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A Uniform-Increment Process from a Random Linear Slope

Article Quant Q&A · Author: Zhiyuan Wang
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Summary

The document answers whether a process can have increments distributed uniformly over an interval whose width depends on the elapsed time. It constructs a single random variable, uniform on the interval from minus one to one, and uses it as the slope of a line through the origin. Scaling that shared random slope by time makes each value uniform over the requested time-scaled interval. Subtracting the value at an earlier time cancels the earlier part of the line, leaving the same random slope multiplied by the elapsed interval.

This establishes the requested marginal increment distribution with a very simple construction. The explanation is deliberately narrow: all increments share the same underlying random slope, so the construction does not imply independent increments or describe a typical diffusion process. No applications, empirical evidence, or broader conditions on stochastic processes are discussed.

Key ideas

  • A single uniform random variable on the unit interval from minus one to one can serve as a random slope.
  • Multiplying that slope by time gives a value uniform over an interval that expands with time.
  • The increment between two times depends only on their separation under this linear construction.
  • All increments are linked through the same random slope, so the construction does not establish independent increments.

Tags

Statistics
Full text
# Is there uniform stochastic process?


# Is there uniform stochastic process?












Shall I construct a stochastic process $X(t)$ such that $X(s+t)-X(s)\sim U(-t,t)$ ? Or is there already any similar formula?

## Answer by Bjørn Kjos-Hanssen (score 3)

https://quant.stackexchange.com/a/36666

Let $U$ be uniform in $[-1,1]$ and let $X_t=Ut$, which is uniform in $[-t,t]$. Then $$X_{t+s}-X_s=U(t+s)-Us=Ut$$ so this works. So it's not so much a stochastic process as just a random variable giving the slope of a line.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.