-dimensional [[vector (mathematics and physics)|vector]] (e.g. d-dimensional space or (1-dimensional) time and (''d''&nbsp;&minus;&nbsp;1)-dimensional space) assigning a real value to each of the points in the field. In its most general form, the ambit field, <math>Y</math>, is defined by a constant plus a [[stochastic integral]], where the [[integral|integration]] is done with respect to a ''[[Lévy basis]]'', plus a smooth term given by an ordinary [[Lebesgue integration|Lebesgue integral]]. The integrations are done over so-called ''ambit sets'', which is used to model the [[sphere of influence]] (hence the name, ambit, [[latin]] for "sphere of influence" or "boundary") which affect a given point.
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The use and development of ambit fields is motivated by the need of flexible stochastic models to describe [[turbulence]]<ref name="turb">Barndorff-Nielsen, O. E., Schmiegel, J. [http://data.imf.au.dk/publications/thiele/2005/imf-thiele-2005-14.pdf "Ambit processes; with applications to turbulence and tumour growth"], ''Research report, Thiele Centre'', December 2005</ref> and the evolution of [[electricity]] prices<ref name="forw">Barndorff-Nielsen, O. E., Benth, F. E., and Veraart, A., [http://pure.au.dk/portal/files/21655743/rp10_41.pdf "Modelling electricity forward markets by ambit fields"], ''CREATES research center'', 2010</ref> for use in e.g. [[risk management]] and [[derivative pricing]]. It was pioneered by [[Ole E. Barndorff-Nielsen]] and [[Jürgen Schmiegel]] to model turbulence and tumour growth.<ref name="turb" />
 
Note, that this article will use notation that includes time as a dimension, i.e. we consider (''d''&nbsp;&minus;&nbsp;1)-dimensional space together with 1-dimensional time. The theory and notation easily carries over to ''d''-dimensional space (either including time herin or in a setting involving no time at all).
 
==Intuition and motivation==
In stochastic analysis, the usual way to model a random process, or field, is done by specifying the ''dynamics'' of the process through a [[Stochastic partial differential equation|stochastic (partial) differential equation]] (SPDE). It is known, that solutions of (partial) differential equations can in some cases be given as an integral of a [[Green's function]] [[convolution|convolved]] with another function – if the differential equation is stochastic, i.e. contaminated by random noise (e.g. [[white noise]]) the corresponding solution would be a stochastic integral of the Green's function. This fact motivates the reason for modelling the field of interest ''directly'' through a stochastic integral, taking a similar form as a solution through a Green's Function, instead of first specifying a SPDE and then trying to find a solution to this. This provides a very flexible and general framework for modelling a variety of phenomena.<ref name="forw" />
 
==Definition==
 
A [[Space-time|tempo-spatial]] ambit field, <math>Y</math>, is a random field in [[space-time]] <math>\chi \times \mathbb{R}</math> taking values in <math>\mathbb{R}</math>. Let <math>\mu \in \mathbb{R}, A_t (x), B_t (x)</math> be [[Ambit Field#Ambit Sets|ambit sets]] in <math>\chi \times \mathbb{R}_{+}, g, q</math> [[deterministic]] [[Integral kernel|kernel]] functions, <math>a</math> a [[stochastic]] [[function (mathematics)|function]], <math>\sigma \geq 0</math> a [[Random field|stochastic field]] (called the ''energy dissipation field'' in [[turbulence]] and ''[[volatility (finance)|volatility]]'' in [[finance]]) and <math>L</math> a Lévy basis. Now, the ambit field <math>Y</math> is
 
: <math>Y_t = \mu + \int_{A_{t}(x)} g(\eta,s,x,t)\sigma_{s}(\eta) L(d\eta,ds) + \int_{B_t(x)} q(\eta,s,x,t)a_{s}(\eta) \, d\eta \, ds</math>
 
===Ambit sets===
In the above, the ''ambit sets'' <math>A_t(x)</math> and <math>B_t(x)</math> describe the sphere of influence for a given point in space-time. I.e. at a given point, <math>(t,x) \in \chi \times \mathbb{R}</math> the sets <math>A_t(x)</math> and <math>B_t(x)</math> are the points in space-time which affect the value of the ambit field at <math>(t,x), Y_t(x)</math>. When time is considered as one of the dimensions, the sets are often taken to only include time-coordinates which are at or prior to the current time, t, so as to preserve [[causality]] of the field (i.e. a given point in space-time can only be affected by events that happened prior to time <math>t</math> and can thus not be affected by the future).
 
The ambit sets can be of a variety of forms and when using ambit fields for modelling purposes, the choice of ambit sets should be made in a way that captures the desired properties (e.g. [[stylized facts]]) of the system considered in the best possible way. In this sense, the sets can be used to make a particular model fit the data as closely as possible and thus provides a very flexible – yet general – way of specifying the model.
 
