Hi Julia,
This is indeed very strange notation. I've never seen this kind of thing before, and no, there is no guideline or rule for this.
At first it looked like it was referring to "high" and "low" fingering which most often is notated with an H, L, minus sign or plus sign following the finger number, but then I noticed an arrow not following a fingering (measure 6 after the 2nd G) and an arrow after a 0 or open string (measure 13).
I don't think I would use up bow and down bow signs for these, since the down bow is used for its true purpose on the first note. So I think you'll have to create something to represent those arrows. You may choose to use word sign expressions where the arrows appear: "ua" for up arrow and "da" for down arrow. (I wouldn't use just u and d since the d stands for decrescendo.) And be sure you include a transcriber's note indicating what you are doing!
I know that will make the measures even more cluttered than they already are with position signs and fingerings on almost every note! But I can't think of a clearer way. If you think of something else that works, please share!
Hope that helps a bit!