This is a question from a quant test I am perusing trying to learn about stochastic processes. I am far removed from probability theory and could use some guidance.
Let $A_t = e^{\sqrt{t}Z}$, where $Z$~$N(0,1)$.
Suppose you invest \$5000 at time $t=0$. What is the probability that the value of the investment is less than \$20,000 at time $t = 5$?
My attempt:
The value of the investment at every time step $t$ will be equal to the principal times the value function. Let's call the value $V_t(Z)$, a random variable as a function of $Z$.
Then, $$V_5(Z) = 5000 * e^{Z\sqrt{t}}$$.
Now, to calculate $P(V_5 < 20000)$ we need the PDF of $V_5(Z)$.
I presume that $A_t$ is the value at time $t$ of one dollar invested at time zero? In that case you want
$$P\left(5000 A_5 \leq 20000\right) = P\left(A_5 \leq 4\right) = P\left(Z \leq \frac{\log 4}{\sqrt{5}}\right),$$
which is a simple application of the normal cdf. There is no need to discretize because $e^{\sqrt{t}\cdot Z}$ is the value of one dollar exactly at time $t$, taking into account all of the changes in value that occurred during the time interval $\left[0,t\right]$.