I learned an interesting continuous time trick recently. The context is a note, "
The fragile benefits of endowment destruction" that I wrote with John Campbell, about how to extend our habit model to jumps in consumption. The point here is more interesting than that particular context.
Suppose one time series
x, which follows a diffusion, drives another
y. In the simplest example,
dxt=σdzt
dyt=ytdxt.
In our example, the second equation describes
how habits
y respond to consumption
x. The same kind of structure might describe how invested wealth
y responds to asset prices
x, or how option prices
y respond to stock prices
x.
Now, suppose we want to extend the model to handle jumps in
x,
dxt=σdzt+dJt.
What do we do about the second equation?
yt now can jump too. On the right hand side of the second equation, should we use the left limit, the
right limit, or something in between?