Stochastic Analysis: Martingale and Gaussian Processes

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Course: Stochastic Analysis
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Problem 3.
A martingale is a scholastic process such that
E[X(t)] <∞ for all t.
E[X(t)/f(Xs)] = Xs for all s<t
Let Xt=2Bt +∫
0
t
¿ ¿ +S2)dBs
We need to show that, when
S<t: E Xt/Fs] = Fs
E[2Bt + ∫
0
t
¿ ¿ + S2)d Bs /Fs
By expressing Bt as Bs+[Bt-Bs] it becomes
E: [2(Bs +(Bt-Bs)] + ∫
0
t
¿ ¿ S2)dBs
Hence
E[Xt/Fs] = E[e−2 μ ( μt +Bt ) /Fs]
Opening the brackets, we now have
=E[e−2 μ2−2 μBt
/ Fs ¿
¿ e−2 t μ2
E [e−2 Bt μ2
/ Fs]