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Let S 0 0 be given and deﬁne the stopping time
= minft 0; S t = Lg: Theorem 2.61 The process e,rt^ v t ^ ; S t ^ ; 0 t T;
is a martingale.
Proof: First note that S T L T:
Let ! 2 be given, and choose t 2 0; T . If ! t, then IE e,rT S T , K +1fS T Lg F t ! = 0:
But when ! t, we have vt ^ !; S t ^ !; ! = v t ^ !; L = 0;
so we may write IE e,rT S T , K + 1fS T Lg F t ! = e,rt^ !v t ^ !; S t ^ !; ! :
! t, then the Markov property implies On the other hand, if IE e,rT S T , K + 1fS T Lg F t !
h
i
= IE t;S t;! e,rT S T , K + 1fS T Lg
= e,rt v t; S t; !
= e,rt^ !v t ^ ; S t ^ !; ! :
In both cases, we have e,rt^ vt ^ ; S t ^ = IE e,rT S T , K + 1fS T Lg F t :
Suppose 0 u t T . Then IE e,rt^ vt ^ = IE IE ; S t ^ F u e,rT S T , K + 1fS T Lg F t = IE e,rT S T , K + 1fS T Lg F u
= e,ru^ v u ^ ; S u ^ : F u ...
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 Fall '15
 mr.somebody
 Andrey Markov, Martingale, e,rT 1CS

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