Apologies in advance, my mathematics is likely to be very ad hoc, but I hope it makes sense...
I have a software package that models data using multivariate stochastic differential equations, however the following non-stochastic differential is all that is necessary for my query:
$$dy(t) = Ay(t)dt+x(t)\delta(t-S)dt$$
Based on observations $1,2,3,...,n$ of vectors $y(t)$ and $x(t)$, which were made at times $s \in S$.
I am concerned about the Dirac delta here - this is the best representation I have been able to come up with for what the model is doing, in that the impact of $x(t)$ on $y(t)$ is an instantaneous impulse of magnitude $x(t)$ whenever $t \in S$.
The solution of this with values known at $t_{0}$, $t_0 >= 0$, $t > t_0$, and all $S >= t_0$, should then be:
$$y(t) = e^{A(t-t_0)}y(t_0) + \sum_{s= \min (S \in [t_0,t])}^{max(S \in [t_0,t])} e^{A(t-s)} x_s $$
So, first I would ask: Is my representation appropriate and correct?
Then: Is there a better / more parsimonious way to convey such a model / phenomenon?