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, T_{r}:=inf{s>=0:X_{s}=r} and φ(r):=exp(−2μr). Show that Mt:=φ(X_{t}) for t>=0 is martingale.
To show that M_{t} is martingale, I have to show that:
1. M_{t} is adapted to the filtration {F_{t}}_{t>=0}
2. For every t>=0, E(|M_{t}|)<oo
3. E(M_{t}|F_{s})=M_{s}, for every 0<=s<=t
Right???
To find E(|M_{t}|) and E(M_{t}|F_{s}) do I have to use the property E(B_{t}-B_{s})=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 e^{0}=1???