Suppose we are given a real-valued Gaussian Markov process with initial condition zero
$$Y_t = \int_0^t f(s) dW_s \quad t \in [0,T], f\in L^2([0,T].$$
Following [Example 4.5, 1], $Y$ is distributed according to Gaussian measure on the canonical path space $C[0,T]$ with Cameron-Martin space given by
$$H_Y =\left\{ h: h(t)= \int_0^t f(s)l(s)\: ds, l\in L^2([0,T]) \right\}$$
I would like to show that for a process
$$Y_t = Y_0 +\int_0^t f(s) dW_s \quad t \in [0,T]$$
with initial condition $Y_0 \sim \mathcal N(0, \sigma), \sigma >0$, the Cameron-Martin space changes as one would expect
$$H_Y =\left\{ h: h(t)= y_0+ \int_0^t f(s)l(s)\: ds, l\in L^2([0,T]), y_0\in \mathbb R \right\}$$
This holds true in a couple of examples, however, I cannot seem to prove it in the general case. Any help is appreciated!
[1] Lifshits (2012). Lectures on Gaussian processes.
