From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration

Details From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration

2007-06-16

arxiv.org/abs/0706.2391v1

Mathematics

Probability

Scientific paper

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two such extensions, corresponding to the \Ito-Skorokhod and the Stratononovich integrals, and provides an efficient tool to study these integrals, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields.

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