Integration with respect to a Lévy basis / Ambit fields

In summary, a Lévy basis is a collection of independent, identically distributed random variables used to model the behavior of a random variable over time. Integration with respect to a Lévy basis involves integrating a function with respect to a stochastic process, which is used to model the evolution of complex financial assets subject to random fluctuations. Ambit fields are a generalization of Lévy bases that allow for more complex stochastic processes, and are used to model financial assets with long-range dependence and non-Gaussian behavior. The applications of integration with respect to a Lévy basis / Ambit fields include option pricing, risk management, and portfolio optimization in mathematical finance, as well as modeling complex systems in other fields. Challenges associated with this type
  • #1
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Familiar with basics of stochastic calculus and integration over a Brownian motion. Trying to get a sense of Ambit Fields https://en.wikipedia.org/wiki/Ambit_field

which mention an integration over a Lévy basis:

1603115912317.png

Curious if anyone familiar with this? A Brownian motion is a Levy process, assuming this is something more general. The three conditions above just seem to be saying the basis is continuous and the elements are disjoint
 
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  • #2
, which is typical for integration in a Levy process. In any case, it's an interesting concept and I'm curious to learn more.
 

1. What is a Lévy basis?

A Lévy basis is a set of functions that can be used to represent a stochastic process. It is named after the mathematician Paul Lévy and is often used in the study of probability and statistics.

2. How is integration with respect to a Lévy basis different from traditional integration?

Integration with respect to a Lévy basis involves integrating a function with respect to a set of functions, rather than a single variable. This allows for a more flexible and powerful approach to integration, particularly in the study of stochastic processes.

3. What are Ambit fields?

Ambit fields are a type of stochastic process that can be represented using a Lévy basis. They are used to model complex systems and are particularly useful in the fields of finance and physics.

4. Why is integration with respect to a Lévy basis important?

Integration with respect to a Lévy basis allows for a more comprehensive and flexible approach to analyzing stochastic processes. It also allows for the study of complex systems that cannot be easily represented using traditional integration methods.

5. How is integration with respect to a Lévy basis used in practical applications?

Integration with respect to a Lévy basis is used in a variety of fields, including finance, physics, and engineering. It is particularly useful in modeling and analyzing complex systems, such as stock prices, weather patterns, and particle interactions.

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