Doesn't the gps data contain position $p$?  I would have thought that, not only is $v_{i+1}$ dependent upon $v_{i}$ and $a_{i}$ but is would also be dependent upon $p_{i}$.  Consider:  in any road network there are bottlenecks, speed limits, signals, intersections, steep gradients, etc. that are geolocated.  So something like an ensemble (distribution) defined by:

$F_{a} = Pr ( A_{i+1} \le a_{i+1}\ |\ a_{i},v_{i},p_{i} )$  
$v_{i+1} = v_{i} + a_{i}dt$  

For such an ensemble, the difficulty will lay in the nature of the data.  It is likely that the true population will be asymmetric, non-linear (piece-wise) and may not have defined moments.  These characteristics may not be evident within the sample you have at hand.

As @whuber has stated, the problem, ie exactly what you are seeking to produce, does not yet seem fully and clearly defined.  It is not clear as to whether you are interested in the ensemble or more so the individuals.