optimistically

US /ˌɑːp.təˈmɪs.tɪ.kəl.i/
UK /ˌɑːp.təˈmɪs.tɪ.kəl.i/
1.

lạc quan

in a way that shows that you hope or believe that good things will happen in the future:

:
He optimistically predicted success.