Predicting account balance based on historical data
Given the strong correlation of the downward trend in each cycle, I offer the following that would offer an improved prediction over one linear regression, but not be significantly more complicated (but still would like to see what a precise "best estimator" would be):


Do a linear regression to determine the general upward trend. This would be the Income - Spend = Savings rate.
Do a least mean square linear fit for all the downward trend cycles. This would be the spend rate. (my temptation to do this easily would be detect the upward spike and superimpose the datasets, find the mean downward trend and do a linear regression to that.)
Determine the time of each upward spike to determine the best estimate of the frequency of occurrence of the upward spikes (clock recovery). For this the strongest tone in the FFT should reveal this.  This would be the pay dates.
Determine the mean of the magnitude of the upward spikes.  This would be the income per pay date.


From the information above you can create a future projection of continued upward spikes at the best estimated times along an overall upward trend that will provide a better estimate than one linear regression over the whole dataset.