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 ${(X_t)}_{t\in 𝕋}$ is a real-valued stochastic process with totally ordered time index set $𝕋$, 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 $s_k,t_k\in 𝕋$ satisfying

(1)

and for which . The number of upcrossings is denoted by $U[a,b]$. Note that this is either a nonnegative integer or is infinite. Similarly, the number of downcrossings, denoted by $D[a,b]$, is the maximum nonnegative integer $n$ such that there are times $s_k,t_k\in 𝕋$ satisfying (1) and such that $X_{s_k}>b>a>X_{t_k}$.

Figure 1: Process with 3 upcrossings of the interval $[a,b]$.

Note that between any two upcrossings there is a downcrossing. Given $s_k,t_k$ satisfying (1) and , we can set $s_k^{\prime }=t_k$ and $t_k^{\prime }=s_{k+1}$ for $k=1,\mathrm{\dots },n-1$. Then, $X_{s_k^{\prime }}>b>a>X_{t_k^{\prime }}$ from which we conclude that $D[a,b]\ge U[a,b]-1$. Similarly, we have $U[a,b]\ge D[a,b]-1$. Hence, the number of upcrossings and the number of downcrossings of an interval differ by at most $1$.

For a finite index set $𝕋=\{1,2,\mathrm{\dots },N\}$, the number of upcrossings can be computed as follows. Set $t_0=0$ and define $s_1,s_2,\mathrm{\dots }$ and $t_1,t_2,\mathrm{\dots }$ by

$t_k=inf\{t\in 𝕋:t\ge s_k,X_t>b\}\in 𝕋\cup \{\mathrm{\infty }\}.$

The number of upcrossings of $[a,b]$ is then equal to the maximum $n$ such that (see Figure 1).