850 CHAPTER 31. OPTIONAL SAMPLING THEOREMS

Then UnM[a,b] is clearly a random variable which is at least as large as the number of

upcrossings occurring for t ≤M using only 2n+1 of the stopping times. From the optionalsampling theorem,

E (X (τ2k))−E (X (τ2k−1)) =∫

Ω

X (τ2k)−X (τ2k−1)dP

=∫

Ω

E(X (τ2k) |Fτ2k−1

)−X (τ2k−1)dP

≥∫

Ω

X (τ2k−1)−X (τ2k−1)dP = 0

Note that X (τ2k) = a while X (τ2k−1) = b so the above may seem surprising. However,the two stopping times can both equal M so this is actually possible. For example, it couldhappen that X (t) = a for all t ∈ [0,M].

Next, take the expectation of both sides,

E(

UnM[a,b]

)≤ 1

b−a

n

∑k=0

E (X (τ2k+1))−E (X (τ2k))+1

≤ 1b−a

n

∑k=0

E (X (τ2k+1))−E (X (τ2k))+1

b−a

n

∑k=1

E (X (τ2k))−E (X (τ2k−1))+1

=1

b−a(E (X (τ1))−E (X (τ0)))+

1b−a

n

∑k=1

E (X (τ2k+1))−E (X (τ2k−1))+1

≤ 1b−a

(E (X (τ2n+1))−E (X (τ0)))+1

≤ 1b−a

(E (X (M))−a)+1

which does not depend on n. The last inequality follows because 0≤ τ2n+1 ≤M and X (t)is a sub-martingale. Let n→ ∞ to obtain

E(

UM[a,b]

)≤ 1

b−a(E (X (M))−a)+1

where UM[a,b] is an upper bound to the number of upcrossings of {X (t)} on [0,M] . This

proves the following interesting upcrossing estimate.

Lemma 31.5.2 Let {Y (t)} be a continuous sub-martingale adapted to a normal filtra-tion Ft for t ∈ [0,M] . Then if UM

[a,b] is defined as the above upper bound to the number ofupcrossings of {Y (t)} for t ∈ [0,M] , then this is a random variable and

E(

UM[a,b]

)≤ 1

b−a

(E (Y (M)−a)++a−a

)+1

=1

b−aE (|Y (M)|)+ 1

b−a|a|+1

With this it is easy to prove a continuous sub-martingale convergence theorem.