What does "Lévy Process" mean?
Table of Contents
A Lévy process is a type of mathematical model used to describe random movements that can change suddenly. These processes are often used in finance, insurance, and various fields of science to represent things like stock prices or the size of claims in insurance.
Key Features
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Continuous Time: Lévy processes are defined over continuous time, meaning they can change at any moment rather than just at set intervals.
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Independent Increments: The changes that occur in the process over different time periods are independent of each other. This means that knowing what happens in one period doesn't help predict what will happen in another.
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Stationary Increments: The size of changes only depends on how long the time period is, not on where that period falls in time. For example, a change from time 0 to time 1 looks the same as a change from time 5 to time 6.
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Jump Behavior: Lévy processes can have sudden jumps, which means they can move up or down quickly and unexpectedly. This is different from other models that only allow gradual changes.
Applications
Lévy processes are useful in various areas, especially in finance where they can model unpredictable events, such as sudden market shifts or unexpected risks. They also apply to physical sciences, where sudden changes can occur in natural phenomena.
In summary, Lévy processes are a flexible and powerful way to model randomness and change over time, especially when sudden shifts are involved.