Consider integrable submartingale $(X_r)_{r \in \mathbb{R}_+}$ relative to $(\mathcal{F}_{r})_{r \in \mathbb{R}_+}$ and such that $$\exists c \in \mathbb{R}_+^*,\forall k \in \mathbb{N},E[|X_{k+1}-X_k||\mathcal{F}_k] \leq c \ \ \ \ \ \ (P)$$ $\theta_1$ and $\theta_2$ are two $(\mathcal{F}_r)_r$-stopping times (taking values in $\overline{\mathbb{R}}_{+}$) such that $\theta_1 \leq \theta_2$ and $\theta_2 \in L^1.$
If $\theta_1$ and $\theta_2$ take values in $\overline{\mathbb{N}},$ then we can prove that $X_{\theta_1},X_{\theta_2} \in L^1$ and using the optional stopping theorem that $E[X_{\theta_2}|\mathcal{F}_{\theta_1}] \geq X_{\theta_1}$ a.s.
We suppose next that $(X_r)_{r \in \mathbb{R}_+}$ is right-continuous and that $\theta_1$ and $\theta_2$ take values in $\overline{\mathbb{R}}_+$. The following pictures are taken from stochastic processes, where various conditions imply the continuous optional stopping theorem.
Can we say that condition $C_3$ (picture below) is the continuous version of $(P)$? If so, how can we prove, under $C_3$ or $(P),$ that $X_{\theta_2} \in L^1$ ? What about the following property (following the construction of $(P)$):
$$\exists c \in \mathbb{R}_+^*,\forall(p,r) \in (\mathbb{R}_+)^2,E[|X_{p+r}-X_r||\mathcal{F}_r] \leq c \ \ \ \ \ \ (P')$$
