What is a stochastische dynamische systeme?

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Discussion Overview

The discussion revolves around the concept of "stochastic dynamical systems," exploring its definitions, implications, and examples. Participants examine the nature of randomness in these systems and how they can be modeled, particularly in the context of financial products.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants suggest that "stochastic dynamical system" encompasses various meanings and implies complex behavior in trajectories.
  • Others clarify that "stochastic" refers to randomness, which can be inherent or perceived due to incomplete knowledge of a deterministic system.
  • A participant provides an example of financial products, noting that their evolution can be modeled using Wiener processes and Ito-calculus, which differ from traditional differential equations.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of stochastic dynamical systems, indicating that multiple interpretations and models are present in the discussion.

Contextual Notes

There are limitations regarding the assumptions made about randomness and determinism, as well as the specific mathematical frameworks referenced, which may not be universally accepted or understood.

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what is stochastische dynamische systeme ?
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The term "stochastic dynamical system" has a lot of different meanings. Very loosely speaking, stochastic dynamical system is a system that trajectories behave to be very complicated
 
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"Stochastic" means "random"

We can postulate the randomness directly. Alternatively, a deterministic system may appear random to us because we lack sufficiently complete knowledge of it to enable us to predict its behaviour.
 
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A typical example would be the evolution of financiel products like stock prices or derivatives. These are dynamical systems but are described by so-called Wiener processes, which are described by continuous functions which are nowhere differentiable. The resulting differential equations are not the ones you are used to, but are defined via integral equations in the so-called Ito-calculus. See e.g.

https://arxiv.org/abs/cond-mat/0408143
 
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