What is the Role of Epsilon in Stochastic Continuity?

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Discussion Overview

The discussion centers around the role of epsilon (ε) in the context of stochastic continuity, particularly in relation to its intuitive understanding and mathematical definitions. Participants explore the implications of ε in stochastic processes and its comparison to concepts in calculus.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that ε is an arbitrary small number greater than 0, used in the context of stochastic continuity.
  • There is confusion regarding whether ε is represented on the y-axis, with some participants questioning this visualization.
  • One participant emphasizes that mathematical definitions do not assign ε to any axis, contrasting intuitive visualizations with formal definitions.
  • Another participant expresses frustration with understanding ε without visual aids and seeks clarification on its intuitive meaning in relation to jump discontinuities.
  • Some participants clarify that ε serves as an upper bound for the magnitude difference between random variables, suggesting that the concept does not inherently involve axes.
  • A participant reflects on their previous understanding of ε in calculus, noting that they initially conflated its role in stochastic continuity with its use in limits.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement regarding the visualization and interpretation of ε. While some clarify its mathematical role, others maintain differing views on its intuitive representation.

Contextual Notes

There are unresolved aspects regarding the visualization of ε and its relationship to stochastic continuity, as well as the potential differences between its use in calculus and stochastic processes.

woundedtiger4
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ε is an arbitrary (small) number > 0.
If you are hung up on using axes, s and t are points on the x axis. Xt is a point on the y axis, but it is a random variable rather than just a number.
 
Then, is epsilon on y-axis?
 
woundedtiger4 said:
Then, is epsilon on y-axis?

You are confusing intuitive visualizations of mathematics with the content of mathematical definitions. Even in calculus, there is nothing in the definition of limit that says that epsilon in "on the y-axis".
 
I know what you mean actually I can't understand without visualising therefore it is irritating me that what is epsilon intuitively in continuity of stochastic process? I know the op is about jump discontinuity which is RCLL function so is the epsilon shows any point between the jump ?
 
woundedtiger4 said:
Then, is epsilon on y-axis?

ε is a positive number. It is used as an upper bound of the magnitude difference of two random variables. There is no axis involved. If you insist on thinking "axis", then you may consider everything on the y axis. However, I suggest you try to understand the main point, there is no axis involved, just numbers.
 
  • #10
No, I wasn't thinking epsilon on y-axis,I just tried to give an example that like in calculus I used to think epsilon (not the same epsilon shown in stochastic continuity) as on y-axis for which the delta exists. I didn't mean that the epsilon in stochastic continuity is on y-axis .
Thanks a tonne because now I have understood it.
 

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