# Ambit field

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 (e.g. d-dimensional space or (1-dimensional) time and (d − 1)-dimensional space) assigning a real value to each of the points in the field. In its most general form, the ambit field, ${\displaystyle Y}$, is defined by a constant plus a stochastic integral, where the integration is done with respect to a Lévy basis, plus a smooth term given by an ordinary 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.

The use and development of ambit fields is motivated by the need of flexible stochastic models to describe turbulence[1] and the evolution of electricity prices[2] 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.[1]

Note, that this article will use notation that includes time as a dimension, i.e. we consider (d − 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 (SPDE). It is known, that solutions of (partial) differential equations can in some cases be given as an integral of a Green's function 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.[2]

## Definition

A tempo-spatial ambit field, ${\displaystyle Y}$, is a random field in space-time ${\displaystyle \chi \times \mathbb {R} }$ taking values in ${\displaystyle \mathbb {R} }$. Let ${\displaystyle \mu \in \mathbb {R} ,A_{t}(x),B_{t}(x)}$ be ambit sets in ${\displaystyle \chi \times \mathbb {R} _{+},g,q}$ deterministic kernel functions, ${\displaystyle a}$ a stochastic function, ${\displaystyle \sigma \geq 0}$ a stochastic field (called the energy dissipation field in turbulence and volatility in finance) and ${\displaystyle L}$ a Lévy basis. Now, the ambit field ${\displaystyle Y}$ is

${\displaystyle Y_{t}(x)=\mu +\int _{A_{t}(x)}g(\eta ,s,x,t)\sigma _{s}(\eta )L(d\eta ,ds)+\int _{B_{t}(x)}q(\eta ,s,x,t)a_{s}(\eta )\,d\eta \,ds}$

### Ambit sets

In the above, the ambit sets ${\displaystyle A_{t}(x)}$ and ${\displaystyle B_{t}(x)}$ describe the sphere of influence for a given point in space-time. I.e. at a given point, ${\displaystyle (t,x)\in \chi \times \mathbb {R} }$ the sets ${\displaystyle A_{t}(x)}$ and ${\displaystyle B_{t}(x)}$ are the points in space-time which affect the value of the ambit field at ${\displaystyle (t,x),Y_{t}(x)}$. 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 ${\displaystyle t}$ 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.[2] This motivates the following definition:

Let the ambit field, Y, be given as above and consider a curve in space-time ${\displaystyle \tau (\theta )=(x(\theta ),t(\theta ))\in \chi \times \mathbb {R} }$. An ambit process is defined as the value of the field along the curve, i.e.

${\displaystyle X_{\theta }=Y_{t(\theta )}(x(\theta ))}$

### Stochastic intermittency/volatility

The energy dissipation field/volatility, ${\displaystyle \sigma }$, 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.

${\displaystyle \sigma _{t}^{2}(x)=\int _{C_{t}(x)}h(\eta ,s,x,t){\tilde {L}}(d\eta ,ds)}$

where ${\displaystyle {\tilde {L}}}$ is a non-negative Lévy basis.

## Integration with respect to a Lévy basis

The stochastic integral, ${\displaystyle \int _{A_{t}(x)}g(\eta ,s,x,t)\sigma _{s}(\eta )L(d\eta ,ds)}$, 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)[3] where integration is done with respect to random fields and this theory can be extended to integration with respect to so-called Lévy bases,[4] which is the main building block of the ambit field.

### Definition of Lévy basis

A family ${\displaystyle (\Lambda (A):A\in \mathbb {B} _{b}(S))}$ of random vectors in ${\displaystyle \mathbb {R} ^{d}}$ is called a Lévy basis on ${\displaystyle S}$ if:

1. The law of ${\displaystyle \Lambda (A)}$ is infinitely divisible for all ${\displaystyle A\in \mathbb {B} _{b}(S)}$.
2. If ${\displaystyle A_{1},A_{2},\ldots ,A_{n}\in \mathbb {B} _{b}(S)}$ are disjoint, then ${\displaystyle \Lambda (A_{1}),\Lambda (A_{2}),\ldots ,\Lambda (A_{n})}$ are independent.
3. If ${\displaystyle A_{1},A_{2},\ldots \in \mathbb {B} _{b}(S)}$ are disjoint with ${\displaystyle \bigcup _{i=1}^{\infty }A_{i}\in \mathbb {B} _{b}(S)}$, then
${\displaystyle \Lambda (\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }\Lambda (A_{i})}$, a.s.

where the convergence on the right hand side of 3. is a.s.

Note that properties 2. and 3. define an independently scattered random measure.

## A stationary example

In some data (e.g. commodity prices) there is often found a stationary component, which a good model should be able to capture. The ambit field can be made stationary in a straightforward way. Consider the ambit field ${\displaystyle Y}$, defined as

${\displaystyle Y_{t}=\mu +\int _{A_{t}(x)}g(\eta ,t-s,x)\sigma _{s}(\eta )L(d\eta ,ds)+\int _{B_{t}(x)}q(\eta ,t-s,x)a_{s}(\eta )\,d\eta \,ds}$

where the ambit sets, ${\displaystyle A_{t}(x),B_{t}(x)}$ are of the form ${\displaystyle A_{t}(x)=A+(x,t)}$ where the time-coordinates of ${\displaystyle A}$ are negative (same for ${\displaystyle B}$). Furthermore, we take ${\displaystyle g(\eta ,t,x)=q(\eta ,t,x)=0}$ for ${\displaystyle t\leq 0}$ and that ${\displaystyle \sigma }$ and ${\displaystyle a}$ are also stationary random variables/fields. In particular, we can take ${\displaystyle \sigma }$ to be a stationary ambit field itself:

${\displaystyle \sigma _{t}^{2}(x)=\int _{C_{t}(x)}h(\eta ,t-s,x){\tilde {L}}(d\eta ,ds)}$

where ${\displaystyle {\tilde {L}}}$ is a non-negative Lévy basis and ${\displaystyle h}$ is a positive function.

## References

1. ^ a b Barndorff-Nielsen, O. E., Schmiegel, J. "Ambit processes; with applications to turbulence and tumour growth", Research report, Thiele Centre, December 2005
2. ^ a b c Barndorff-Nielsen, O. E., Benth, F. E., and Veraart, A., "Modelling electricity forward markets by ambit fields", CREATES research center, 2010
3. ^ Walsh, J., "An introduction to stochastic partial differential equations", Lecture Notes in Mathematics, 1986
4. ^ Barndorff–Nielsen, O. E., Benth, F. E., and Veraart, A., "Ambit processes and stochastic partial differential equations", CREATES research center, 2010