91Ó°ÊÓ

Find the period of the LCG defined by \(a=4, b=0, m=9\). Does the period depend on the seed?

Approximate the area under the curve \(y=x^{2}\) for \(0 \leq x \leq 1\), using the LCG with (a) \(a=2, b=0, m=5\) (b) \(a=4, b=1, m=9\).

Use the Euler-Maruyama Method to solve the SDE initial value problem \(d y=y d t+y d B_{t}, y(0)=1\). Plot the approximate solution and the correct solution \(y(t)=e^{\frac{1}{2} t+B_{t}}\). Use a step size of \(h=0.1\) on the interval \(0 \leq t \leq 2\).

Perform a Monte Carlo estimate of \(y(1)\), where \(y(t)\) is the Euler-Maruyama solution of the Langevin equation $$ \left\\{\begin{array}{l} d y=-y d t+d B_{t} \\ y(0)=e \end{array}\right. $$ Average \(n=1000\) realizations with step size \(h=0.01\). Compare with the expected value of \(y(1)\), which is 1 .