This makes a lot of sense, Thanks for highlighting the need to define value more explicitly. I’ll have a look into this stuff!
on the math point—I don’t think that IV would be continuous is the problem, but in general this would mean the noise is present in both frameworks! The case of x^2 sin(1/​x) shows the integral of a function with a discontinuity is not necessarily discontinuous but in general discontinuous functions would have a discontinuous integral so the noise doesn’t distinguish the frameworks.
I thought IV was assumed continuous based on your drawing. Still, I’d be surprised—and I would love to know about it—if you could find an function with a discontinuous integral and does not seem unfit to correctly model IV to me—both out of interest for the mathematical part and of curiosity about what functions we respectively think can correctly model IV.
I think that piecewise continuity and local boundedness are already enough to ensure continuity and almost-everywhere continuous differentiability of the integral. I personally don’t think that functions that don’t match these hypotheses are reasonable candidates for IV, but I would allow IV to take any sign. What are your thoughts on this ?
This makes a lot of sense, Thanks for highlighting the need to define value more explicitly. I’ll have a look into this stuff!
on the math point—I don’t think that IV would be continuous is the problem, but in general this would mean the noise is present in both frameworks! The case of x^2 sin(1/​x) shows the integral of a function with a discontinuity is not necessarily discontinuous but in general discontinuous functions would have a discontinuous integral so the noise doesn’t distinguish the frameworks.
Thank you!
Thanks for your answer !
I thought IV was assumed continuous based on your drawing. Still, I’d be surprised—and I would love to know about it—if you could find an function with a discontinuous integral and does not seem unfit to correctly model IV to me—both out of interest for the mathematical part and of curiosity about what functions we respectively think can correctly model IV.
I think that piecewise continuity and local boundedness are already enough to ensure continuity and almost-everywhere continuous differentiability of the integral. I personally don’t think that functions that don’t match these hypotheses are reasonable candidates for IV, but I would allow IV to take any sign. What are your thoughts on this ?