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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, , 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〔Barndorff-Nielsen, O. E., Schmiegel, J. ("Ambit processes; with applications to turbulence and tumour growth" ), ''Research report, Thiele Centre'', December 2005〕 and the evolution of electricity prices〔Barndorff-Nielsen, O. E., Benth, F. E., and Veraart, A., ("Modelling electricity forward markets by ambit fields" ), ''CREATES research center'', 2010〕 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.〔
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.〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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