Question Let W={Wt:t≥0}W={Wt:t≥0} be a Brownian motion on (Ω,F,F=(Ft)t≥0,P)(Ω,F,F=(Ft)t≥0,P).   Show that WW is an FF-martingale. Show that for every α∈Rα∈R, the process  Xαt:=ex

Question 

Let W={Wt:t≥0}W={Wt:t≥0} be a Brownian motion on 

(Ω,F,F=(Ft)t≥0,P)(Ω,F,F=(Ft)t≥0,P).

1. Show that WW is an FF-martingale. 
2. Show that for every α∈Rα∈R, the process 

Xαt:=exp(αWt−12α2t)Xtα:=exp⁡(αWt−12α2t)

    is an FF-martingale. 

    Hint: The MGF of X∼N(μ,σ2)X∼N(μ,σ2) is MX(α)=exp(μα+12σ2α2)MX(α)=exp⁡(μα+12σ2α2)

 .

3. Define the polynomials Hn(x,y);n=0,1,2,…Hn(x,y);n=0,1,2,… by

Hn(x,y)=∂n∂αnexp(αx−12α2y) at α=0.Hn(x,y)=∂n∂αnexp⁡(αx−12α2y) at α=0.

         For example,

H0(x,y)=1, H1(x,y)=x, H2(x,y)=x2−y, H3(x,y)=x3−3xy,H0(x,y)=1, H1(x,y)=x, H2(x,y)=x2−y, H3(x,y)=x3−3xy, 

H4(x,y)=x4−6x2y+3y2, etc.H4(x,y)=x4−6x2y+3y2, etc. 

        It can be shown (using Taylor 

series) that

Xαt=exp(αWt−12α2t)=∑n=0∞αnn!Hn(Wt,t).Xtα=exp⁡(αWt−12α2t)=∑n=0∞αnn!Hn(Wt,t).

        We now show that Hn(Wt,t)Hn(Wt,t) is a martingale for each nn.

(a) Let 0≤s≤t0≤s≤t and α∈Rα∈R. Explain why for each F∈FsF∈Fs

∫FXαtdP=∫FXαsdP∫FXtαdP=∫FXsαdP.

(b) By differentiating (1) on both sides nn times with respect to

αα 

and interchanging the derivative with the integral (no need to 

justify this step), show that 

E(Hn(Wt,t)|Fs)=Hn(Ws,s).E(Hn(Wt,t)|Fs)=Hn(Ws,s).

(c) Conclude that {Hn(Wt,t):t≥0}{Hn(Wt,t):t≥0} is a martingale.

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