I am studying multidimensional SDEs on the form $$ dX_t = a(X_t, t) dt + b(X_t, t) dW_t $$ where $a$ is a vector valued function and $b$ is a matrix valued function. All calculations are performed in a standard (multidimensional) Euclidean space. What norm is used on the matrix $b$ in the Lipschitz condition $$ \| a(x,t)-a(y,t) \| + \| b(x,t)-b(y,t) \| \leq K \| x-y \| ? $$ I haven't been able to find is directly spelled out. Regarding $\| a(x,t)-a(y,t) \|$ I assume it is just the Euclidean norm.
The particular Lipschitz condition is taken from Monte Carlo Methods in Financial Engineering by Paul Glasserman, appendix B.2 on SDEs.
I will state here the initial part of the fundamental theorem of existence and uniqueness of stochastic differential equations taken from Oksendal (Th. 5.2.1.).
Let $T>0$ and $b(\cdot,\cdot):[0,T]\times \mathbb{R}^n\to\mathbb{R}^n$ and $\sigma(\cdot,\cdot):[0,T]\times \mathbb{R}^n\to\mathbb{R}^{n \times m}$ be measurable functions satisfying $$|b(t,x)|+|\sigma(t,x)|\leq C(1+|x|), \ \ \ x \in \mathbb{R}^n,t \in [0,T]$$ for some constant $C$ and where $|\sigma|^2=\sum |\sigma_{ij}|^2$ and such that $$|b(t,x)-b(t,y)|+|\sigma(t,x)-\sigma(t,y)|\leq D|x-y|$$ For some constant $D$. Let (...)
I think this answers yours question.