# Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)

Given a filtered probability space $$( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$$ and an $$\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$$-measurable bounded process $$X: \Omega \times \mathbb{R}_+ \rightarrow \mathbb{R}$$, we know by the predictable projection theorem that there exists a process $$^pX$$ (unique up to evanescence) such that

1. $$^pX$$ is predictable
2. $$(^pX)_T = \mathbb{E}(X_T | \mathcal{F}_{T-})$$ on $$\{T<\infty\}$$ for every predictable stopping time $$T$$

Taking for granted this result (I am not sure whether of not it requires any assumption on the filtration), my question is about the predictable projection of a stopped process.

In particular, take $$X$$ measurable as in the theorem and consider the stopped process $$X^S$$, where $$S$$ is any optional time. How can we express $$^p(X^S)$$ in terms of $$^pX$$?

As far as I understand (Jacod & Shiryaev, page 23), it is the case that

$$\begin{equation} ^p(X^S) = (^pX) \mathbb{I}_{[\![0, S]\!]} + X_S \mathbb{I}_{]\!]S, \infty ]\!]} \tag{*} \end{equation}$$

but I cannot prove either of the defining properties.

With regard to (1), I was hoping to show that both terms on the rhs of (*) are predictable.

$$(^pX) \mathbb{I}_{[\![0, S]\!]}$$ is predictable, because it is the product of two predictable processes.

The problem is that I do not see why $$X_S \mathbb{I}_{]\!]S, \infty ]\!]}$$ needs to be predictable. In order to have $$Y \mathbb{I}_{]\!]S, \infty ]\!]}$$ predictable if $$S$$ is optional I need $$Y$$ to be $$\mathcal{F}_S$$-measurable, but $$X_S$$ is not necessarily $$\mathcal{F}_S$$-measurable.

What is even more puzzling is that - if I interpret Jacod & Shiryaev correctly - we can deduct from (*) that $$X_S \mathbb{I}_{]\!]S, \infty ]\!]}$$ is predictable (since it is the difference of two predictable processes) for every bounded measurable process $$X$$ and optional time $$S$$.

Would not $$X(\omega, t) = \mathbb{I}_A(\omega)$$, with, say, $$A \in \mathcal{F} \setminus \mathcal{F}_1$$ and $$S(\omega) = \frac12$$ be a trivial counterexample to this (in this example $$X_S \mathbb{I}_{]\!]S, \infty ]\!]}$$ would not even be adapted)?

With regard to (2), I am even more clueless.

Apologies for the macroscopic error that I am somewhere certainly making.

It seems to me that there is an error in formula 2.29 on page 23 of J&S. (Amazingly, the error persists in the second edition, which appeared 23 years after the first.) The formula is correct provided $$X$$ is optional. More generally, it would be correct if $$X$$ were replaced in the second term on the right side by the optional projection $$\,{}^o\!X$$ of $$X$$.