In a random walk (Wiener Process, Brownian Motion) $W_t$, the expected change over time is given by a normal distribution. That means the larger the absolute value of the changing during an interval, the smaller the probability that that change will occur.
However, can we also create a stochastic differential equation that results in small Brownian-like motion most of the time, but then at certain rare occasions results in a sudden very large change?
e.g. like:
Such a large crash compared to the variance of the rest of the curve would not have a significant probability of occurring under Brownian motion (i.e. you can't get this large decrease with brownian motion without also increasing the variance at the rest of the curve).
More generally, can we create SDE's with arbitrary such behavior, or does this become complex to an unmanageable degree?