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    Hi!!!
    I need some help at the following exercise...
    Let B be a typical brownian motion with μ>0 and x ε R. Xt:=x+Bt+μt, for each t>=0, a brownian motion with velocity μ that starts at x. For r ε R, Tr:=inf{s>=0:Xs=r} and φ(r):=exp(2μr). Show that Mt:=φ(Xt) for t>=0 is martingale.

    To show that Mt is martingale, I have to show that:
    1. Mt is adapted to the filtration {Ft}t>=0
    2. For every t>=0, E(|Mt|)<oo
    3. E(Mt|Fs)=Ms, for every 0<=s<=t
    Right???

    To find
    E(|Mt|) and E(Mt|Fs) do I have to use the property E(Bt-Bs)=0?

    E(|e-2μ(x+Bt+μt)|)=E(|e-2μ(x+Bt+Bs-Bs+μt)|)=E(|e-2μ(x+Bs)e-2μ(Bt-Bs)e-2μ^2t|)=e-2μ(x+Bs)E(|e-2μ(Bt-Bs)|)E(|e-2μ^2t|).

    Is the mean value E(|e-2μ(Bt-Bs)|) equal to e0=1???


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    I have no idea what any of this means.
 
 
 
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