In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process.
Later (pp. 329-330) for a process $\lambda\in\mathbb{L}^2_{loc}$ the exponential martingale associated to $\lambda$ is defined as
$$ Z_t^{\lambda} = \exp\left(-\int_{0}^t\lambda_s\,dW_s-\frac{1}{2}\,\int_{0}^t\left|\lambda_s\right|^2\,ds\right),\quad t\in\left[0,T\right], \quad (10.1). $$
What puzzles me is Lemma 10.1:
Since the exponential martingale is defined only for $\lambda$ in the class $\mathbb{L}^2_{loc}$, condition (10.3) in the Lemma 10.1 should be immediately guaranteed by (4.34) in Definition 4.33. Where am I wrong?
Fix a random variable $X$ (measurable at time zero if you like) which is finite but not in $L^\infty$, then define $\lambda_t = X$ for all $t$. Then
$$ \int_0^T \lambda_t^2\,dt = T X^2. $$
By assumption $T X^2 < \infty$ for all $T, \omega$, but there is no $C$ such that $TX^2 \leq C$ almost surely, as this would imply $X \in L^\infty$.
The difference is that (4.34) lets the constant depend on $\omega$, whereas (10.3) does not.