===Ambit process===
Often, the object of interest is not the ambit field itself, but instead a process taking a particular path through the field. Such a process is called an ''ambit process''. As an example such a process can represent the price of a particular financial object – e.g. the price of a [[forward contract]] for a certain time and point in space, space representing things such as time to delivery, [[spot price]], period of delivery etc.<ref name="forw" /> This motivates the following definition:
 
Let the ambit field, ''Y'', be given as above and consider a curve in space-time <math>\tau(\theta) = (x(\theta), t(\theta)) \in \chi \times \mathbb{R}</math>. An ambit process is defined as the value of the field along the curve, i.e.
 
: <math>X_\theta = Y_{t(\theta)}(x(\theta))</math>
 
===Stochastic intermittency/volatility===
The energy dissipation field/volatility, <math>\sigma</math>, is, in general, stochastic (called ''[[intermittency]]'' in the context of ''[[turbulence]]''), and can be modelled as a stochastic variable or field. Particularly, it may itself be modelled by another ambit field, i.e.
 
: <math>\sigma^2_t(x) = \int_{C_t(x)} h(\eta,s,x,t) \tilde{L}(d\eta,ds)</math>
 
where <math>\tilde{L}</math> is a non-negative Lévy basis.
 
==Integration with respect to a Lévy basis==
The stochastic integral, <math>\int_{A_{t}(x)} g(\eta,s,x,t)\sigma_{s}(\eta) L(d\eta,ds)</math>, in the definition of the ambit process is an integral of a stochastic field (the [[integrand]]) over Lévy basis (the [[integrator]]), and is thus more complicated than the usual stochastic [[Itô-integral]]. A new theory of integration was provided by Walsh (1987)<ref>Walsh, J., "An introduction to stochastic partial differential equations", ''Lecture Notes in Mathematics'', 1986</ref> where integration is done with respect to random fields and this theory can be extended to integration with respect to so-called Lévy bases,<ref name="spde">Barndorff–Nielsen, O. E., Benth, F. E.,
and Veraart, A., [ftp://ftp.econ.au.dk/creates/rp/10/rp10_17.pdf "Ambit processes and stochastic partial differential equations"], ''CREATES research center'', 2010</ref> which is the main building block of the ambit field.
 
===Definition of Lévy basis===
A family <math>(\Lambda(A) : A \in \mathbb{B}_b(S))</math> of random vectors in <math>\mathbb{R}^d</math> is called a ''Lévy basis'' on <math>S</math> if:
: 1. The law of <math>\Lambda(A)</math> is [[infinitely divisible]] for all <math>A \in \mathbb{B}_{b}(S)</math>.
: 2. If <math>A_1 , A_2, \ldots, A_n \in \mathbb{B}_b(S)</math> are disjoint, then <math>\Lambda(A_1), \Lambda(A_2),\ldots, \Lambda(A_n)</math> are independent.
: 3. If <math>A_1 , A_2, \ldots \in \mathbb{B}_b(S)</math> are disjoint with <math>\bigcup_{i=1}^\infty A_i \in \mathbb{B}_b(S)</math>, then
::: <math>\Lambda(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \Lambda(A_i)</math>,   a.s.
 
where the [[convergence (mathematics)|convergence]] on the right hand side of 3. is a.s.
 
Note that proporties 2. and 3. define an independently scattered [[random measure]].
 
==A stationary example==
In some data (e.g. commodity prices) there is often found a [[Time-invariant|stationary]] component, which a good model should be able to capture. The ambit field can be made stationary in a straight forward way. Consider the ambit field <math>Y</math>, defined as
 
: <math>Y_t = \mu + \int_{A_{t}(x)} g(\eta,t-s,x)\sigma_{s}(\eta) L(d\eta,ds) + \int_{B_{t}(x)} q(\eta,t-s,x)a_{s}(\eta) \, d\eta \, ds</math>
 
where the ambit sets, <math>A_{t}(x), B_{t}(x)</math> are of the form <math>A_{t}(x) = A + (x,t)</math> where the time-coordinates of <math>A</math> are negative (same for <math>B</math>). Furthermore, we take <math>g(\eta,t,x) = q(\eta,t,x) = 0 </math> for <math> t \leq 0</math> and that <math>\sigma</math> and <math>a</math> are also stationary random variables/fields. In particular, we can take <math>\sigma</math> to be a stationary ambit field itself:
 
: <math>\sigma^2_{t}(x) = \int_{C_{t}(x)} h(\eta,t-s,x) \tilde{L}(d\eta,ds)</math>
 
where <math>\tilde{L}</math> is a non-negative Lévy basis and <math>h</math> is a positive function.
 
==References==
<references/>
 
==External links==
* [http://www.ambitprocesses.au.dk/ Ambit Processes at University of Aarhus]
 
[[Category:Probability theory]]

Latest revision as of 16:43, 15 October 2014

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