Most of the work you will find on jump diffusion models will be in derivative pricing or related work on insurance. In essence, they tend to be interesting ways to think about future distributions.
If your performance metric is the mean squared error, we can easily show that what you should be trying to estimate is the conditional expected value of the process. Jump diffusions are designed to think about higher moments. They seldom are very sophisticated ways to think about conditional expectations, unless you try to look at time-varying dynamics for jump intensities and/or sizes. It wouldn't be especially interesting, in other words.
Now, if you want measures of conditional quantiles to predict intervals, that would tap directly into the interesting properties of jump diffusion models -- people use them to force skewness and kurtosis, mostly on shorter horizons to force agreement with empirical volatility surfaces. You can set up a pinball loss type function as the objective function of your neural network and it would force it to predict quantiles from which you can build prediction intervals.
In that case, maybe your comparison would be interesting: you could get build things like value-at-risk type measures for future withdrawals, or predict things like between X_1 and X_2 number of people will make withdrawals over some period of time in the future, with a confidence of Z%. Both types of models can do this, just as could a linear regression (if you change the least square loss for a pinball loss).
