For the embroidery on my keikogi, I go with ジョシュ, which is "Josh" in phonetic katakana.
Why not ジャシュ? It's much closer phonetically.
I spent a few decades writing my name in kana as ジョシュア, but when I had to get some business cards printed up a few years ago with my name in kana, I decided to go with ジャシュア as an experiment. ジョシュア for Joshua and ジョン for John probably get chosen by native English speakers because in romaji they become "joshua" and "jon", even though they are pronounced "joe-shoe-ah" and "joan".
I had become very used to Japanese pronouncing my name that way, but over the last two years when I've handed a Japanese person my card with ジャシュア written on it, they've pronounced my name much more like a native English speaker, and it's actually been a bit of a shock to me.
I suppose to be really historically accurate, we should write our name the way the Joshua is written in the Old Testament in Japanese, which I think is ヨシュア.