What is the Role of Epsilon in Stochastic Continuity?

[woundedtiger4]

 

[mathman]

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

 

[woundedtiger4]

Then, is epsilon on y-axis?

 

[Stephen Tashi]

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".

 

[woundedtiger4]

https://www.youtube.com/watch?v=-ejyeII0i5c&feature=youtube_gdata_player

At 5:20

 

[Stephen Tashi]

https://www.youtube.com/watch?v=-ejyeII0i5c&feature=youtube_gdata_player

At 5:20

That's an intuitive presentation. Such presentations are not mathematical definitions.

 

[woundedtiger4]

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]

For example:

http://math.stackexchange.com/quest...and-adapted-process-is-x-t-always-predictablehttp://math.stackexchange.com/questions/6502/about-stochastic-continuity

 

[mathman]

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.

 

[woundedtiger4]

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.