# Need for left limits in Stochastic Calculus theorems

A lot of the theorems in stochastic analysis are stated for cadlag processes (i.e. right continuous processes with left limits), but I am having a hard time seeing why the "left limits" part is important. It seems like for the most part just right continuity is enough, so I was wondering if anybody had a general explanation for why the assumption of left limits is usually included.

For a specific example, Proposition 2.3.5 in Revuz and Yor's "Continuous Martingales and Brownian Motion" states

A cadlag adapted process $$X$$ is a martingale if and only if for every bounded stopping time $$T$$ the random variable $$X_T \in L^1$$ and $$\mathbb{E}[X_T] = \mathbb{E}[X_0]$$.

The "only if" part comes from the optional stopping theorem, which did not include the assumption that $$X$$ is cadlag (because martingales have cadlag modifications anyway when the filtration satisfies the usual conditions). The proof for the converse direction is to fix $$s < t$$ and $$A \in \mathcal F_s$$ and define $$T = t 1_{A^c} + s 1_A$$ and use that $$\mathbb{E}[X_t] = \mathbb{E}[X_T]$$ to show $$\mathbb{E}[X_t 1_A] = \mathbb{E}[X_s 1_A]$$ and hence $$\mathbb{E}[X_t | \mathcal F_s] = X_s$$, but this also doesn't seem to use the left limits assumption. I originally thought it was to ensure $$X$$ is progressively measurable so that $$X_T$$ is measurable, but being right continuous and adapted is enough to conclude $$X$$ is progressively measurable so I'm still confused on why we need left limits.

In the context of semimartingales, left limits come for free: a right-continuous finite variation process automatically has left limits, and ditto (a.s.) for a local martingales. There is probably some "historical accident" at play here too. Stochastic Calculus originally grew out of the theory of Markov processes, where cadlag was a default.

• Right, but I'm not sure why they're even worth mentioning? Is it just that left limits are an additional nice property that comes along for free, so they're included even though they aren't needed? Commented Jul 3, 2020 at 19:59
• A cadlag function is a simpler object that a right-continuous function (e.g., a cadlag function is the uniform limit of step functions, a sometimes useful property). Commented Jul 4, 2020 at 15:50