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Let's say I have an Ito process $X_t$, and another process $Y_t=e^{\int_t^{t+\delta} X_s ds}$. I want to know that quadratic variation of $Y_t$ and another process.

I know that the quadratic variation of a smooth function, with any other function is $0$, but I'm wondering if $Y_t$ counts as a smooth function in this case, since it is a function of a stochastic process $X_t$, rather than being a function of $s$: $e^{\int f(s) ds}$. So, is the quadratic variation of $Y_t$ with another process, $=0$ even though it is a function of a random varible $X_t$? Or not necessarily?

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