I am seeking clarification on the existence and uniqueness of a strong solution to a stochastic differential equation (SDE) in the context of a Brownian motion. Let $\mathrm{W}$ be a $q$-dimensional Brownian motion on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}=(\mathcal{F}_t)_{t \in[0, T]}, \mathbb{P})$, and let $x_0 \in \mathbb{R}^d$. Consider measurable functions $b: [0, T] \rightarrow \mathbb{R}^d$ and $\sigma: [0, T] \rightarrow \mathbb{R}^{d \times q}$, where $b$ and $\sigma$ are locally bounded.
I am struggling to find a reference or proof for the existence of a unique strong solution to the SDE: $$ d X_t=b(t)X_t dt+\sigma(t)X_t d W_t, \quad X_0=x_0. $$
Clearly, the drift and the diffusion are locally Lipschitz, but we can't show the linear growth condition.
I would greatly appreciate any references or insights on this matter.
See theorem 3.1 in Stochastic Differential Equations and Diffusion Processes (Volume 24) (North-Holland Mathematical Library, Volume 24) 0th Edition by S. Watanabe (Author), N. Ikeda (Author)
The extension to inhomogeneous $\sigma(t,x)$, follows the same proof since you have locally bounded.