I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. Besides the weakenening of the Lipschitz condition in the state space, what struck me is that it appears there are sets of structural assumptions for solution existence requiring only integrability in $t$ of some bounding functions, instead of uniformity of the Lipschitz constant in the state space.
More precisely let, on a filtered space with usual assumptions $(\Omega, P, \mathcal F, (\mathcal F_t)_{t \geq 0})$
$$dX_t=b(t,X_t)dt+ \sigma(t, X_t)d W_t, \qquad X_0 \in \mathbb R^d, \qquad \qquad (1) $$
for a $d$-dimensional $\mathcal F_t$-adapted Brownian motion $W$, and $b:[0,\infty) \times \mathbb R^d \rightarrow \mathbb R^d$, $b:[0,\infty) \times \mathbb R^d \rightarrow \mathbb R^d$ $\sigma:[0,,\infty) \times M^d(\mathbb R) \rightarrow \mathbb R^d$ are respecitvely progressively measurable and continuous with respect to the second argument. Then if for all $t>0$ and $R>0$
$$\int_0^T \sup_{|x| \leq R}|b(s,x)|+||\sigma(s,x)||ds <\infty, \qquad \qquad (2)$$
as a consequence of Theorem 1.1 in Lan and Wu, with $\eta_R(x)=x$ for all $R$ - thus getting rid of "locality" we have that, for all $x,y \in \mathbb R^d, t \geq 0$ if $$ ||\sigma(t,x)-\sigma(t,y)||^2+ 2 \langle x-y, b(t,x)-b(t,y) \rangle \leq g(t)|x-y|^2 \qquad (3) $$ with $g$ a meausrable function such that $$\int_0^t g(s) ds <\infty $$ then a unique (local, in principle) strong solution of (1) exists.
A similar condition, of slow growth type, requiring only integrability of the constant leads to globality of solution.
I would like to understand this a little bit better. I expect that integeability of the Lipschitz constant can replace in some cases uniformity of the Lipschitz condition is well established, since Lan and Wu do not put any special emphasis on this implication of their work. Can anyone reccommend some references?
EDIT
To be specific, a possible claim that would be interesting is the following.
Q: if we replace the constant C in the usual local or Global textbook Lipschitz conition with a locally integrable function b of time, does a unique (strong/weak) solution still exists? At the very least, if $b=0$ in $(1)$ this would appear to follow from Lan and Wu 2014 $(by (3))$.
