on the interval $\left[0,1\right]$ with initial condition $X\left(0\right)=1$ and step size $\Delta t=1∕10$. Use the results of your coin-tossing record which was rescaled to ${Ŵ}_{10}\left(10t\right)∕\sqrt{10}$ as an approximation of $W\left(t\right)$ to create the 10 increments $dW$ needed for the simulation.

by guessing and checking with Itô’s Formula.

1. Suppose you buy a stock (or more properly an index fund) whose value $S\left(t\right)$ is described by the stochastic differential equation $dS=0\cdot Sdt+\sigma \cdot SdW$ corresponding to a zero compounded growth and pure market fluctuations proportional to the stock value. What are your chances to double your money?
2. Suppose you buy a stock (or more properly an index fund) whose value $S\left(t\right)$ is described by the stochastic differential equation $dS={\sigma }^2∕2\cdot Sdt+\sigma \cdot SdW$ corresponding to a non-zero compounded growth and pure market fluctuations proportional to the stock value and the two parameter values are coincidentally connected in their value. What are your chances to double your money?