TO HAND IN Some practice with It's formula Let (Bt, t > 0) be a standard...

 

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TO HAND IN Some practice with Itô's formula Let (Bt, t > 0) be a standard Brownian motion. For each of the processes (Xt, t≤T) below: • Determine if they are martingales for the Brownian filtration. If not, find a compensator for it. • Find the mean, the variance and the covariance. • Is the process Gaussian? Argue briefly. (a) X₁ = ft² cos s dB. (b) X₁ = B. (c) Xt=et/2 cos Bt. Hint: If Z is standard Gaussian, then E[sin² (σZ)] = 1-e-202 (d) Zt = (B₁ +t) exp(−B₁ — ½). 2 TO HAND IN Some practice with Itô's formula Let (Bt, t > 0) be a standard Brownian motion. For each of the processes (Xt, t≤T) below: • Determine if they are martingales for the Brownian filtration. If not, find a compensator for it. • Find the mean, the variance and the covariance. • Is the process Gaussian? Argue briefly. (a) X₁ = ft² cos s dB. (b) X₁ = B. (c) Xt=et/2 cos Bt. Hint: If Z is standard Gaussian, then E[sin² (σZ)] = 1-e-202 (d) Zt = (B₁ +t) exp(−B₁ — ½). 2