I'm about 10 hours in the game and my stallion fell in some water and he died. So now I have this horse with terrible stamina and I was wondering if there is a way to train horses to increase their stamina.
thanks for any help

Using the same horse for a long period of time will marginally increase its stamina, not enough to be worth training for. 
 
Next time you get a stallion make sure you buy the deed to it at the general store. That way if you drown another horse you can instantly call another one.

I don't think there is. I mean there was the once when I got a notice saying ''your horse now trusts you so he has more stamina'' but that literally only happened once, and at the exact same moment during my multiple playthroughs. I think it was something they started but then they just cut it short. 
 
But anyway you can buy the deed to the Golden Pearl horse now from a general store once you caught it in that story mission. If you have high enough honour it'll only be around 500$, aswell.

Thanks a lot guys.