upcrossings and downcrossings
Inequalities involving the number of times at which a stochastic process^{} passes upwards or downwards through a bounded interval play an important role in the theory of stochastic processes. Whether or not a sequence of real numbers converges^{} to a limit can be expressed in terms of the finiteness of the number of upcrossings or downcrossings, leading to results such as the martingale convergence theorem and regularity of martingale^{} sample paths. As the main applications are to stochastic processes, we suppose that ${({X}_{t})}_{t\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}}$ is a real-valued stochastic process with totally ordered time index set^{} $\mathrm{\pi \x9d\x95\x8b}$, usually a subset of the real numbers. However, for the definitions here, it is enough to consider ${X}_{t}$ to be a sequence of real numbers, and the dependence on any underlying probability space^{} is suppressed.
For real numbers $$, the number of upcrossings of $X$ across the interval $[a,b]$ is the maximum nonnegative integer $n$ such that there exists times ${s}_{k},{t}_{k}\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}$ satisfying
$$ | (1) |
and for which $$. The number of upcrossings is denoted by $U\beta \x81\u2019[a,b]$. Note that this is either a nonnegative integer or is infinite^{}. Similarly, the number of downcrossings, denoted by $D\beta \x81\u2019[a,b]$, is the maximum nonnegative integer $n$ such that there are times ${s}_{k},{t}_{k}\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}$ satisfying (1) and such that ${X}_{{s}_{k}}>b>a>{X}_{{t}_{k}}$.
Note that between any two upcrossings there is a downcrossing. Given ${s}_{k},{t}_{k}$ satisfying (1) and $$, we can set ${s}_{k}^{\beta \x80\xb2}={t}_{k}$ and ${t}_{k}^{\beta \x80\xb2}={s}_{k+1}$ for $k=1,\mathrm{\beta \x80\xa6},n-1$. Then, ${X}_{{s}_{k}^{\beta \x80\xb2}}>b>a>{X}_{{t}_{k}^{\beta \x80\xb2}}$ from which we conclude that $D\beta \x81\u2019[a,b]\beta \x89\u20afU\beta \x81\u2019[a,b]-1$. Similarly, we have $U\beta \x81\u2019[a,b]\beta \x89\u20afD\beta \x81\u2019[a,b]-1$. Hence, the number of upcrossings and the number of downcrossings of an interval differ by at most $1$.
For a finite index set $\mathrm{\pi \x9d\x95\x8b}=\{1,2,\mathrm{\beta \x80\xa6},N\}$, the number of upcrossings can be computed as follows. Set ${t}_{0}=0$ and define ${s}_{1},{s}_{2},\mathrm{\beta \x80\xa6}$ and ${t}_{1},{t}_{2},\mathrm{\beta \x80\xa6}$ by
${t}_{k}=inf\beta \x81\u2018\{t\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}:t\beta \x89\u20af{s}_{k},{X}_{t}>b\}\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}\beta \x88\u037a\{\mathrm{\beta \x88\x9e}\}.$ | ||
$$ |
The number of upcrossings of $[a,b]$ is then equal to the maximum $n$ such that $$ (see Figure 1).
References
- 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
Title | upcrossings and downcrossings |
---|---|
Canonical name | UpcrossingsAndDowncrossings |
Date of creation | 2013-03-22 18:49:33 |
Last modified on | 2013-03-22 18:49:33 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 5 |
Author | gel (22282) |
Entry type | Definition |
Classification | msc 40A05 |
Classification | msc 60G17 |
Related topic | MartingaleConvergenceTheorem |
Related topic | ConvergenceOfASequenceWithFiniteUpcrossings |
Defines | upcrossing |
Defines | downcrossing |