lognormal process and its multiplier

Question

I define a process $$X(t)=Y(t)exp(-0.5\sigma^2t+\sigma W(t)),$$ where $W(t)$ is a standard BM, and $\sigma$ is const. $Y(t)$ is meant to be a time dependent function. What I can't wrap my head around is when $Y(t)=0$ for some $t$. What's the intuition for this case? All the paths of $X$ process cross? I really don't mean to have them cross and want to have them run wild, so does mean I have to put constraints on this function? I already have this definition for $X$ process, so I am trying to make sense as to what is the role of $Y$ here.

Answer 1

It may be illustrative to examine its SDE. Let $Z(t)=\exp(-0.5\sigma^2t+\sigma W_t)$. Then,

$$dX(t) = d(Y(t)Z(t)) = Z(t)(Y'(t)dt + Y(t)\sigma dW_t)$$

Note that the time-varying function $Y(t)$ effectively modifies the standard Brownian process to a process with time-varying volatility, i.e $Y(t)\sigma$. At $Y(t) = 0$, the modification depresses the volatility to zero.

So, all the paths at that moment do not run wild. Instead, they all stay where they are and calmly drift pass the moment before running randomly again.