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- Apr 14, 2013

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Hi!!!

I need some help at the following exercise...

Let \(\displaystyle B\) be a typical brownian motion with \(\displaystyle μ>0\) and \(\displaystyle x\) ε \(\displaystyle R\). \(\displaystyle X_{t}:=x+B_{t}+μt\), for each \(\displaystyle t>=0\), a brownian motion with velocity \(\displaystyle μ\) that starts at \(\displaystyle x\). For \(\displaystyle r\) ε \(\displaystyle R\), \(\displaystyle T_{r}\):=inf{\(\displaystyle s>=0:X_{s}=r\)} and \(\displaystyle φ(r):=exp(-2μr)\). Show that \(\displaystyle M_{t}:=φ(X_{t})\) for t>=0 is martingale.

Could you tell me the purpose of \(\displaystyle T_{r}\)??

I need some help at the following exercise...

Let \(\displaystyle B\) be a typical brownian motion with \(\displaystyle μ>0\) and \(\displaystyle x\) ε \(\displaystyle R\). \(\displaystyle X_{t}:=x+B_{t}+μt\), for each \(\displaystyle t>=0\), a brownian motion with velocity \(\displaystyle μ\) that starts at \(\displaystyle x\). For \(\displaystyle r\) ε \(\displaystyle R\), \(\displaystyle T_{r}\):=inf{\(\displaystyle s>=0:X_{s}=r\)} and \(\displaystyle φ(r):=exp(-2μr)\). Show that \(\displaystyle M_{t}:=φ(X_{t})\) for t>=0 is martingale.

Could you tell me the purpose of \(\displaystyle T_{r}\)??

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