stochastic.217

# stochastic.217 - 216 Let S 0 0 be given and deﬁne the...

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Unformatted text preview: 216 Let S 0 0 be given and deﬁne the stopping time = minft  0; S t = Lg: Theorem 2.61 The process e,rt^ 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 vt ^ !; S t ^ !; ! = v t ^ !; L = 0; so we may write IE   e,rT S T  , K + 1fS T  Lg F t ! = e,rt^ !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,rt^ !v t ^ ; S t ^ !; ! : In both cases, we have   e,rt^ vt ^ ; S t ^  = IE e,rT S T  , K + 1fS T  Lg F t : Suppose 0  u  t  T . Then IE  e,rt^ vt ^   = 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,ru^ v u ^ ; S u ^  : F u  ...
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