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In the definition of martingales, one finds in Stroock and Varadhan (Multidimensional Diffusion processes - page 20) the strange request that it be right-continuous process.

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However no such requirement is made in the wiki https://en.wikipedia.org/wiki/Martingale_%28probability_theory%29

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nor in Kallenberg (Foundations of modern probability - page 96)

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Isn't this strange of the part of Stroock and Varadhan? Has the notion of martingales evolved?

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These still aren't the most general definitions for martingales. The most general definition I know of is:

Let $I$ be a poset. A Family $F$ of sigma-algebras is said to be a filtration if for all $s, t \in I$ with $s \le t$ the inclusion $F_s \subset F_t$ holds. A family $X_i$ of (quasi)integrable random variables is called a martingale with respect to $F$, if:

  1. $X_i$ is $F_i$-measurable for all $i \in I$
  2. If $s \le t$, then $E[X_t | F_s] = X_s$

The problem with this definition: It is rarely necessary to use it. In most cases $I = \mathbb{N}$, $I = \mathbb{Z}$ or $I$ is an interval in $\mathbb{R}$. It is not uncommon for an author to omit the most general definition, if he only uses some special instances.

For example, your quote from Kallenberg's book is only an introduction. He later defines martingales on the real numbers and only needs these types of martingales in the book. The book of Stroock and Varadhan mainly deals with stochastic analysis. Herein, the assumption of right-continuity is useful and a prerequisite for many theorems. They need the special assumptions, so they simply add them to the definition of martingales.

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Processes that are not right-continuous are a gigantic pain to deal with. Most likely they include right-continuous in their definition because they always intend to work under that assumption, and don't want to have to keep writing it. As with any mathematical term, there doesn't have to be a universal "correct" definition of martingale; authors are free to adopt their own variants.

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