62.11. HITTING THIS BEFORE THAT 2107

of generality in assuming T = ∞ since if it is less than ∞, you could just let M (t)≡M (T )for all t > T. In this case, the equiintegrability of the M (t) follows because for t < T,∫

[|M(t)|>λ ]|M (t)|dP =

∫[|M(t)|>λ ]

|E (M (T ) |Ft)|dP

≤∫[|M(t)|>λ ]

|M (T )|dP

and from Theorem 62.9.5,

P(|M (t)|> λ )≤ P([M∗ (t)> λ ])≤ 1λ

∫Ω

|M (T )|dP.

Definition 62.11.1 Let M be a process adapted to the filtration Ft and let τ be a stoppingtime. Then Mτ , called the stopped process is defined by

Mτ (t)≡M (τ ∧ t) .

With this definition, here is a simple lemma.

Lemma 62.11.2 Let M be a right continuous martingale adapted to the normal filtrationFt and let τ be a stopping time. Then Mτ is also a martingale adapted to the filtration Ft .

Proof:Let s < t. By the Doob optional sampling theorem,

E (Mτ (t) |Fs)≡ E (M (τ ∧ t) |Fs) = M (τ ∧ t ∧ s) = Mτ (s) .

Theorem 62.11.3 Let {M (t)} be a continuous real valued martingale adapted to the nor-mal filtration Ft and let

M∗ ≡ sup{|M (t)| : t ≥ 0}

and M (0) = 0. Lettingτx ≡ inf{t > 0 : M (t) = x}

Then if a < 0 < b the following inequalities hold.

(b−a)P([τb ≤ τa])≥−aP([M∗ > 0])≥ (b−a)P([τb < τa])

and(b−a)P([τa < τb])≤ bP([M∗ > 0])≤ (b−a)P([τa ≤ τb]) .

In words, P([τb ≤ τa]) is the probability that M (t) hits b no later than when it hits a. (Notethat if τa = ∞ = τb then you would have [τa = τb] .)

Proof: For x ∈ R, define

τx ≡ inf{t ∈ R such that M (t) = x}

with the usual convention that inf( /0) = ∞. Let a < 0 < b and let

τ = τa∧ τb