Is there a way to actually write a compiler in its own language?
You have to have some existing language to write your new compiler in. If you were writing a new, say, C++ compiler, you would just write it in C++ and compile it with an existing compiler first. On the other hand, if you were creating a compiler for a new language, let's call it Yazzleof, you would need to write the new compiler in another language first. Generally, this would be another programming language, but it doesn't have to be. It can be assembly, or if necessary, machine code.
If you were going to bootstrap a compiler for Yazzleof, you generally wouldn't write a compiler for the full language initially. Instead you would write a compiler for Yazzle-lite, the smallest possible subset of the Yazzleof (well, a pretty small subset at least). Then in Yazzle-lite, you would write a compiler for the full language. (Obviously this can occur iteratively instead of in one jump.) Because Yazzle-lite is a proper subset of Yazzleof, you now have a compiler which can compile itself.
There is a really good writeup about bootstrapping a compiler from the lowest possible level (which on a modern machine is basically a hex editor), titled Bootstrapping a simple compiler from nothing. It can be found at https://web.archive.org/web/20061108010907/http://www.rano.org/bcompiler.html.
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