Let the process (Xt), t 0, solve the SDE dXt =-Xidt + dB, Xo E R....

 

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Let the process (Xt), t≥ 0, solve the SDE dXt =-Xidt + dB, Xo E R. (a) Show that the process Y₁ = et X, has independent Gaussian increments, and give the distribution of the increments over the time interval [s, t]. (b) Show that the process (X) has Gaussian increments, but they are not independent. (c) State with reason whether the process (Xt) is a Gaussian process and give its mean and covariance functions. (d) Derive the conditional expectation E(XX.) for s <t. (e) Show that if Xo has distribution N(0, 1/2) and is independent of the pro- cess (B₁), then for any time t, Xt has N(0, 1/2) distribution. Let the process (Xt), t≥ 0, solve the SDE dXt =-Xidt + dB, Xo E R. (a) Show that the process Y₁ = et X, has independent Gaussian increments, and give the distribution of the increments over the time interval [s, t]. (b) Show that the process (X) has Gaussian increments, but they are not independent. (c) State with reason whether the process (Xt) is a Gaussian process and give its mean and covariance functions. (d) Derive the conditional expectation E(XX.) for s <t. (e) Show that if Xo has distribution N(0, 1/2) and is independent of the pro- cess (B₁), then for any time t, Xt has N(0, 1/2) distribution.

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a Let Yt eXt Then dYt eXtdXt deXt eXtXtdt dBt deXt eXtdBt Xtdt Since dBt has in... View the full answer