29.4. OPTIONAL SAMPLING AND STOPPING TIMES 801

Proof: Consider X (1) ,X (τ ∧σ) ,X (τ) . Then

E (X (τ) |Fτ∧σ )≥ X (σ ∧ τ ∧ τ) = X (σ ∧ τ) .

Also E (X (σ ∧ τ) |F1) ≥ X (σ ∧ τ ∧1) = X (1) so this stochastic process is a sub-martin-gale. ■

This optional sampling theorem gives a convenient way to consider the Doob maximalestimate presented earlier.

Proposition 29.4.12 Let {X (k)} be a real valued sub-martingale, and let λ > 0. Thenfor X∗n ≡max{Xk : k ≤ n} as earlier,∫

[X∗n≤λ ]X (n)+ dP≥ λP([X∗n > λ ]) = λP

([(X∗n )

+ > λ])

Proof: Let τ = n and let σ be the first hitting time of the set (λ ,∞) by X (k) . Thenω ∈ [X∗n > λ ] if and only if for some k≤ n,Xk > λ if and only if σ (ω) = k for some k≤ n.By Proposition 29.4.11 X (1) ,X (σ ∧n) ,X (n) is a sub-martingale and the set of interest[X∗n > λ ] is the one where σ < ∞. Then

E (X (n)) ≥ E (X (σ ∧n)) =∫[σ<∞]

X (σ ∧n)dP+∫[σ=∞]

X (σ ∧n)dP

=∫[σ<∞]

X (σ ∧n)dP+∫[σ=∞]

X (n)dP

≥ λP([σ < ∞])+∫[σ=∞]

X (n)dP

Therefore,

E(X[X∗n >λ ]X (n)+

)≥ E

(X[X∗n >λ ]X (n)

)≥ λP([σ < ∞]) = λP([X∗n > λ ]) .■

Now let Xn∗ = min{X (k) : k ≤ n} . What about P([Xn∗ <−λ ])? Let σ be the first hit-ting time for (−∞,−λ ) and note that [Xn∗ <−λ ] consists of the set of ω where σ (ω)< ∞.As noted in Proposition 29.4.11, X (1) ,X (σ ∧n) ,X (n) is a sub-martingale. Thus∫

X (1)dP ≤∫[σ<∞]

X (σ ∧n)dP+∫[σ=∞]

X (σ ∧n)dP

=∫[σ<∞]

X (σ ∧n)dP+∫[σ=∞]

X (n)dP

It follows that∫X (1)dP−

∫[σ=∞]

X (n)dP≤∫[σ<∞]

X (σ ∧n)dP≤−λP([σ < ∞])

and so,

λP([σ < ∞]) = λP([Xn∗ < λ ])≤∫|X (1)|+ |X (n)|dP

Therefore, we obtain the following theorem which is a maximal estimate for sub-martin-gales.