31.6. HITTING THIS BEFORE THAT 853

31.6 Hitting This Before ThatLet {M (t)} be a real valued continuous martingale for t ∈ [0,T ] where T ≤∞ and M (0) =0. In case T = ∞, assume the conditions of Theorem 31.5.4 are satisfied. Thus, accordingto these conditions, there exists M (∞) and the M (t) are equi-integrable. With the Dooboptional sampling theorem it is possible to estimate the probability that M (t) hits a beforeit hits b where a < 0 < b. There is no loss of generality in assuming T = ∞ since if it is lessthan ∞, you could just let M (t)≡M (T ) for all t > T. In this case, the equiintegrability ofthe M (t) follows because for t < T,∫

[|M(t)|>λ ]|M (t)|dP =

∫[|M(t)|>λ ]

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

≤∫[|M(t)|>λ ]

E (|M (T )| |Ft)dP =∫[|M(t)|>λ ]

|M (T )|dP

and from Theorem 31.4.4,

P(|M (t)|> λ )≤ P([M∗ (t)> λ ])≤ 1λ

∫Ω

|M (T )|dP.

Definition 31.6.1 Let M be a process adapted to the filtration Ft and let τ be astopping time. Then Mτ , called the stopped process is defined by

Mτ (t)≡M (τ ∧ t) .

With this definition, here is a simple lemma. I will use this lemma whenever convenientwithout comment.

Lemma 31.6.2 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 .The same is true for a sub-martingale.

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) .

As for a sub-martingale X (t) , for s < t

E (Xτ (t) |Fs)≡ E (X (τ ∧ t) |Fs)≥ X (τ ∧ t ∧ s)≡ Xτ (s) . ■

Theorem 31.6.3 Let {M (t)} be a continuous real valued martingale adapted to thenormal 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] .)