I try to understand the Proof of Theorem 4.21 in Carmona Delarue (2018). In the following, what I don't understand:

Processes are assumed to be defined on a complete filtered probability space $(\Omega, \mathcal F, \mathbb F=(\mathcal F_t)_{t \in [0,T]},\mathbb P)$ supporting a $d$-dimensional Wiener process $W=(W_t)_{t\in [0,T]}$ wrt. $\mathbb F$, the filtration $\mathbb{F}$ satisfying the usual conditions. We denote by $\mathbb{H}^{2,n}$ the Hilbert space $$\mathbb{H}^{2,n} = \{ Z \in \mathbb{H}^{0,n}: \mathbb{E} \int_0^T |Z_s|^2 dx < \infty \},$$ where $\mathbb{H}^{0,n}$ stands for the collection of all $\mathbb{R}^n$-valued progressively measurable processes on $[0,T]$.

Assume that, for each $(x,\mu) \in \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$, the processes $B(\cdot,\cdot,x,\mu):[0,T]\times \Omega \to \mathbb{R}^d, (t,\omega) \mapsto B(t,\omega,x,\mu)$ and $\Sigma (\cdot,\cdot,x,\mu):[0,T]\times \Omega \to \mathbb{R}^{d \times d}, (t,\omega) \mapsto \Sigma(t,\omega,x,\mu)$ are $\mathbb{F}$-progressively measurable and belong to $\mathbb{H}^{2,d}$ and $\mathbb{H}^{2,d\times d}$ respectively. Furthermore, assume that for any $t \in [0,T], \omega \in \Omega, x, x' \in \mathbb{R}^d$ and $\mu, \mu'\in \mathcal{P}_2(\mathbb{R}^d)$, $$|B(t,x,\mu)-B(t,x',\mu')|+|\Sigma(t,x,\mu)-\Sigma(t,x',\mu')| \leq L (|x-x'|+W_2(\mu,\mu')).$$

Temporarily fix some $\mu = (\mu_t)_{t \in [0,T]} \in \mathcal{C}([0,T],\mathcal{P}_2(\mathbb{R}^d))$ and let $X_0 \in L^2(\Omega,\mathcal{F}_0,\mathbb{P},\mathbb{R}^d)$. Then "the classical existence result for Lipschitz SDE guarantees existence and uniqueness of a strong solution of the classical stochastic differential equation with random coefficients": $$dX_t = B(t,X_t,\mu_t)dt + \Sigma(t,X_t,\mu_t)dW_t$$

My problem: I don't find such a classical existence result. I think the $\mu$ part is mostly irrelevant here. It comes from McKean-Vlasov setting that is treated originally. I didn't want to omit it here in case it is relevant in some way. Crossing it out simplifies things a little. Still, when I look for the standard existence result (e.g. Karatzas Shreve, Chapter 5, Theorem 2.9) there is always some linear growth condition like $$ \|B(t,x)\|^2 + \|\Sigma(t,x)\|^2 \leq K^2(1+\|x\|^2).$$ I do not see, why this is implied by the conditions.