Modeling when the response variable has too many 0's and few continuous values? For problems where the data represents online fraud or insurance (where each row represents a transaction), it is typical for the response variable to denote the value of fraud committed in dollars. Such a response value might have less than 5% non-zero values denoting fraudulent transactions. 

I have two questions regarding such a dataset: 


What algorithms can we use to ensure that the model not only predicts the fraudulent transactions accurately, but also predicts the value of fraud associated with these.  
Assuming that we can quantify the cost involved in each false positive (tagging a non-fraudulent transaction as fraudulent) and cost incurred due to a false negative (tagging a fraudulent transaction as non-fraudulent), how can we optimize the model to maximize savings (or minimize losses)?
How about


Ordinary Least Square (OLS) regression? Since you have a class imbalance, you might want to combine that with boosting algorithms.
If you have a function to quantify the cost involved with FP's and FN's, use any optimization technique you can find. My favorite is genetic algorithms. You may also try linear programming.