Ambit field

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.

Definition

A tempo-spatial ambit field, Y , is a random field in space-time χ × R taking values in R . Let μ ∈ R , A t ( x ) , B t ( x ) be ambit sets in χ × R + , g , q deterministic kernel functions, a a stochastic function, σ ≥ 0 a stochastic field (called the energy dissipation field in turbulence and volatility in finance) and L a Lévy basis. Now, the ambit field Y is

Y t ( x ) = μ + ∫ A t ( x ) g ( η , s , x , t ) σ s ( η ) L ( d η , d s ) + ∫ B t ( x ) q ( η , s , x , t ) a s ( η ) d η d s

Ambit sets

In the above, the ambit sets A t ( x ) and B t ( x ) describe the sphere of influence for a given point in space-time. I.e. at a given point, ( t , x ) ∈ χ × R the sets A t ( x ) and B t ( x ) are the points in space-time which affect the value of the ambit field at ( 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 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. This motivates the following definition:

Let the ambit field, Y, be given as above and consider a curve in space-time τ ( θ ) = ( x ( θ ) , t ( θ ) ) ∈ χ × R . An ambit process is defined as the value of the field along the curve, i.e.

X θ = Y t ( θ ) ( x ( θ ) )

Stochastic intermittency/volatility

The energy dissipation field/volatility, σ , 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.

σ t 2 ( x ) = ∫ C t ( x ) h ( η , s , x , t ) L ~ ( d η , d s )

where L ~ is a non-negative Lévy basis.

Integration with respect to a Lévy basis

The stochastic integral, ∫ A t ( x ) g ( η , s , x , t ) σ s ( η ) L ( d η , d s ) , 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) where integration is done with respect to random fields and this theory can be extended to integration with respect to so-called Lévy bases, which is the main building block of the ambit field.

Definition of Lévy basis

A family ( Λ ( A ) : A ∈ B b ( S ) ) of random vectors in R d is called a Lévy basis on S if:

1. The law of Λ ( A ) is infinitely divisible for all A ∈ B b ( S ) . 2. If A 1 , A 2 , … , A n ∈ B b ( S ) are disjoint, then Λ ( A 1 ) , Λ ( A 2 ) , … , Λ ( A n ) are independent. 3. If A 1 , A 2 , … ∈ B b ( S ) are disjoint with ⋃ i = 1 ∞ A i ∈ B b ( S ) , then

where the 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 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 Y , defined as

Y t = μ + ∫ A t ( x ) g ( η , t − s , x ) σ s ( η ) L ( d η , d s ) + ∫ B t ( x ) q ( η , t − s , x ) a s ( η ) d η d s

where the ambit sets, A t ( x ) , B t ( x ) are of the form A t ( x ) = A + ( x , t ) where the time-coordinates of A are negative (same for B ). Furthermore, we take g ( η , t , x ) = q ( η , t , x ) = 0 for t ≤ 0 and that σ and a are also stationary random variables/fields. In particular, we can take σ to be a stationary ambit field itself:

σ t 2 ( x ) = ∫ C t ( x ) h ( η , t − s , x ) L ~ ( d η , d s )

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