2104 CHAPTER 62. STOCHASTIC PROCESSES

≤ 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 submartingale. 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 62.10.2 Let {Y (t)} be a continuous submartingale adapted to a normal filtrationFt 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 submartingale convergence theorem.

Theorem 62.10.3 Let {X (t)} be a continuous submartingale adapted to a normal filtra-tion such that

supt{E (|X (t)|)}=C < ∞.

Then there exists X∞ ∈ L1 (Ω) such that

limt→∞

X (t)(ω) = X∞ (ω) a.e. ω.

Proof: Let U[a,b] be defined by

U[a,b] = limM→∞

UM[a,b].

Thus the random variable U[a,b] is an upper bound for the number of upcrossings. FromLemma 62.10.2 and the assumption of this theorem, there exists a constant C independentof M such that

E(

UM[a,b]

)≤ C

b−a+1.

Letting M→ ∞, it follows from monotone convergence theorem that

E(U[a,b]

)≤ C

b−a+1