Ambit field

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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 (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, , 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[edit]

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[edit]

A tempo-spatial ambit field, , is a random field in space-time taking values in . Let be ambit sets in deterministic kernel functions, a stochastic function, a stochastic field (called the energy dissipation field in turbulence and volatility in finance) and a Lévy basis. Now, the ambit field is

Ambit sets[edit]

In the above, the ambit sets and describe the sphere of influence for a given point in space-time. I.e. at a given point, the sets and are the points in space-time which affect the value of the ambit field at . 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 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[edit]

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 . An ambit process is defined as the value of the field along the curve, i.e.

Stochastic intermittency/volatility[edit]

The energy dissipation field/volatility, , 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.

where is a non-negative Lévy basis.

Integration with respect to a Lévy basis[edit]

The stochastic integral, , 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[edit]

A family of random vectors in is called a Lévy basis on if:

1. The law of is infinitely divisible for all .
2. If are disjoint, then are independent.
3. If are disjoint with , then
, 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[edit]

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 , defined as

where the ambit sets, are of the form where the time-coordinates of are negative (same for ). Furthermore, we take for and that and are also stationary random variables/fields. In particular, we can take to be a stationary ambit field itself:

where is a non-negative Lévy basis and is a positive function.

References[edit]

  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

External links[edit]