Question
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)
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.
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