Algebra of random variables

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 , 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 and the evolution of electricity prices for use in e.g. risk management and derivative pricing. It was pioneered by Ole E. Barndorff-Nielsen and 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). (Wikipedia).

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Related pages

Integral | Integrator | Risk management | Random field | White noise | Mathematics | Function (mathematics) | Intermittency | Random measure | Causality | Vector (mathematics and physics) | Volatility (finance) | Lebesgue integration | Stochastic | Stochastic partial differential equation | Green's function | Convolution | Turbulence