(15 points total). a. Finding the Stochastic Differential Equation (SDE). Find \\( d\\left(e^{\\sigma W_{t}-\\frac{1}{2} \\sigma^{2} t}\ ight) \\). Hint: set \\( f(x, t)=e^{\\sigma x-\\frac{1}{2} \\sigma^{2} t} \\), compute partial derivatives, use Ito's Lemma with \\( d f=\\frac{\\partial f}{\\partial x} d x+ \\) \\( \\frac{\\partial f}{\\partial t} d t+\\frac{1}{2} \\frac{\\partial^{2} f}{\\partial x^{2}}(d x)^{2} \\) where \\( (d x)^{2}=d t \\), then set \\( x=W_{t} \\). 11 b. Finding the Stochastic Integral. Evaluate \\( \\int_{0}^{t} e^{-s} d W_{s} \\). Hint: let \\( f(x, t)=e^{-t} x \\), compute partial derivatives, use Ito's Lemma with \\( d f=\\frac{\\partial f}{\\partial x} d x+\\frac{\\partial f}{\\partial t} d t+\\frac{1}{2} \\frac{\\partial^{2} f}{\\partial x^{2}}(d x)^{2} \\) where \\( (d x)^{2}=d t \\), set \\( x=W_{t} \\), set the differential \\( d\\left(e^{-t} W_{t}\ ight) \\) equal to your result from Ito and integrate from 0 to \\( t \\)