Martingale Theorems � � • Stopping Time: Given a filtered measurable space Ω , F , {F n } n ≥ 0 , a stopping time is a random variable T : Ω �→ Z + with the property that { ω : T ( ω ) ≤ n } ∈ F n for all n ≥ 0. • In words: we can decide whether T ≤ n based on information available at time n . 1
• If T is a stopping time, then equivalently θ n = 1 T>n , n ≥ 0 , is adapted to {F n } n ≥ 0 . • Consequently, if { X n } n ≥ 0 is a martingale, then so is n − 1 � � � Z n = X j +1 − X j θ j . j =0 • But Z n = X T ∧ n − X 0 , where T ∧ n = min( T, n ). • So { X T ∧ n } n ≥ 0 is a martingale. 2
� � • Optional Stopping Theorem. Let Ω , F , {F n } n ≥ 0 , P be a fil- tered probability space. Suppose that the process { X n } n ≥ 0 is � � a P , {F n } n ≥ 0 -martingale, and that T is a bounded stopping time. Then E [ X T |F 0 ] = X 0 and hence E [ X T ] = E [ X 0 ] . • Boundedness of T gives a simple proof, but is not necessary. – But if T is unbounded , conditions on { X n } are needed for the theorem to hold. 3
• Positive Supermartingale Convergence Theorem: If { X n } n ≥ 0 � � is a P , {F n } n ≥ 0 -supermartingale and P [ X n ≥ 0] = 1 for all n ≥ 0, then there exists a F ∞ -measurable X ∞ such that P [ X n → X ∞ as n → ∞ ] = 1 . – Note that a martingale is also a supermartingale, so this theorem also applies to positive martingales. 4
• Compensation: Submartingales tend to increase over time, and supermartingales tend to decrease. � � • Suppose that { X n } n ≥ 0 is a P , {F n } n ≥ 0 -submartingale. Then there is a previsible non-decreasing process { A n } n ≥ 0 such that � � { X n − A n } n ≥ 0 is a P , {F n } n ≥ 0 -martingale. – We simply set A 0 = 0 and A n − A n − 1 = E [ X n − X n − 1 |F n − 1 ]. � � • Similarly, if { X n } n ≥ 0 is a -supermartingale, then P , {F n } n ≥ 0 there is a previsible non-decreasing process { A n } n ≥ 0 such that � � { X n + A n } n ≥ 0 is a P , {F n } n ≥ 0 -martingale. • In each case, { A n } n ≥ 0 is the compensator . 5
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