# When does the predictable $\sigma$-algebra $\mathcal{P}$ coincide with the optional $\sigma$-algebra $\mathcal{O}$?

The setup of my question is the following: Suppose that we have a measurable space $$(\Omega,\mathcal{F})$$ and a filtration $$\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$$ on it. Let $$\mathcal{P}(\mathbf{F})$$ be the predictable $$\sigma$$-algebra, that is, the $$\sigma$$-algebra generated by all real-valued left-continuous processes adapted to the filtration $$\mathbf{F}$$. Let $$\mathcal{O}(\mathbf{F})$$ be the optional $$\sigma$$-algebra, that is, the $$\sigma$$-algebra generated by all real-valued right-continuous process with left-limits adapted to the filtration $$\mathbf{F}$$.

I was trying to find conditions on the underlying measurable space $$(\Omega,\mathcal{F})$$ or the filtration $$\mathbf{F}$$, under which the predicatble $$\sigma$$-algebra $$\mathcal{P}(\mathbf{F})$$ coincides with the optional $$\sigma$$-algebra $$\mathcal{O}(\mathbf{F})$$. For example, if the filtered space $$(\Omega,\mathcal{F})$$ supports a process $$X=(X_t)_{t \geq 0}$$ with values in a metric space $$(E,d)$$ for example, such that the paths $$\mathbb{R}_+ \ni t \mapsto X_t \in E$$ are continuous, and the filtration $$\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$$ is generated by $$X$$, that is, $$\mathcal{F}_t = \sigma(X_r : 0 \leq r \leq t),$$ does the equality $$\mathcal{P}(\mathbf{F}) = \mathcal{O}(\mathbf{F})$$ hold?

If this condition is too weak, does there exist some other general condition under which the predictable $$\sigma$$-algebra coincides with the optional $$\sigma$$-algebra?

Off the top of my head, I'd look in vol.2 of Probabilités et Potentiel (Dellacherie & Meyer) or in Limit Theorems for Stochastic Processes (Jacod & Shiryaev). Another convenient resource is the blog https://almostsure.wordpress.com of Geo. Lowther.

The key is that for a bounded rc martingale $$M$$, the predictable projection $${}^p\!M$$ coincides with the left limit process $$(M_{t-})$$. Thus, if $$\mathcal P =\mathcal O$$ then each such $$M$$ coincides with its predictable projection and so is left continuous, hence continuous.

Conversely, if each bounded rc martingale is continuous, then each such martingale is predictable. In particular, for a bounded r.v. $$Z$$ the rc version of the martingale $$t\mapsto E[Z|\mathcal F_t]$$ must be predictable. Using this, Theorem IV-62 of Dellacheie & Meyer cited above yields that every stopping time is predictable, and this in turn implies that $$\mathcal O =\mathcal P$$.

A sufficient (and necessary) condition is that each bounded right-continuous martingale is continuous. This is true, for example, if the filtration is that of a Brownian motion.

The condition you suggest is too weak. Example: Let $$U$$ be uniformly distributed on $$(0,1)$$ and let $$\xi$$ be an independent random variable taking the two values $$1$$ and $$-1$$ with equal likelihood, both defined on some probability space $$(\Omega,\mathcal F,\Bbb P)$$. Take your filtration to be $$\mathcal F_t:=\cap_{\epsilon>0}\mathcal F_{t+\epsilon}^o$$, where $$\mathcal F_t^o$$ is defined by $$\mathcal F^o_t:=\sigma\{1_{\{U\le s\}}, 0\le s\le t; \xi1_{\{U\le t\}}),\qquad t\ge 0.$$ Notice that $$U$$ is an $$(\mathcal F_t)$$-stopping time. Now define a continuous-path process $$X$$ by
$$X_t=\cases{0,&0\le t This process generates $$(\mathcal F_t)$$ (modulo null sets) but the (right-continuous) martingale $$M_t:=\Bbb E[X_1|\mathcal F_t] = \cases{0,&0\le t is not continuous.

• can you recommend a reference for the sufficient and necessary condition that each bounded right-continuous martingale has to be continuous?
– vaoy
Aug 4 '20 at 20:22