\\[ d r_{t}=\\kappa\\left(\\theta-r_{t}\ ight) d t+\\sigma d B_{t}, r(0)=r_{0} \\] where \\( \\kappa>0 \\) is the mean-reversion speed, \\( \\theta>0 \\) the mean-reversion level, and \\( \\sigma>0 \\) the volatility. Use the function \\( f\\left(t, r_{t}\ ight):=e^{k t} r_{t} \\) and It's formula to show that \\[ r_{T}=\\theta+e^{-\\kappa T}\\left(r_{0}-\\theta\ ight)+\\sigma \\int_{0}^{T} e^{\\kappa(t-T)} d B_{t} . \\] b) Today's \\( (t=0) \\) price of a zero-coupon bond \\( P(0, T) \\) with maturity \\( T>0 \\) is given by \\[ P(0, T)=E\\left[e^{-\\int_{0}^{T} r_{i} d t}\ ight] \\] Calculate the price of \\( P(0, T) \\) in Vasicek's model. Instructions: Integrate equation (3) directly, from 0 to \\( T \\), and use the previous solution (4), to get an expression for \\( \\int_{0}^{T} r_{t} d t \\). Next, calculate the mean and variance of \\( \\int_{0}^{T} r_{t} d t \\). For the variance, use the Ito isometry