1.The exponential function in Newton calculus has the unique property that it is the only function that is equal to its derivative so that f'(x) = f(x) f(x) = e^x (up to a multiplicative constant). What deterministic equation does it satisfy? Derive the Ito calculus exponential function using Ito Lemma.

2. The answer obtained in the previous step should be e^Wt 1/2 t , with Wt the Wiener process or the standard Brownian motion. Derive the stochastic differential equation (SDE) whose solution is e^Wt and an SDE whose solution is e^Wt 1/2t

3. Simulate one path for e^Wt and, using the same path, for e^Wt 1/2 t.

4. Using Monte Carlo simulation with 1000 paths, calculate and plot the moving average of each process at the 100 points on the interval [0, 1]; i.e. t = 0.01.