be any sequence, finite or infinite, of random variableswith values in ℝ which are defined on some probability space,
(Ω,S,P)
. We say
{Xn }
isa Martingale if
E (Xn |xn− 1,⋅⋅⋅,x1) = xn−1
and we say
{Xn }
is a submartingale if
E (Xn |xn−1,⋅⋅⋅,x1) ≥ xn− 1.
Next we define what is meant by an upcrossing.
Definition 58.10.4Let
{xi}
_{i=1}^{I}be any sequence of real numbers, I ≤∞. Definean increasing sequence of integers
{mk }
as follows. m_{1}is the first integer ≥ 1
such that x_{m1}≤ a, m_{2}is the first integer larger than m_{1}such that x_{m2}≥ b, m_{3}is the first integer larger than m_{2}such that x_{m3}≤ a, etc. Then each sequence,
{ }
xm2k−1,⋅⋅⋅,xm2k
, is called an upcrossing of
[a,b]
.
Proposition 58.10.5Let
{Xi}
_{i=1}^{n}be a finite sequence of real random variablesdefined on Ω where
(Ω, S,P)
is a probability space. Let U_{[a,b]
}
(ω )
denote the numberof upcrossings of X_{i}
(ω)
of the interval
[a,b]
. Then U_{[a,b]
}is a random variable.
Proof:Let X_{0}
(ω)
≡ a + 1, let Y_{0}
(ω )
≡ 0, and let Y_{k}
(ω)
remain 0 for k = 0,
⋅⋅⋅
,l
until X_{l}
(ω)
≤ a. When this happens (if ever), Y_{l+1}
(ω)
≡ 1. Then let Y_{i}
(ω)
remain 1 for i = l + 1,
⋅⋅⋅
,r until X_{r}
(ω )
≥ b when Y_{r+1}
(ω)
≡ 0. Let Y_{k}
(ω)
remain 0 for k ≥ r + 1 until X_{k}
(ω)
≤ a when Y_{k}
(ω)
≡ 1 and continue in this
way. Thus the upcrossings of X_{i}
(ω )
are identified as unbroken strings of ones
with a zero at each end, with the possible exception of the last string of ones
which may be missing the zero at the upper end and may or may not be an
upcrossing.
Note also that Y_{0} is measurable because it is identically equal to 0 and that if Y_{k} is
measurable, then Y_{k+1} is measurable because the only change in going from k to k + 1 is
a change from 0 to 1 or from 1 to 0 on a measurable set determined by X_{k}. Now
let
{
Z (ω) = 1 if Yk(ω) = 1 and Yk+1(ω) = 0,
k 0 otherwise,
if k < n and
{
1 if Yn(ω) = 1 and Xn (ω) ≥ b,
Zn (ω) = 0 otherwise.
Thus Z_{k}
(ω)
= 1 exactly when an upcrossing has been completed and each Z_{i} is a
random variable.
∑n
U[a,b](ω) = Zk(ω)
k=1
so U_{}
[a,b]
is a random variable as claimed.
The following corollary collects some key observations found in the above
construction.
Corollary 58.10.6U_{}
[a,b]
(ω)
≤ the number of unbroken strings of ones in the sequence,
{Yk(ω)}
there being at most one unbroken string of ones which produces no upcrossing.Also
( i−1)
Yi(ω) = ψi {Xj (ω)}j=1 , (58.10.16)
(58.10.16)
where ψ_{i}is some function of the past values of X_{j}