Roses store a lot of their energy in the canes. Sometimes they'll use that stored energy to produce new growth. I don't see any rooting going on at the bottom though. To try and root this I would take it out of the water and get it into some soil instead.

Get a clear plastic cup and punch a hole in the bottom. Fill it with some potting soil and moisten it well but let all the excess water drain out. With a pencil poke a hole down into the soil. At the bottom of the stem of the rose scrape off some of the green outside skin so that some of the white inner layer shows. Dip the end into some type of rooting hormone. Put the cutting into the hole in the soil you made being careful not to rub off the hormone. Gently pat the soil back around the stem. I like to tap the cup on the counter a couple of times to make sure there aren't any air pockets too. At this point methods can differ greatly. A lot of people will place a baggie over the cup to create a little greenhouse and keep the humidity up around the cutting. I don't do this by I only do mine outside during the warmer months. If it's warm enough to get yours outside in your area now go ahead and put it outside in a place where it will get partial shade or only dappled sunlight. If not put it in a sunny window or under lights. You need to keep the soil moist but not soggy wet. Drying out is the worst thing for them. It will take several weeks for it to form roots. That's why I like the clear cups. You'll be able to see the roots in the cup when they form,. If the top part of the cane above the soil gets shriveled or blackened then the cutting didn't take.

I hope it works for you!