Let W = {W :t 0} be a Brownian motion on (2, F, F = (Ft)tzo,...

 

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Let W = {W₁ :t≥ 0} be a Brownian motion on (2, F, F = (Ft)tzo, P). Fix a, ß ER and consider the following SDE: dX₁ = -x dt+dW₁, 0<t<T T-t and Xo = a, XT = B. A solution to this SDE, with the given boundary conditions, is called a Brownian bridge. By applying Ito's lemma to Y₁ = f(t, X₁) = solve this SDE and find the distribution, mean, and variance of Xt, where 0 < t < T. Let W = {W₁ :t≥ 0} be a Brownian motion on (2, F, F = (Ft)tzo, P). Fix a, ß ER and consider the following SDE: dX₁ = -x dt+dW₁, 0<t<T T-t and Xo = a, XT = B. A solution to this SDE, with the given boundary conditions, is called a Brownian bridge. By applying Ito's lemma to Y₁ = f(t, X₁) = solve this SDE and find the distribution, mean, and variance of Xt, where 0 < t < T.